ANALYSIS AND SUPPRESSION OF INSTABILITIES IN VISCOELASTIC FLOWS

Transcription

1 ANALYSIS AND SUPPRESSION OF INSTABILITIES IN VISCOELASTIC FLOWS By Karkala Arun Kumar A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy (Chemical Engineering) at the UNIVERSITY OF WISCONSIN MADISON 2001

2 c Copyright 2001 by Karkala Arun Kumar All Rights Reserved

3 i Abstract The viscoelastic character of polymer solutions and melts gives rise to instabilities that are not seen in the flows of Newtonian liquids. In industrial applications such as coating and extrusion, these so-called elastic instabilities can impose a limitation on the throughput. Hence, it is important to understand, and if possible, to suppress them. The first instability we study is the phenomenon of melt fracture, which occurs in the extrusion of polymer melts and takes the form of gross distortions of the surface of the extrudate. This instability is linked to the phenomenon of wall-slip, i.e., the velocity of the polymer at the wall relative to the velocity of the wall itself (also called the slip velocity) is non-zero. Several slip relations based on microscopic theories for polymers predict regions in which the slip velocity is multivalued. The expectation is that a multivalued slip relation will result in a multivalued flow curve, which in turn causes melt fracture. Using a simple slip relation, we show that when the dependence of the slip velocity on the pressure is taken into account, this is not necessarily true: a multivalued slip law does not necessarily imply a multivalued flow curve. The second instability we study is the filament stretching instability, which occurs in the extension of a polymeric liquid bridge between two parallel plates. This instability takes the form of a bifurcation to a non-axisymmetric shape near the endplates at

4 ii high extensions. Motivated by the idea of stress localization near the free surface, we model the portion of the filament near the endplates as an elastic membrane enclosing an incompressible fluid and show this is unstable to non-axisymmetric disturbances. The third instability that we present results for occurs when polymeric liquids flow along curved streamlines. Such flows are common in industrial coating operations, and concentric cylinder geometries where the fluid flows along curved streamlines, such as circular Couette flow (where the flow is driven by the rotation of one of the cylinders) and Dean flow (where an azimuthal pressure gradient drives the flow), serve as model geometries for the more complicated coating flows. In the context of Dean flow, we show how the addition of a steady or time-periodic axial flow of small magnitude compared to the primary azimuthal flow, and applied either in Couette or Poiseuille form, can significantly delay the onset of the instability. The stabilization mechanism is related to the generation of axial normal stresses induced by the secondary flow which suppress radial velocity perturbations. Recent experimental observations by Groisman and Steinberg (1997) and Baumert and Muller (1999) in the nonlinear regime of viscoelastic circular Couette flow have shown the formation of stationary, spatially isolated, axisymmetric patterns, termed diwhirls or flame patterns. These patterns are very long wavelength vortex pairs, with a core region of strong radial inflow, surrounded by a larger region of much weaker radial outflow. These structures may be connected to localized defects seen in coating flows, and may also form the building blocks for more complicated patterns seen in viscoelastic flows. Modeling of these patterns is complicated by the absence of a stationary bifurcation in isothermal circular Couette flow. We show how these solutions can be accessed by numerical continuation from stationary bifurcations in flows that are geometrically

5 iii similar to circular Couette flow using a simple constitutive equation for the polymer. Although the stationary solutions we compute are unstable, they are very similar, both qualitatively and quantitatively, to the experimentally observed diwhirls and flame patterns. We also use the results from our computations to propose a fully nonlinear self-sustaining mechanism for these patterns.

6 iv Acknowledgments I wish to express my deep gratitude to my advisor, Professor Michael Graham, without whose guidance this work would not have been possible. The breadth and depth of his knowledge, his scientific curiosity, and his attention to detail have been, and will continue to be, a source of inspiration for me. I would also like to thank Bill Black and Venkat Ramanan for their patience in answering my questions, and for helping me when I first started my research. It was a privilege to have worked with such talented colleagues as John Kasab, Richard Jendrejack, Gretchen Baier, Philip Stone, Guiyu Bai, and Jun Sato. Along with Bill and Venkat, they helped create a stimulating atmosphere for research and contributed immensely to my learning experience. Outside of work, Prasad, Shreyas, Rahul, Sirjana, Ramesh, Kamal, Sanjay, Mukund, Mala, and others too numerous to name made my stay in Madison enjoyable. None of what I have accomplished would have been possible without the support of my parents and my brother. For their love, and for my parents' belief in the value of a good education, I shall be forever grateful.

7 v Contents Abstract i Acknowledgments iv 1 Overview Constitutive equations for polymeric liquids Instabilities in polymeric liquids Pressure dependent slip and flow curve multiplicity Introduction Mathematical Model Results and Discussion Conclusions The filament stretching instability Introduction Planar elongation: a model problem Elongation of a truncated cone Results and Discussion

8 vi 3.5 Conclusions Stabilization of Dean flow instability Instabilities in coating flows Elastic instability in Dean flow Formulation Stability and Numerical Analysis Results and Discussion Conclusions Localized solutions in viscoelastic shear flows Introduction Formulation Discretization and solution methods Results and discussion Conclusions Conclusions and future work 168 A Introduction to finite elasticity 175 A.1 The basic equations of finite elasticity B Base state solutions and matrix components in Dean flow 182 B.1 Base state solutions B.2 Disturbance equations C Velocity and Stress scalings for Couette-Dean flow 189

9 vii D Time Integration of viscoelastic flows 193 D.1 Governing equations D.2 Numerical Method D.3 Results and Discussion E Branch tracing in three dimensional plane Couette flow 205 E.1 Introduction and Formulation E.2 Discretization E.3 Preliminary Results E.4 Proposals for future work Nomenclature 216 Bibliography 226

10 viii List of Figures 1 An illustration of the rod climbing effect. The beaker in (a) contains a Newtonian liquid, glycerin, which shows a vortex. The beaker in (b) contains a solution of polyacrylamide in glycerin, which climbs the rod (figure scanned from Bird et al. (1987a)) Diagram of simple steady shear flow. As shown here, the two plates are moved with equal speed in opposite directions Diagram of elongational flow with l =exp( ɛt) (a) A random walk calculation showing one of a very large number of conformations of a polymer molecule (b) A dumbbell model which only captures the longest relaxation time Pictorial representation of the two types of pitchfork bifurcations: (a) supercritical and (b) subcritical. [y] is some measure of y that captures the features of the transition. A solid line indicates a stable branch while a dashed line indicates an unstable branch Extrusion related instabilities (a) sharkskin (b) melt fracture. (Agassant et al., 1991)

11 ix 7 A typical flow curve for polymers exhibiting spurt and melt fracture. The y coordinate 8V/Dis proportional to the exit flow rate (Kalika and Denn, 1987) The filament stretching instability (a) The non-axisymmetric bifurcation seen from below the bottom plate (b) side view of the instability at a later stage (Spiegelberg and McKinley, 1996) Flow visualization of the purely elastic Taylor-Couette instability (Larson et al., 1990) Flow visualization of a blade coating geometry. The figure clearly shows the presence of an upstream recirculation region with curved streamlines (Davard and Dupuis, 2000) Sequence of snapshots showing the transition from non-axisymmetric disordered flow to solitary vortex structures. On the left in (a), the entire flow geometry is shown, with the box showing the cross section being visualized. On the right, in (b), the actual transition sequence is shown (Groisman and Steinberg, 1998) Sequence of snapshots showing the transition from non-axisymmetric flow to the predominantly axisymmetric and localized flame patterns (Baumert and Muller, 1999). The flow geometry is the same as in figure Plot of equation 30 for different values of pressure (A 2 =4, A 3 = , β =0.0102) Schematic diagram of the constant piston speed experiment

12 x 15 Flow curve for a Newtonian Fluid (Λ = 10, A 2 = 3, A 3 = , β =0.0102, H =1). The profiles at the points marked `X' are shown in figures 18 and Behavior of critical stresses τ c2 and τ c3 with increasing Λ (A 2 =3, A 3 = ): (a) Newtonian model (β =0.0102, H =1) (b) shear thinning model (β =0.0102, H =1, n =0.56) (c) UCM model (β =0.0102, H =1) (d) PTT model (ɛ =10 1, β =0.0102, H =1). The upper curve corresponds to τ c = τ c2 and the lower one to τ c = τ c3. The curves are not extended to Λ=0because the approximations used are not valid for small Λ Flow curve for a Newtonian fluid showing no multiplicity (Λ = , A 2 =3, A 3 = , β =0.0102, H =1) Profiles of pressure, slip velocity and shear stress on the low flow rate branch for the Newtonian fluid of figure 15. ( P =5.22, Λ=10, A 2 = 3, A 3 = , β =0.0102, H =1) Profiles of pressure, slip velocity and shear stress on the high flow rate branch for the Newtonian fluid of figure 15 ( P =6.39, Λ=10, A 2 =3, A 3 = , β =0.0102, H =1) Flow profiles on the three dimensional slip surface for a shear thinning fluid: (a) upper and lower limit points ( P =6.39, Λ=10) (b) cusp point ( P = , Λ = ) (c) point in the central portion without multiplicity ( P = 140.0, Λ = ). Other parameters are A 2 =3, A 3 = , β =0.0102, H =1, and n =0.56 for all three cases

13 xi 21 Oscillatory flow for a Newtonian fluid with Q p =0.21:(a) Pressure drop and exit flow rate vs. time (b) Plot of Q e vs. P b superimposed on the steady state flow curve Non-oscillatory flow for a Newtonian fluid with Q p =0.15: Pressure drop and exit flow rate vs. time Non-oscillatory flow of a Newtonian fluid with Λ = 250, Q p = and other parameters as in figure 21: Pressure drop and exit flow rate vs. time Oscillatory flow of a UCM fluid (C =10 6, Λ=10, Re =10 4, κ = 10 5, De = De = 10, λ 1 = λ 2 = 0.01, Q p = 0.081): (a) Barrel pressure and exit flow rate vs. time (b) Plot of Q e vs. P b superimposed on the steady state flow curve Schematic of a filament stretching rheometer. The setup on the left shows the undeformed state of the liquid bridge Schematic of a planar elongation setup and flow field Coordinate system for the truncated cone Evolution of the shape of a truncated cone under elongation: (a) undeformed configuration (b) axisymmetric configuration at l = 0.2 (c) axisymmetric configuration at l = l c = 0.5 (d) post bifurcation nonaxisymmetric shape at l = l c =0.5. The figures on the right track the change in a cross section originally at a distance of 0.75 units from the left edge. In (d), the perturbation has been exaggerated for clarity Spatial profile of the hoop stress, τ θθ, and the amplitude of the bifurcating solution ˆλ

14 xii 30 Some commonly used coating industrial coating processes: (a) Dip coating and rod coating. (b) Blade coating and air knife coating. (c) Gravure coating. (d) Reverse roll coating. (e) Extrusion coating. (f) Slide coating and curtain coating (Cohen, 1992) Photograph of ribbing instability in forward roll coating (Coyle et al., 1990b) Mechanism of the elastic instability in Dean flow Illustration of the mechanism by which additional axial stresses generated by a superimposed axial flow stabilize the viscoelastic Dean flow instability Dean flow geometry, shown with superimposed Poiseuille flow Neutral stability curves for DAC flow (S=0). In each case, the position of Wp c,min is denoted by a Neutral stability curves for DAP flow (S=0). In each case, the position of Wp c,min is denoted by a Plot of Wp c,min vs. We z for DAC and DAP flows (S=0). Note the linear scaling at high We z Plots of Wp c,min vs. We z for S =0and S =10, displaying the stabilizing influence of solvent viscosity Plot of Wp c,min vs. ñ for different values of We z for DAC flow (S =0) Plot of Wp c,min vs. ñ for different values of We z for DAP flow (S =0) Plot of Wp c,min vs. We z for DAC flow with ñ =1.0 (S =0). Note the linear scaling for We z >

15 xiii 42 Neutral stability curve of pure Dean flow at high α. The Takens-Bogdanov bifurcation point is indicated by a Decay of the perturbation hoop stress τ θθ when axial flow is imposed. Parameters are: We z =1.0, ω 1 =0.5, Wp =4.06, S = Time sequence of density plots of the perturbation hoop stress τ θθ when axial flow is imposed. The parameters are identical to figure 43, so that without axial flow, the flow is neutrally stable. Each frame shows a z r cross-section of the geometry Plot of the magnitude of the Floquet multiplier β vs. ω 1 for different values of α. (We z =0.5, Wp =4.06, S =0) Plot of the magnitude of the Floquet multiplier β vs. ω 1 for different values of We z. (Wp =4.06, α =6.6, S =0). β asymptotes to 1 at large ω 1 in agreement with the asymptotic prediction Plot of the magnitude of the periodic component of the hoop stress τ θθ over a cycle of the forcing for ω 1 1 (We z =1.0, Wp =4.06, α =6.6, ω 1 =0.01). Note the large increases in magnitude Plot of the magnitude of the periodic component of the hoop stress τ θθ over a cycle of the forcing for ω 1 = O(1) (We z =1.0, Wp =4.06, α =6.6, ω 1 =1.0). The magnitude remains O(1) over the entire cycle Geometry of Couette-Dean flow in an annulus A spectral element mesh with 16 axial and 16 radial elements with fifth order polynomials in each direction in each domain. Note the dense concentration of points near r = 1and z = L/2. The high resolution is necessary to capture the intense stress localization in these regions

16 xiv 51 Comparison of ILUT and ILU(0) preconditioners. The test problem was the calculation of the unit tangent for a point on the nontrivial branch in Dean flow. The matrix A had a dimension of Linear stability curves at δ =1(Dean flow) computed using the FENE-P model. The points marked TB are Takens-Bogdanov points. The lines correspond to points where the base state flow loses stability to stationary axisymmetric perturbations Linear stability curves at δ =1(Dean flow) computed using the FENE- CR model. As in figure 52, only stationary axisymmetric perturbations are considered. Note the complete absence of non-stationary bifurcations Continuation in We θ of a stationary solution in Dean flow. The parameter values are L =1.05, b = 700, ɛ =0.20, and S =1.2. At the Hopf point, a pair of complex conjugate eigenvalues become unstable. These collide and form two real eigenvalues, one of which re-crosses the imaginary axis at We θ =29.57, where the stationary branch originates. The solution amplitude used here differs from that used in subsequent figures and is defined in equation A path to stationary solutions in circular Couette flow. The parameters are We θ =25.15, L =3.07, b = 1830, ɛ =0.2, and S = Results from continuing the stationary circular Couette flow solutions in L. The parameters are We θ =25.15, b = 1830, ɛ =0.2, and S =1.2. The gaps in the lower branch correspond to places where we changed the mesh. Note the flatness of the branches as L increases. We have computed extensions of the upper branch at lower values of We θ

17 xv 57 Density plot of QQ θθ (white is large stretch, black small) and contour plot of the streamfunction at L = (We θ = 24.29, b = 1830, S = 1.2, and ɛ = 0.2). For clarity, most of the flow domain is not shown. Note the very strong localization of QQ θθ near the center. The maximum value of QQ θθ at the core is 1589 which gives τ θθ = Compared to this, the maximum value of QQ θθ in the circular Couette base state is 706, which gives τ θθ = Away from the core of the diwhirl, the structure is pure circular Couette flow. The streamlines show striking similarity to those in figure 10 of Groisman and Steinberg (1998). This point was generated by stretching the point at the corresponding We θ on the upper branch of the curve for L =9.11 in figure Diwhirl solution amplitudes as functions of We θ and L. Note that the curves at L =9.11 and L =4.74 are very close together, while both curves are well separated from the curve at L =3.07 (b = 1830, S =1.2, and ɛ =0.2) Plot of the location of the turning point, We θ,c, versus L at S =1.2and ɛ =0.2. Note the flatness of the curve at large L Plot of the position of the linear stability limit in circular Couette flow with respect to axisymmetric disturbances and the turning point in We θ for the diwhirls as a function of b. The parameters are S = 1.2 and ɛ =0.2. The computations for the diwhirls were performed at L =4.74, which is close to the minimum in figure

18 xvi 61 (a) The axial variation of v r at r =0.6for L =4.74 and We θ =23.50 on the upper branch. (b) Figure 9 on page 2457 of Groisman and Steinberg (1998), shown here for purposes of comparison. We have shifted the axial coordinate so that the symmetry axis of the computed diwhirl in (a) is at z =0, to make comparison with (b) easier Variation of solution amplitudes with ɛ. Here, We θ =25.15, and the other parameters as in figure Diwhirl solution amplitudes as a function of the parameter c r for two different wavelengths. We θ =25.15 and the other parameters are as in figure 58. The existence of turning points demonstrates that these solutions cannot be extended to the FENE-CR model Intensity plot of the dimensionless viscous dissipation for We θ =23.73 on the upper branch at L =4.74. Light areas represent areas of large viscous dissipation and dark areas represent regions of low viscous dissipation. The horizontal axis is stretched by a factor of two relative to the vertical axis for clarity (a) Vector plot of v near the outer cylinder at the center of the diwhirl structure (oblique arrows) and the base state (straight arrows). The length of the arrows is proportional to the magnitude of the velocity. The axial velocity is identically zero in the base state, and is zero by symmetry at the center of the diwhirl. (b) Principal stress directions at the same location as for (a). The Couette flow stress is not shown because it is very small in comparison. This figure shows how fluid elements at larger radii are pulled down and forward sustaining the increase in v θ...157

19 xvii 66 Nonlinear self-sustaining mechanism for the diwhirl patterns Density plot of v r showing the (a) real and (b) imaginary parts of the destabilizing axisymmetric disturbance, and (c) the streamlines of the base diwhirl. Note that the core of the diwhirl is entirely unaffected by the disturbance. The parameters are We θ =23.87, S =1.2, ɛ =0.20, b = 1830, and L = Density plot of the perturbation radial velocity for the three non-axisymmetric unstable eigenmodes with n =1. The parameters are identical to those in figure Density plot of the perturbation radial velocity for the three non-axisymmetric unstable eigenmodes with n =2. The parameters are identical to those in figure Bifurcation diagram for Dean flow at b = 1830, S =1.2, ɛ =0.2, and L = We could not continue the branch beyond We θ =13.94, for reasons discussed in section Radial velocity profiles corresponding to the points marked by the filled and open circles are shown in figure Density plots of the radial velocity at the points circled in figure 70. (a) We θ = (b) We θ = Linear stability curves and existence boundaries for nonlinear solutions in Dean flow at S =1.2, ɛ =0.2, and b = The filled triangle shows the data at L =1.795 where the solution terminates via collision with the L/2 branch in a pitchfork bifurcation before the turning point Streamlines for equation 214 with A =1and k = π

20 xviii 74 Poincare map of the trajectory of a point in the flow field given by equation 215. The parameters chosen were k = π, ω =2π, and ɛ = Illustration of the effect of adding a diffusion term to the constitutive equation on a mesh of The parameter values used were We =1, Re =1, ɛ =0.2, k = π, ω =2π, b =10, and S = Effect of mesh refinement on numerical stability. Runs with both meshes were performed with a = 10 4 and the same parameter values as in figure Effect of artificial diffusion when equation 223 in integrated in a known velocity field. The parameters and mesh are identical to those in figure 75. Note the earlier blow up when compared to the coupled case Contours and density plot of the streamwise velocity u in the y z plane in the base state. The value of Re = 150, Λ=0.002, and F r = Contours and density plot of the u component of the two unstable eigenvectors for the flow shown in figure 78 in the x z plane at y =0. The mode shown in (a) has the higher growth rate of the two unstable modes Plot of the profile of the u component of the streamwise velocity of the two unstable eigenvectors for the flow shown in figure 78 at y =0, and z = π/γ. The eigenvalue corresponding to eigenvector 1 has the higher growth rate. Eigenvector 2 is simply a vertically shifted and scaled version of eigenvector Contours and density plot in the x z plane of the u component of the real and imaginary part of the unstable eigenvector at Re = 150, Λ=0.1, and F r =

21 1 Chapter 1 Overview Unlike Newtonian fluids, polymeric liquids have some memory of the deformation they have experienced. Newtonian fluids respond virtually instantaneously to an imposed deformation rate, whereas polymeric fluids respond on a macroscopically large time scale, known as the relaxation time. When subjected to deformation rates much larger than the inverse relaxation time, their behavior resembles that of elastic solids, whereas their response to deformation rates much smaller in magnitude than the inverse relaxation time resembles that of viscous liquids. For this reason, they are known as viscoelastic liquids. The viscoelastic nature of polymer melts and solutions causes them to exhibit behavior not seen in Newtonian liquids. An interesting example of this behavior is the so called rod-climbing effect. An illustration of this effect is shown below in figure 1. The figure shows two liquids being stirred in two different beakers. The liquid in figure 1(a) is Newtonian, while the liquid in figure 1(b) is a polymer solution. The Newtonian liquid shows a vortex, with free surface being depressed in the region near the stirrer. We would expect this intuitively, based on the fact that the centrifugal force pushes liquid near the

22 2 stirrer outward. In contrast, the polymer solution shows the opposite effect, with the liquid climbing up the stirring rod. This is because the polymer molecules are stretched along the circular streamlines of the flow. The extra tension in the streamlines exerts an inward force on the fluid, which acts against the centrifugal force and gravity and pushes the liquid up the rod. Other examples of viscoelastic behavior include die swell and elastic recoil (Bird et al., 1987a). (a) (b) Figure 1: An illustration of the rod climbing effect. The beaker in (a) contains a Newtonian liquid, glycerin, which shows a vortex. The beaker in (b) contains a solution of polyacrylamide in glycerin, which climbs the rod (figure scanned from Bird et al. (1987a)). In industrial applications involving polymeric liquids, instabilities that arise from their viscoelastic nature can have important consequences, and this document details a computational investigation of a few of these instabilities. In this chapter, we present an overview of the topics covered in the rest of the thesis. Before discussing instabilities in polymer flows however, it is instructive to look at some of the elementary approaches taken to modeling polymeric liquids, which lead to the constitutive equations that we use in our analyses.

23 3 1.1 Constitutive equations for polymeric liquids Unlike Newtonian liquids, which are very well described by Newton's law of viscosity, there is no single constitutive equation that describes the entire range of polymeric liquids. Consequently, the approach taken is to use simple flows to check the predictions of constitutive equations with the actual behavior of the polymeric liquid being studied. We will, for the most part, be interested in the behavior of dilute solutions of linear polymers. In this section, we discuss some of the flows used to characterize these liquids, and then proceed to describe a few of the simple constitutive equations used to model them. y z x Figure 2: Diagram of simple steady shear flow. As shown here, the two plates are moved with equal speed in opposite directions. The simplest flow used for rheological characterization is the steady shear flow shown in figure 2. Denoting the velocity components in the three coordinate directions as v x, v y, and v z, the only component that is not zero is v x. For a Newtonian fluid, the force required to maintain the flow would only have a component in the flow direction. The only nonzero components of the stress tensor are the shear stresses, τ yx and τ xy. Since the stress tensor is symmetric, these are equal to each other and are given by Newton's

24 4 law of viscosity as τ xy = τ yx = η γ yx, (1) where γ yx = dv x /dy is the shear rate, γ = γ yx, and η is the viscosity, which is independent of the shear rate. For a polymer solution, the viscosity η, would, in general, depend on the shear rate, i.e., η = η( γ). In addition, the stress components τ xx, τ yy, and τ zz are not all zero, unlike in the shear flow of Newtonian liquids. A direct physical consequence of this is that there are components of the fluid stresses that act to push the plates apart. Therefore, in addition to the shearing force, a force acting normal to the plates has to be applied to maintain the flow. The additional nonzero stress components allow us to define two new fluid properties using the relations τ xx τ yy = Ψ 1 ( γ) γ 2 yx, (2) τ yy τ zz = Ψ 2 ( γ) γ 2 yx. (3) The quantity τ xx τ yy is known as the first normal stress difference, while τ yy τ zz is known as the second normal stress difference. Ψ 1 and Ψ 2 are known as the first and second normal stress coefficients respectively, and are functions of γ. Another flow used for rheological characterization is elongational flow (figure 3). Here, the velocity field is given by v x = 1 2 ɛx v y = 1 2 ɛy v z = ɛz, (4) so that fluid elements are being stretched exponentially in one direction and compressed exponentially in the two directions perpendicular to it. The material function of relevance

25 5 y 1/ l l 1/ l x z Figure 3: Diagram of elongational flow with l =exp( ɛt). here is the elongational viscosity η, which is defined by the relation η = τ zz τ xx. (5) ɛ For Newtonian liquids, η =3η, and the ratio η/η is called the Trouton ratio. Q coarse graining (a) (b) Figure 4: (a) A random walk calculation showing one of a very large number of conformations of a polymer molecule (b) A dumbbell model which only captures the longest relaxation time.

26 6 The steady shear and elongational flows are used as model flow fields for testing constitutive equations for polymers. Experimental measurements of the behavior of specific polymer solutions and melts can be compared with the predictions of the constitutive equations. The expectation is that constitutive equations that can capture the behavior of a polymer in these model flows might reasonably be expected to work in more complicated flow fields as well. Also, the model flows are used to obtain rheological parameters needed in constitutive equations. For example, simple shear flow can be used to obtain the zero shear viscosity and the relaxation time. We now proceed to discuss simple constitutive equations that are used for modeling dilute solutions of linear polymers. The constitutive equations that we describe below are based on molecular models. Figure 4(a) shows a random walk model of a polymer molecule, which represents one of a large number of conformations that the polymer molecule can take. These conformations change continually due to thermal motion, with different time scales associated with motion on different length scales. For example, it is easy to see that small internal reorientations would take place on a shorter time scale than a change involving a large number of segments. Thus, the polymer molecule responds on a whole spectrum of time scales. For the simple models that we will use in this work, we only consider the upper limit of this spectrum, i.e., the longest relaxation time. At the level of coarse graining described above, the polymer molecules are modeled as elastic dumbbells (a pair of beads connected by a spring), as shown in figure 4(b), with Q representing the end to end vector of the dumbbell. The elastic force here represents the restoring force due to entropy. A solution of such dumbbells is described by the probability distribution function for Q, denoted by ψ(q,t). For a dilute solution of these dumbbells, neglecting inertia and hydrodynamic interaction between the beads allows us

27 7 to use kinetic theory (Bird et al., 1987b) to derive a diffusion equation for ψ(q,t), which is given by ( { ψ t + v ψ = Q [κ Q]ψ 2kT ζ ψ Q 2 }) ζ F c ψ. (6) Here, v is the velocity vector, k is Boltzmann's constant t represents time, T represents temperature, F c the spring force, ζ the friction coefficient due to hydrodynamic drag, and κ is a second order tensor which specifies the local velocity field. Equation 6 serves as the starting point for the calculation of the averages of quantities of physical interest. For the purpose of this work, we are mainly interested in the stress tensor τ. For elastic dumbbells, this is given by τ = N QF c NkT I, (7) where N is the number density of dumbbells, and we have once again assumed that the external forces acting on the two beads of the dumbbell cancel each other out. For simple models, τ can be approximated from the ensemble average of the second order tensor QQ. This quantity, QQ, is given by the equation QQ (1) = 4kT ζ I 4 ζ QF c. (8) Here, QQ (1) is the upper convected derivative of QQ given by QQ (1) = QQ t + v QQ { ( v) t QQ + QQ ( v) }, (9) where v is the velocity vector, and the superscript t is used to denote the transpose. In our discussion above, we have not specified a form for the spring force. The simplest case occurs when we assume that the spring is Hookean, i.e., F c = HQ, where H is the spring constant. This assumption yields the upper convected Maxwell (UCM)

28 8 equation. This equation assumes that the beads can stretch to an infinite extent, which is clearly not a reasonable assumption, especially for so called strong flows, where fluid elements undergo exponentially large degrees of stretching. However, the UCM equation has the virtue of being simple, and for this reason is often used to get qualitative information on the effect of viscoelasticity. This simplicity of form is in some respects deceptive, for it masks the fact that the UCM equation is extremely difficult to use in the numerical simulation of strong flows. This in turn gives rise to another important use of the UCM equation: as a test case for numerical methods for solving viscoelastic flow problems. Since polymers have finite maximum lengths, a nonlinear spring law makes for a more realistic approximation. The Finitely Extensible Nonlinear Elastic (FENE) spring law takes the form F c Q = H,Q Q 1 Q 2 /Q 2 0, (10) 0 where Q 2 = tr(qq) is the square of the dumbbell extension with tr used to denote the trace of its argument. For small extensions, the spring is nearly Hookean, but increases in stiffness for larger extensions, up to a maximum extension of Q 0. In the limit Q 0, equation 10 becomes Hooke's law. Substituting equation 10 into equation 8 gives QQ (1) = 4kT ζ I 4H ζ QQ 1 Q 2 /Q 2 0. (11) The second term on the right hand side prevents equation 11 from being an explicit equation for QQ. To obtain a closed form equation for QQ, it is necessary to make a closure approximation. One such approximation is the Peterlin closure (Bird et al., 1987b) which takes the form QQ 1 Q 2 /Q 2 0 QQ. (12) 1 tr( QQ )/Q 2 0

29 9 Substituting this relation into equation 11 gives the FENE-P constitutive equation. A related FENE type equation is the FENE-CR equation (Chilcott and Rallison, 1988), which is a good model for dilute solutions of some polymers. This equation is ( 1 4kT QQ (1) = 1 tr( QQ )/Q 2 0 ζ I 4 ) ζ QQ. (13) In all three equations, knowing QQ enables us to get expressions for the stress tensor. For the UCM equation, this is given by the Kramers expression τ = N H QQ NkT I. (14) The quantity N kt is equal to G, the relaxation modulus of the polymer. Substituting equation 14 and the Hookean spring law into equation 8 yields an explicit equation for τ, given by which is linear in τ. Here, λ is the relaxation time, given by η is the viscosity, given by and τ + λ τ (1) = η γ, (15) λ = ζ 4H, (16) η = λg, (17) γ = I (1) = v +( v) t (18) is the shear rate tensor. For the FENE-P equation, the Kramers expression for the stress tensor is QQ τ = N H NkT I, (19) 1 tr( QQ )/Q 2 0

30 10 and for the FENE-CR equation, the stress tensor can be calculated using the relation τ = 1 (N H QQ NkT I). (20) 1 tr( QQ )/Q 2 0 In the latter two cases, it is possible to substitute equations 19 and 20 into the constitutive equations for QQ and get explicit constitutive equations for τ. However, it is much simpler to work with QQ, and use equation 19 or 20 to get τ when needed. Table 1 summarizes the qualitative behavior of the UCM, FENE-P and FENE-CR equations in shear and elongation. Also presented in table 1 is the behavior of the exact FENE model in these flows, obtained from Brownian dynamics simulations. In this work, we will use the FENE-P and UCM equations for most of our calculations. Occasionally, we will use the FENE-CR equation when testing for model dependence. Property Model η Ψ 1 η UCM Constant Constant Blows up at finite elongation rate FENE Shear thins Shear thins Saturates at high elongation rate FENE-P Shear thins Shear thins Saturates at high elongation rate FENE-CR Constant Shear thins Saturates at high elongation rate Table 1: Summary of behavior of the UCM, FENE, FENE-P, and FENE-CR constitutive equations in shear and elongation. All four models predict Ψ 2 = Instabilities in polymeric liquids Before embarking on a discussion of flow instabilities in polymeric liquids, it is instructive to clarify what we mean by the term flow instability. This is best done with the aid

31 11 of a simple example. Consider the flow of a Newtonian liquid (such as water) in the gap between two very long concentric cylinders, driven by the motion of the inner cylinder. At low rotation rates, we would see a flow where the velocities and pressure only varied with radial position and are constant along the azimuthal and axial directions. We will refer to this flow as the base flow. As the rotation speed is increased, a transition occurs to an axisymmetric (i.e. no variation in the azimuthal direction), axially periodic vortex flow. This flow is qualitatively different from the base flow: in particular, unlike in the base flow, the velocities and stresses vary in the axial direction. What is interesting is that the base flow is an admissible solution to the governing equations at all values of the rotation rate. Beyond a critical value of the rotation rate however, perturbations (even infinitesimal ones) applied to the base flow grow in magnitude until a qualitatively different steady state is reached. We say that the base flow is unstable beyond the critical rotation speed, hence the phrase flow instability. The qualitative change caused by the variation of a parameter (in this case, the rotation rate of the inner cylinder) is called a bifurcation. In this work, we will mainly be interested in two types of bifurcations: pitchfork and Hopf. Pitchfork bifurcations generally occur in systems with symmetry. As a simple one-dimensional example, consider the differential equation ẏ = µy y 3. (21) This equation is invariant with respect to the transformation y y, i.e., replacing y by y results in the same equation. Steady states are obtained by setting ẏ to zero. They are solutions to f(y) =µy s y s 3 =0, (22) which are y s =0and y s = ± µ, with the subscript s being used to denote a steady state.

32 12 While y s =0is a solution for all values of µ, the solutions y s = ± µ are valid only for µ>0. In this problem, y s = µ and y s = µ bifurcate from the solution y s =0 at µ =0. In a small neighborhood of the bifurcation point, the branches y s = µ and y s = µ are one sided, i.e., they only exist for µ>0. This type of bifurcation is called a pitchfork. Note that the bifurcating branches y s = µ and y s = µ are related by symmetry. For the one-dimensional example presented here, the stability of the bifurcating branches is determined by the sign of the linearization of the steady state equations, evaluated at the point whose stability is to be determined. If the linearization is positive, the solution is unstable, because small disturbances grow. If the linearization is negative, small disturbances decay, and the solution is stable. For equation 21, the linearization is given by f y = f y = µ 3y 2 s. (23) Substituting y s =0gives f y = µ. Thus, the solution y s =0is stable for negative values of µ and unstable for positive µ. Note that the change in stability occurs at the bifurcation point, µ =0. Substituting y s = ± µ in equation 23 gives f y = 2µ, which is negative for µ>0, the only regime where these solutions exist. Therefore, these solutions are stable. When the bifurcating branch is stable, the bifurcation is said to be supercritical. The opposite case, a subcritical bifurcation, occurs when the bifurcating branches are unstable. An example of this is the system ẏ = y 3 + µy, (24) which has the steady states y s =0and y s = ± µ. An analysis similar to the one presented above shows that the steady states y s = ± µ bifurcate from y s =0at µ =0,

33 13 only exist for negative values of µ and are unstable. Figure 5 shows a pictorial representation of supercritical and subcritical bifurcations. For higher dimensional systems, stability is determined by the eigenvalues of the matrix that results from linearizing the steady state governing equations, with the linearization being performed about the point whose stability is being determined. This matrix is called the Jacobian. If all the eigenvalues of the Jacobian are negative, the solution is stable. If one or more of the eigenvalues are positive, the solution is unstable. A bifurcation occurs when the real part of one or more eigenvalues changes sign. [y] [y] µ (a) µ [y] [y] µ (b) µ Figure 5: Pictorial representation of the two types of pitchfork bifurcations: (a) supercritical and (b) subcritical. [y] is some measure of y that captures the features of the transition. A solid line indicates a stable branch while a dashed line indicates an unstable branch. The other bifurcation that we will present here is the Hopf bifurcation. A Hopf bifurcation occurs when a pair of complex conjugate eigenvalues of the Jacobian crosses

34 14 the imaginary axis. In this case, the bifurcating branches are not steady states, but timeperiodic oscillations (also called limit cycles). As with the pitchfork bifurcation, a supercritical Hopf bifurcation occurs if the bifurcating branch of periodic solutions is stable, and a subcritical Hopf bifurcation occurs when they are unstable. Since a Hopf bifurcation requires that a pair of eigenvalues cross the imaginary axis, it follows that a system must be at least two-dimensional to show a Hopf bifurcation. An example of a system that undergoes a Hopf bifurcation is ẏ 1 = y 2 + y 1 (µ y 1 2 y 2 2 ), ẏ 2 = y 1 + y 2 (µ y 1 2 y 2 2 ), (25) which can be written in vector form as ẏ = f(y). (26) It is trivial to show that (y 1s,y 2s )=(0, 0) is a steady state. Linearizing about this steady state, we get the Jacobian matrix J = f y = µ 1 1 µ, (27) which has the complex conjuagte pair of eigenvalues µ±i. This pair crosses the imaginary axis at µ =0, giving rise to a Hopf bifurcation. It can be shown (see Seydel (1994)) that the bifurcating branch of limit cycles is stable, so this is a supercritical Hopf bifurcation. With this background, we can now move on to a discussion of flow instabilities in polymer solutions. The flow instability that we described at the beginning of this section as an example is called the Taylor-Couette instability. It is driven by inertial effects, specifically, the unstable stratification of angular momentum. Another example of an

35 15 inertia driven instability is the phenomenon of turbulence in pipe and plane Couette flow at high Reynolds numbers. In most practical applications, the high viscosity of polymer melts and solutions results in a low Reynolds number, which means that inertial effects are of secondary importance. However, polymeric fluids display an entirely separate class of instabilities which have their origin in the elastic nature of these fluids. In an industrial setting, these so called elastic instabilities, seen in many commercially important flows, are detrimental to the quality of the final product, and avoiding them involves imposing limitations on throughput, or modifying the flow apparatus. It is therefore of practical importance to understand these instabilities, and if possible, come up with methods to delay their onset. The work presented in this document focuses on four of these: melt fracture, the filament stretching instability, the viscoelastic Couette-Dean instability, and nonlinear pattern formation in Couette-Dean flow. These instabilities are seen either in industrial processing operations or in simple flows used to characterize the rheological properties of viscoelastic liquids. The first part of our work concentrates on melt fracture, which is an instability seen in extrusion processes. In an extrusion process, a polymer melt is forced out of a die by applying a pressure gradient or a constant volumetric flow rate. At low flow rates (or pressure drops), the shape of the extrudate is smooth. As the flow rate is increased, the extrudate, for linear polymers, begins to show surface distortions. These are first seen in the form of small amplitude, small wavelength distortions parallel to the surface (figure 6(a)). The effect is called sharkskin, and a detailed review of this phenomenon may be found in Graham (1999). At somewhat higher pressure drops, if the flow is being driven by a constant pressure gradient, the flow rate shows a sudden large jump to a higher value. This phenomena is known as spurt flow. If the flow is driven by an

36 16 imposed constant flow rate, the pressure drop and exit mass flow rate show oscillations. These are accompanied by gross distortions in the shape of the extrudate (figure 6(b)). This phenomena is known as melt fracture. The dependence of the onset flow rate (or pressure drop) for both sharkskin as well as melt fracture (Ramamurthy, 1986; Wang and Drda, 1997; Piau et al., 1995) suggests that both phenomena have their origin in the fact that flows of polymeric fluids exhibit slip at the wall, i.e., the velocity of the polymer at the wall is non-zero relative to the velocity of the wall itself. This is different from Newtonian liquids which obey at no-slip condition at a solid boundary. Currently, there exist several slip expressions relating the slip velocity to the wall shear stress which are based on microscopic theories for polymers (Leonov, 1990; Brochard and de Gennes, 1992; Adjari et al., 1994; Yarin and Graham, 1998; Mhetar and Archer, 1997; Hill, 1998). These theories predict certain regions in which the slip velocity is multivalued. They do not, however, consider the effect of pressure on slip velocity. (a) (b) Figure 6: Extrusion related instabilities (a) sharkskin (b) melt fracture. (Agassant et al., 1991). While sharkskin is a phenomenon related to flow at the exit of the die, melt fracture is related to the flow profile over the entire length of the die. A flow curve is generated by

37 17 plotting the flow rate (or equivalently, the shear rate) versus the pressure drop (figure 7), and if the slip relation is multivalued (i.e., it has a region where more than one slip velocity is possible for a given shear rate), the flow curve itself can have a multivalued region. Spurt flow results from a jump from the lower to the upper branch of the flow curve when the operation is at constant pressure. Melt fracture arises due to relaxation oscillations arising from the interaction between fluid compressibility and slip, when an attempt is made to operate on the decreasing part of the flow curve at constant imposed volumetric flow rate. Since many theoretical models for slip have a multivalued region, the expectation is that the flow curve will also be multivalued. We show in chapter 2 that this is not always the case. If the slip velocity decreases with increasing pressure (for which there is experimental evidence in the literature), we get the surprising result that multivaluedness in the slip relation does not imply multivaluedness in the flow curve. Thus, there could be certain cases where melt fracture would not be seen even if the slip relation is multivalued. 8V/D Shear Stress (MPa) Figure 7: A typical flow curve for polymers exhibiting spurt and melt fracture. The y coordinate 8V/D is proportional to the exit flow rate (Kalika and Denn, 1987).

38 18 The second instability that we present here is related to the flow of polymeric solutions in extension. This so-called filament stretching instability was first observed by Spiegelberg and McKinley (1996) in the extension of a liquid bridge between two parallel plates. This flow has been used to measure the extensional viscosity of polymeric fluids. A visualization of the instability is shown in figure 8. The instability takes the form of a bifurcation to a non-axisymmetric shape near the endplates, followed by a break-up of the filament, and ultimately by complete detachment from one of the endplates. There is evidence to show that elasticity plays a significant role in this instability. The instability is never seen in Newtonian liquids, and always occurs when elastic effects are large (extensional rates on the order of the inverse polymer relaxation time). The mechanism of this instability is not well understood, and it has been hypothesized that it is related to the classical Saffman-Taylor instability in Newtonian liquids (Saffman and Taylor, 1958), which occurs when a less viscous liquid displaces a more viscous one. In chapter 3, we propose a different mechanism, much more closely related to elastic effects. We model the region near the endplates (where the instability is seen), as a membrane in the shape of a thin truncated cone enclosing an incompressible fluid, and show that this is unstable to non-axisymmetric disturbances. (a) (b) Figure 8: The filament stretching instability (a) The non-axisymmetric bifurcation seen from below the bottom plate (b) side view of the instability at a later stage (Spiegelberg and McKinley, 1996).

39 19 The third viscoelastic instability for which we present results is the purely elastic instability in flows of viscoelastic fluids with curved streamlines. This instability was first observed in a circular Couette geometry (two concentric cylinders with liquid filling the annulus, and the flow driven by one of the cylinders) by Larson et al. (1990), and a visualization of the instability is shown in figure 9. At a certain critical flow rate, the smooth homogeneous flow breaks up into axially and azimuthally periodic cells. The Reynolds number at the onset of instability in their experiment was close to zero, so clearly the mechanism is different from the classical inertial Taylor-Couette instability described at the beginning of this section. The mechanism of this instability is a purely elastic one, and is related to the coupling of perturbations in the hoop stress with the base state velocity gradients. Joo and Shaqfeh (1991) showed that a similar instability arises when the flow is driven by imposing an azimuthal pressure gradient (Dean flow). In this case, the mechanism is related to the coupling between a radial velocity perturbation and the base state hoop stress. Figure 9: Flow visualization of the purely elastic Taylor-Couette instability (Larson et al., 1990).

40 20 Flows with curved streamlines are common in coating operations. In coating techniques such as forward and reverse roll coating (see figure 30, and the discussion in section 4.1), they arise as a result of the curvature of the geometry. Even in techniques such as blade coating where the geometry is not curved, curved streamlines can be present in recirculation regions as shown in figure 10. Instabilities that occur in these flows impose a limitation on the operating speed of the processing apparatus and so limit throughput. Flows in a circular Couette geometry are also used to characterize the properties of polymer solutions in shear. Given the importance of these flows, any scheme to suppress the instability may have considerable practical utility. Graham (1998) found that the addition of a steady axial flow of small magnitude (compared to the azimuthal forcing) either in Couette or Poiseuille form significantly delays the onset of the elastic instability. The mechanism is related to the development of an additional axial normal stress induced by the secondary flow, which suppresses radial velocity perturbations. In chapter 4 we build on the work of Graham (1998) by showing that the elastic instability in Dean flow can be suppressed by the same technique, thus demonstrating its utility for a broader class of flows. Further, we demonstrate that an oscillatory axial forcing also yields stabilization. Figure 10: Flow visualization of a blade coating geometry. The figure clearly shows the presence of an upstream recirculation region with curved streamlines (Davard and Dupuis, 2000).

41 21 In chapter 6, we present work on nonlinear pattern formation in viscoelastic circular Couette flow. Recent experimental observations by Groisman and Steinberg (1997) and Baumert and Muller (1999) have shown the formation of stationary, spatially isolated, axisymmetric patterns in circular Couette flow. These patterns have been termed diwhirls by Groisman and Steinberg, and as flame patterns by Baumert and Muller. These patterns are very long wavelength axisymmetric vortex pairs, with a core region of strong radial inflow, surrounded by a much larger region of weaker radial outflow. In the absence of non-isothermal effects, the primary bifurcation in circular Couette flow is to a non-axisymmetric, time dependent mode (i.e., a Hopf bifurcartion). Upon further increasing the strength of the flow, there is a secondary transition to the diwhirl structures or flame patterns. This transition is shown in figure 11 for the diwhirls and in figure 12 for the flame patterns. If, at this point, the flow strength is reduced, these patterns persist, until eventually the base flow is recovered at a shear rate much lower than where the first transition to the non-axisymmetric mode occurred. t z (a) (b) Figure 11: Sequence of snapshots showing the transition from non-axisymmetric disordered flow to solitary vortex structures. On the left in (a), the entire flow geometry is shown, with the box showing the cross section being visualized. On the right, in (b), the actual transition sequence is shown (Groisman and Steinberg, 1998). Localized structures such as those described above are important for several reasons.

42 Figure 12: Sequence of snapshots showing the transition from non-axisymmetric flow to the predominantly axisymmetric and localized flame patterns (Baumert and Muller, 1999). The flow geometry is the same as in figure

43 23 Firstly, they may be connected to localized defects seen in coating flows. Secondly, they may form the building blocks of more complex patterns seen in viscoelastic flows, such as the recently observed phenomenon of elastic turbulence (Groisman and Steinberg, 2000). Finally, they can serve as a test for the ability of the approximate constitutive equations described in the previous section to model complex flows. The modeling of these structures is complicated by the fact that the transition in the space of circular Couette flow occurs from a non-axisymmetric, time dependent state. Rather than undertake the approach of modeling a three-dimensional time dependent flow, we adopt an approach that is computationally simpler. We start from stationary bifurcations in flows that are geometrically similar to circular Couette flow, and continue the bifurcating stationary solutions into the regime of circular Couette flow. These stationary solutions are very similar to the experimentally observed diwhirls and flame patterns. We also compute the stability of these solutions with respect to axisymmetric and non-axisymmetric perturbations. So far, we have discussed purely elastic instabilities, i.e., instabilities that are caused by the viscoelastic character of polymers. Polymers can also have significant effects on instabilities driven by inertia. Of particular interest to us is the transition to turbulence in plane Couette flow. It is well known that adding a small quantity of polymer shifts this transition to higher Reynolds numbers (Giles and Pettit, 1967; White and McEligot, 1970). Since plane Couette flow is stable to small perturbations at all Reynolds numbers, non-trivial solutions that may be related to coherent structures seen in turbulence have to be obtained indirectly. Waleffe (1998) obtained such non-trivial solutions by adding a forcing term to the Navier-Stokes equations, so that the modified flow had a stationary bifurcation. What is interesting is that the non-trivial solutions emanating from the stationary bifurcation in the system with forcing persist even when the forcing is removed:

44 24 they exist as isolated steady states in plane Couette flow, and show similarities to coherent structures seen in turbulence. Determining the behavior of these solutions when a polymer is added can help us understand what effect polymers have on turbulence, and in Appendix E, we present a brief description of some preliminary work done in setting up branch tracing of three dimensional Newtonian plane Couette flow with Waleffe's forcing. Future goals would involve coupling this with a polymer constitutive equation. The main body of this document is divided into four chapters. Chapter 2 describes the work on melt fracture. Chapter 3 concentrates on the instability in filament stretching. Chapter 4 describes the stabilization of the elastic instability in Dean flow. Chapter 5 describes the work on the modeling of the diwhirl and flame patterns in circular Couette flow. In chapter 6, we present a discussion on future directions in the area of elastic instabilities. In Appendix D, we briefly discuss methods to integrate viscoelastic constitutive equations, and finally, in Appendix E, we present an application of continuation methods to track solutions in three dimensional plane Couette flow.

45 25 Chapter 2 Pressure dependent slip and flow curve multiplicity The melt fracture instability in the extrusion of polymer melts is often linked to the fact that the polymer does not obey a no slip condition at the wall of the die. It is thought that if the slip law, which is a relationship between the velocity of the polymer with respect to the wall and the shear stress, has a multivalued region, the flow curve will be multivalued as well. In this chapter, we demonstrate that this is not always the case. In particular, we show that adding a pressure dependence which preserves the multivaluedness of the slip law can give rise to flow curve that is not multivalued. 2.1 Introduction Several flow instabilities occur during the process of extrusion of melts of linear polymers, and are manifested in the form of distortions in the extrudate. A discussion of Most of the material in this chapter has been published in Kumar and Graham (1998a)

46 26 these may be found in several reviews (Denn, 1990; Larson, 1992; Leonov and Prokunin, 1994). In this work, we consider the phenomena of hysteresis and spurt flow. We can get a clearer picture of these phenomena by examining the flow curve, which is a plot of apparent wall shear rate ( γ A ) versus wall shear stress (τ w ) during capillary extrusion. For steady flows in a circular capillary, and γ A = 4Q πr 3 + 4u s R, (28) τ w = pr 2L, (29) with R and L being the radius and length of the capillary respectively, Q the flow rate, p the pressure drop, and u s the slip velocity of the fluid at the wall. Thus, τ w is a measure of the pressure drop, and γ A is a measure of the flow rate. The unusual behavior seen in these curves is usually attributed to the effect of wall slip (i.e., a nonzero value of u s ) because of the dependence of the phenomena on the materials of construction of the die (Ramamurthy, 1986; Wang and Drda, 1997; Piau et al., 1995). A plot of log( γ A ) versus log(τ w ) starts as a straight line, indicating a power law regime. At a certain critical stress τ c1, there is a distinct increase in slope. This is followed by a sharp jump in the value of γ A at a second critical shear stress τ c2. A rheometer operated in the constant pressure mode would show a sudden jump to a higher flow rate at this value of τ w. If the pressure is increased further, γ A continues along the high flow rate branch. At this point, if the pressure is decreased, the flow curve decreases along the high flow rate branch even below τ c2, until there is a jump to the low flow rate branch at a third critical stress τ c3, less than τ c2. Thus, the flow curve exhibits hysteresis (Bagley et al., 1958; Tordella, 1956, 1963; El Kissi and Piau, 1990). This hysteresis implies that the flow curve displays a region of

47 27 multiplicity: two stable flow rates are possible at the same pressure drop. If a rheometer is operated in the constant piston speed mode in the region between τ c2 and τ c3, the pressure and extrudate flow rate show oscillations (Lupton and Regester, 1965; Kalika and Denn, 1987; Ramamurthy, 1986; Hatzikiriakos and Dealy, 1992a). This phenomenon is known as spurt flow. The dependence of the critical shear stress τ c2 on the length to diameter ratio (L/D) of the die can be inferred from several experimental works in the literature. The data of Vinogradov et al. (1984) for polybutadienes and Wang and Drda (1996) for entangled linear polyethylenes, which are taken at low L/D ( 25), suggest that τ c2 is virtually independent of L/D. The data of El Kissi and Piau (1990) taken at somewhat higher L/D ratios (L/D =20and 40) for polydimethylsiloxanes, corrected for entrance losses show that there is a small decrease in this value (from 0.61 bar to 0.59 bar) for the constant pressure experiments where the L/D ratio was varied by fixing D and varying L. Kalika and Denn (1987) report data for the constant piston speed experiments using LLDPE which show that τ c2 decreases as L/D is varied from 33.2 to 66.2, and at L/D = 100.1, their reported value of τ c3 is greater than τ c2 (in this case, the flow curve shows a large jump in flow rate in the relatively small region between τ c2 and τ c3 ) which indicates that there may not be multiplicity if the rheometer is operated in the constant pressure mode at high L/D. Finally, Hatzikiriakos and Dealy (1992a) report that they observe an increase in τ c2 with increasing L/D (from L/D =10to L/D = 100) for linear high density polyethylene, and take this as evidence of a pressure dependence of wall slip. Thus, there seems to be evidence in the literature to support the L/D dependence, but there does not seem to be an agreement on the trend, and this may differ depending on the polymer used and other experimental conditions.

48 28 In addition to experimental evidence for multiplicity in slip behavior, the motivation for using a multivalued slip model arises from the slip relations obtained by several researchers using microscopic theories for polymers. Several of these theories explicitly predict multivaluedness in the relation between slip velocity and shear stress, i.e., there is a region in which three distinct values of slip velocity are possible for the same value of shear stress. Specific examples of such relations include those proposed by Leonov (1990); Yarin and Graham (1998); Mhetar and Archer (1997); Hill (1998). In particular, in the model of Yarin and Graham (1998), the limit point at which the jump from a small slip to a large slip regime occurs arises from an imbalance between the increasing force per adsorbed chain and the decrease in the concentration of adsorbed chains, as shear stress increases. Multivalued slip relations such as these have been used, for example, to model spurt and oscillations in capillary and Couette flows of molten polymers (Adewale and Leonov, 1997). Georgiou and Crochet (1994) have proposed a computationally convenient phenomenological slip equation which also shows a maximum and minimum in wall shear stress. Although not derived from first principles, this model has a closed form expression, and can be viewed as an approximate version of models based on molecular parameters which do not have closed form expressions. They showed that this slip relation, when taken along with finite compressibility of the polymer melt, can lead to self sustained oscillations of the pressure drop and mass flow rate at the exit of the die for Newtonian flow in a slit. However, they do not include a pressure dependence in their slip relation. Several slip models relating the slip velocity to the wall shear stress and pressure (or total compressive normal stress on the wall) have been proposed in the literature. In particular, Hill et al. (1990); Denn (1992); Person and Denn (1997) propose slip relations

49 29 which show a power law dependence on the wall shear stress and an exponential one on pressure. Hatzikiriakos and Dealy (1992b) have proposed a similar slip model which shows saturation at high pressures. They used this model for the low flow rate branch and a simple power law dependence on the shear stress for the upper branch of the flow curve to model flow oscillations in a capillary rheometer (Hatzikiriakos and Dealy, 1992a). Thus, their model is in effect a discontinuous multivalued slip relation. The slip model of Stewart (1993) brings in the pressure dependence through changes in the density. All these models predict a decrease in slip velocity with increase in pressure. There is also experimental evidence from White et al. (1991) that this is indeed the case. In this work, we study the combined effects of adding pressure dependent and multivalued slip on flow curve multiplicity in capillary extrusion. We do this by modifying the approximate slip relation proposed by Georgiou and Crochet and adding a pressure dependence. Consistent with observations, the pressure dependence is such that slip velocity decreases with increasing pressure. As with the model proposed by Georgiou and Crochet, our relation is not derived from first principles, but contains the same qualitative features found in more complex models derived from molecular considerations. We apply this equation to the steady flow of incompressible Newtonian and shear thinning fluids through a cylindrical die at a constant pressure drop. As also done by Person and Denn (1997), we simulate the entire axial profile of pressure, stress and slip velocity. The flow curve obtained shows multiplicity, with the critical shear stress τ c2 showing a small decrease at low L/D ratios and a more pronounced one at higher L/D ratios. Most importantly, we see that at sufficiently high L/D, the flow curve is no longer multivalued, despite the fact that the slip relation is multivalued, i.e., it predicts a maximum and minimum in shear stress at all pressures between the entry and exit pressures in the die. Thus

50 30 we obtain the result that multiplicity in the slip relation does not guarantee multiplicity in the flow curve. 2.2 Mathematical Model Slip Model In terms of dimensionless quantities, our slip model may be written as ) A 2 τ w = u sp (u sp +, (30) 1+A 3 u 2 sp where u sp = u s (1 + exp(βp)). (31) Here, τ w is the shear stress exerted by the fluid on the wall and u s is the slip velocity. The scaling factors used are u = G/a 1, P = G and τ w = G for the velocity, pressure and shear stress respectively, a 1, A 2, A 3 and β are parameters of the slip model and G is the shear modulus of the fluid. For small u sp, this model reduces to the Navier slip condition, τ w = A 2 u sp. If we arbitrarily choose A 3 = , we can calculate that the slip model loses multiplicity for A 2 < Figure 13 shows a plot of the slip relation for various values of P for A 2 =3, A 3 = , and β = Unless otherwise mentioned, we will work with A 2 = 3 and A 3 = in subsequent calculations. Note that the pressure dependence is such that the slip velocity decreases with increase in pressure, as shown in figure 13, but the multiplicity remains. Since the slip model is multivalued by construction, the natural expectation is that the flow curve will be as well. We shall see that this expectation is not always fulfiled.

51 P=50 P=10 τ w Figure 13: Plot of equation 30 for different values of pressure (A 2 =4, A 3 = , β =0.0102). u s Constant Pressure Formulation We now develop the governing equations for the constant pressure case using Newtonian, shear thinning and UCM constitutive equations. We have already described the UCM equation in the introduction as a constitutive equation for obtaining the polymer component of the stress tensor in dilute solutions. It can also be derived using network models for polymer melts (Bird et al., 1987b). In this case, Q can be thought of as the length of a polymer segment between two junctions in the network, with the segment being modeled as a Hookean spring. As shown by Lupton and Regester (1965), fluid compressibility does not play an important part in the constant pressure case, and we can simplify our analysis slightly if we assume that the fluid is incompressible. However, for the constant piston speed case, fluid compressibility plays a crucial role in providing a mode for storing energy, which is essential for the generation of relaxation oscillations.

52 32 We first consider the pressure driven flow of an incompressible Newtonian fluid through a cylindrical die of radius R (diameter D) and length L. The pressure is defined so that it is zero at the exit of the die. Using the lubrication approximation (Pearson and Petrie, 1968), the dimensionless volumetric flow rate is given by Q = u s + τ w H, (32) Here, we assume that the diameter of the die is kept constant and choose to scale the volumetric flow rate using Q = πr 2 u. In addition, we scale the lengths by L and retain the scaling factors for pressure, slip velocity and shear stress discussed above. The dimensionless number H =8ηu /GR, where η is the fluid viscosity. Note that since γ A =4Q/πR 3, QH measures the dimensionless apparent shear rate. The use of the incompressibility condition dq/dz = 0 gives us the following equations for the slip velocity and pressure: du s dz dp dz = dp dz τ w P H + τw u s, (33) = 4Λτ w, (34) where Λ = L/D. Equation 33 is obtained by using equation 32 and differentiating equation 30 with respect to z. This set of equations is to be solved using the boundary conditions P (0) = P, (35) P (1) = 0. (36) Since it will be convenient to calculate the flowrate simultaneously with the other

53 33 quantities, we add the redundant equation and boundary condition dq = 0 (37) dz Q(0) = u s (0) + τ w(0) H. (38) For a power law fluid, the constitutive relation between the shear stress exerted by the fluid on the wall and the radial velocity gradient is given by Bird et al. (1987a) as τ rz = K v z n 1 v z r r. (39) Using the lubrication approximation once again, the dimensionless volumetric flow rate is given by Q = u s + nτ 1/n w H, (40) where τ w is obtained from equation 30. We define the dimensionless number H in a manner similar to the Newtonian case as H = n(3 + 1/n)K1/n a 1 G 1/n 1 R, (41) which results in the following set of equations: du s dz dp dz dq dz = τ w P τ w u s + H τ w (1/n 1) dp dz, (42) = 4Λτ w, (43) = 0. (44) These equations have to be solved together with the boundary conditions: P (0) = P, (45) P (1) = 0, (46) Q(0) = u s (0) + nτ 1/n w (0) H. (47)

54 34 The governing equations for the UCM constitutive equation are similar to those for the Newtonian model. The inclusion of a normal stress only changes the boundary conditions at the entrance and exit. Neglecting the elastic normal stresses in the barrel, and entrance and exit effects, these are given by: P (0) + τ zw (0) = P b, (48) P (1) + τ zw (1) = 0. (49) Here, τ zw is the normal stress τ zz evaluated at the wall. At steady state, τ zw = 2 τw. 2 We can examine the combined effect of viscoelasticity and shear thinning by using the Phan-Thien-Tanner (PTT) equation, which can also be derived from network theory (Bird et al., 1987b). The general form of the PTT equation is given by Bird et al. (1987a) as Z(tr τ )τ + λ τ (1) + ξ λ{ γ τ + τ γ} = η γ. (50) 2 We consider the case of affine motion (ξ =0) and use the linear form of the function Z, i.e, Z =1 ɛλtr τ /η. In this case, the governing equations can be written as du s dz dp dz dq dz = dp dz τ w (1 + 4ɛ/H τ 2 P w ) H + τw u s (1 + 4ɛ/H τw), (51) 2 = 0, (52) = 0. (53)

55 35 The boundary conditions are given by Q(0) u s (0) = τ w(0) H + 4ɛ 3H τ w 3 (0), (54) P (0) + τ zw (0) = P b, (55) P (1) + τ zw (1) = 0. (56) Our simulations show that the addition of viscoelasticity in the form of either the UCM or the PTT equation does not change the qualitative behavior of the model Constant Piston Speed Formulation We now consider the flow of a Newtonian fluid through a capillary die as shown in figure 14. The fluid is driven by a piston moving at constant speed. We assume that the melt has a constant compressibility χ, and use a linear relation for the density, i.e., ρ = ρ 0 (1 + χp ), (57) where ρ 0 is the density at P =0. We first write the governing equations for the Newtonian case. A mass balance on the barrel gives C dp b dt = 1 κ (Q p Q 0 ), (58) where P b is the dimensionless pressure in the barrel (scaled by G), t is the dimensionless time (scaled by the residence time t = Q p /πr 2 L), C = V b /πr 2 u t and κ = Gχ (cf. Lupton and Regester (1965) and Molenaar and Koopmans (1994)). The volumetric rate of displacement of the piston, Q p is constant during the simulation and Q 0 is the volumetric flow rate at the exit of the barrel. We assume that C remains constant in the

56 36 plunger L V b diameter = D die barrel dv b / dt =Q p Figure 14: Schematic diagram of the constant piston speed experiment. time scale of the experiment. A mass balance on the capillary die gives ( P t = λ 1 Q P z + 1 ) Q, (59) κ z with λ 1 = t u /L. The boundary conditions are Q = Q 0, and P = P b at the entrance of the die. Finally, integrating the momentum balance over r gives Re Q ( ) 1 t = λ P 2 4Λ z + τ w, (60) where λ 2 = t G/η, Re = ρ 0 u D/4η is the Reynolds number and η is the fluid viscosity. The momentum equation in this form is also valid for viscoelastic flow if τzz z Numerical results for viscoelastic flow show that this condition is satisfied. P z. We now need an equation relating the volumetric flow rate to the slip velocity. If the compressibility is small, we can assume that equation 32 still holds. Hence, the governing equations for the Newtonian case are equations 58 to 60 together with equation 32 and the slip relation, equation 30. The same set of equations hold for the shear thinning case, with the exception that equation 32 is replaced by equation 40.

57 37 As mentioned before, viscoelasticity does not significantly affect the steady state equations. However, it is possible that using an evolution equation for the normal and shear stresses could yield time-dependent results that are qualitatively different from those for the Newtonian and shear thinning cases. Thus, we felt that it was necessary to examine the time-dependent case using a simple viscoelastic model. This section discusses the formulation of the governing equations using the UCM model for viscoelasticity (Bird et al. (1987a)). Assuming that the Reynolds number is small, the UCM model gives the following evolution equation for τ w : De 2ΛDe τ w t = τ w De 8(Q u s). (61) Evaluating the evolution equation for τ zz at the wall gives De τ zw 2Λ t + De τ w Λ τ w t = 2τ 2 w τ zw. (62) Here, τ zw represents the value of τ zz at the wall of the capillary and De = λu /R. The Deborah number is given by De = λ v /R, where v = Q p /πr 2. The slip velocity and shear and normal stresses are related by the slip model. The boundary conditions are: P (0) + τ zw (0) = P b, (63) P (1) + τ zw (1) = 0. (64) In the first boundary condition, we assume that the elastic normal stresses in the barrel are negligible. These boundary conditions neglect entrance and exit effects.

58 Results and Discussion Constant Pressure Case We wish to determine the parameter regimes in which flow curve multiplicity occurs. A natural way to do this is through bifurcation analysis. At fixed Λ, wefind the turning points of the flow curve (i.e. the points where the curve turns back on itself). These determine the boundaries τ c2 and τ c3 of the multiplicity region at that Λ (in the model we consider, slip occurs at all shear stress levels and hence τ c1 does not exist). Then we track the motion of these points as Λ varies, thus outlining the region in ( P, Λ) space where flow curve multiplicity occurs. The AUTO software package (Doedel, 1981; Taylor and Kevrekidis, 1990) automatically performs this type of analysis for boundary value problems like the one we consider here. AUTO uses a domain decomposition collocation method for spatial discretization and a pseudo-arclength continuation scheme to trace out steady state solution curves in one parameter or curves of bifurcation points in two parameters. Figure 15 shows the flow curve computed by AUTO at Λ=10, A 2 =3, A 3 = , β = and H =1. We see that this curve is multivalued, as expected because the slip model is multivalued. The turning points occur at P = and P = , corresponding to critical shear stresses of τ c2 = and τ c3 = The results of the turning point continuation are shown in figure 16a, where the upper curve corresponds to τ c2, and the lower one to τ c3. Note that our slip law depends on pressure, so the slip velocity, and hence τ w changes through the length of the die. The values of τ c2 and τ c3 that we report here are computed by dividing the pressure drop P by 4Λ. This is what the (constant) wall shear stress would be in the absence of slip,

59 X 0.28 Q X Figure 15: Flow curve for a Newtonian Fluid (Λ =10, A 2 =3, A 3 = , β = , H =1). The profiles at the points marked `X' are shown in figures 18 and 19. P

60 40 under the same pressure drop. As the figure shows, the value of τ c2 shows a decrease and the two critical shear stresses approach one another as the length is increased, and this effect is more pronounced at higher Λ values. Finally, at Λ = , we see that the the two turning points come together in a cusp, indicating that the flow curve beyond this Λ value has no multiplicity. The pressure drop at this value is and the slip relation still predicts a multiple valued curve for all pressures in this range. Thus, the differential effect of pressure in the capillary has resulted in the absence of multivaluedness in the flow curve, although the slip relation itself is multivalued for the entire pressure range experienced in the die. We also note that at the cusp, the critical stresses have a value of , which corresponds to a change of only 1.24% from the value of τ c2 at Λ=10. Finally, for completeness, we show a plot of the flow curve for Λ = in figure 17, where there is no multiplicity, just as shown on figure 16a. Results for the shear thinning case with n =0.56 and the other parameter values as for the Newtonian model are similar. The results of the turning point continuation are shown in figure 16b. The behavior is similar to the Newtonian case where there is a more pronounced drop in τ c2 at higher Λ values. Also, the flow curve loses multiplicity at Λ = and a pressure drop of , which corresponds to τ c2 = τ c3 = As with the Newtonian case discussed in the preceding paragraph, the slip relation itself remains multivalued for all pressures experienced in the die, and the loss of multiplicity in the flow curve is a result of the differential effect of pressure in the capillary. As mentioned earlier, we find that including the effect of viscoelasticity does not affect the qualitative behavior of the model, in particular, the loss of multiplicity at high Λ. The result of a turning point continuation using the UCM constitutive equation is

61 41 (a) τ c One steady state Three steady states One steady state (b) τ c One steady state Three steady states One steady state One steady state (c) τ c Three steady states (d) τ c One steady state One steady state Three steady states One steady state Λ Figure 16: Behavior of critical stresses τ c2 and τ c3 with increasing Λ (A 2 =3, A 3 = ): (a) Newtonian model (β =0.0102, H =1) (b) shear thinning model (β = , H = 1, n = 0.56) (c) UCM model (β = , H = 1) (d) PTT model (ɛ =10 1, β =0.0102, H =1). The upper curve corresponds to τ c = τ c2 and the lower one to τ c = τ c3. The curves are not extended to Λ=0because the approximations used are not valid for small Λ.

62 Q Figure 17: Flow curve for a Newtonian fluid showing no multiplicity (Λ = , A 2 =3, A 3 = , β =0.0102, H =1). P shown in figure 16(c). A similar result for the PTT model, with ɛ =0.1is shown in figure 16(d). In both cases, the same parameters were used as for the Newtonian model. Figures 18 and 19 show the spatial profiles of the pressure, shear stress and slip velocity at points on the low and high flow rate branches of the flow curve for the Newtonian fluid shown in figure 15. The profiles at a point on the low flow rate branch are shown in figure 18 and those at a point on the high flow rate branch are shown in figure 19. In both cases, the pressure profiles are nearly linear and the shear stress decreases as we move towards the exit of the die. The degree of variation in the magnitude of the shear stress is much smaller than that of the pressure and correspondingly, the slip velocity increases towards the exit where the pressure is lowest. This behavior is expected intuitively and also seen by Hatzikiriakos and Dealy (1992b) in their simulations at constant piston speed.

63 43 However, it differs from the results of Person and Denn (1997) where slip velocity has the smallest magnitude at the exit, which is the region of lowest shear stress. They interpret this result as arising from the strong power law dependence of slip velocity on shear stress. Finally, the small variation in τ w with z provides an a posteriori validation of our neglect of axial gradients. 6 P u s τ w z Figure 18: Profiles of pressure, slip velocity and shear stress on the low flow rate branch for the Newtonian fluid of figure 15. ( P =5.22, Λ=10, A 2 =3, A 3 = , β =0.0102, H =1).

64 44 6 P u s τ w z Figure 19: Profiles of pressure, slip velocity and shear stress on the high flow rate branch for the Newtonian fluid of figure 15 ( P =6.39, Λ = 10, A 2 = 3, A 3 = , β =0.0102, H =1).

65 45 It is of interest to observe the location of the profiles of the two turning points τ c2 and τ c3 on the three-dimensional surface defined by the slip model (equation 30). The pressure profile is almost linear for all cases, and hence, the plots may also be viewed as the profiles superimposed on the slip curve in (z, u s,τ w ) space, with the higher pressures corresponding to lower z values, i.e., points close to the entrance of the die. Such a plot is shown for the shear thinning case with Λ=10in figure 20(a). At this value of Λ, the pressure drop is relatively small, and the profiles at the two turning points lie very close to the location of the maxima and minima of equation 30 at all pressures in the die. The location of the profile at the cusp is shown in figure 20(b). Here, the high pressure points (which lie near the entrance of the die) are located to the right of the minimum at the corresponding pressure. As we move down the die to regions of lower pressures, the points tend to move toward the right, i.e., closer to the maxima, till at the exit, the points are located to the left of the maximum. Finally, figure 20(c) shows the profile of a point in the central region (i.e., the region where there is a sharp increase in the slope of the flow curve, although there is no multiplicity) Constant Piston Speed case For the constant piston speed case, we performed a spatial discretization of the governing equations using a Chebyshev collocation scheme (Canuto et al., 1988) with N =30collocation points for the pressure, volumetric flow rate and slip velocity. The resulting system of differential-algebraic equations was solved using the integrator DDASAC (Caracotsios and Stewart, 1985), for imposed values of Q p and zero initial conditions. For the base case, we chose values of χ =10 9 m 2 /N and G =10 5 N/m 2 and Re = In all

66 46 (a) Τ w P u s (b) Τ w P u s 0.1 (c) 0.3 Τ 0.2 w P u s 0.1 Figure 20: Flow profiles on the three dimensional slip surface for a shear thinning fluid: (a) upper and lower limit points ( P =6.39, Λ=10) (b) cusp point ( P = , Λ = ) (c) point in the central portion without multiplicity ( P = 140.0, Λ= ). Other parameters are A 2 =3, A 3 = , β =0.0102, H =1, and n =0.56 for all three cases.

67 47 runs, C was fixed at 10 6 and the parameters λ 1 and λ 2 were fixed at 0.01 and 0.1 respectively. All other parameters were chosen to have the same values as in the constant pressure case. Results for the Newtonian and shear thinning cases are similar and we only show them for the Newtonian case. As expected because of the multiplicity in the constant pressure flow curve, attempting to operate at a flow rate corresponding to the branch between τ c2 and τ c3 of the flow curve (which we call the unstable region) results in oscillations in the pressure drop P b and the exit flow rate Q e. This is shown in figure 21, for a value of Q p = 0.21 where the steady state solution lies in the unstable region of the flow curve. Operating on the low or high flow rate branches results in stable flow as shown in figure 22, for Q p =0.15, where the steady state solution lies on the low flow rate branch of the flow curve. These results are similar to those obtained by Georgiou and Crochet (1994) and Adewale and Leonov (1997), and are to be expected when multivalued slip models are used. As long as Re 1, the Reynolds number has virtually no effect on the frequency of the oscillations. This can be understood by noting that, for Re 1, these are basically classical relaxation oscillations, as occur in problems with widely separated time scales (Nayfeh and Mook, 1979). Here, the time scales correspond to the compressibility (κc) and inertia (Re). Hence, Re must be nonzero for oscillations to be observed, but is otherwise not important as long as it is small. Decreasing the compressibility has the effect of increasing the frequency of the oscillations. This can be understood on the same lines as the Re dependence by noting that the time spent on the high and low flow rate branches of the flow curve is directly related to the compressibility. Fluids with lower

68 Pb 5.7 (a) Qe time (b) 0.24 Q e Figure 21: Oscillatory flow for a Newtonian fluid with Q p =0.21:(a) Pressure drop and exit flow rate vs. time (b) Plot of Q e vs. P b superimposed on the steady state flow curve. Pb

69 P b Q e Figure 22: Non-oscillatory flow for a Newtonian fluid with Q p =0.15: Pressure drop and exit flow rate vs. time. time

70 50 values of χ would spend less time on these branches and hence exhibit oscillations of a higher frequency. Figure 23 shows the results for a simulation at Λ = 250, where the steady state flow curve shows no multiplicity. Recall, however, that the slip model itself remains multivalued, and as in the constant pressure case, the loss of multiplicity is a result of the differential effect of pressure along the capillary. The value of Q p was chosen to be , which lies in the steepest region of the flow curve. As expected, no oscillations are found. We conclude with a brief discussion of the results obtained using the UCM constitutive equation, with the same slip model parameters as for the Newtonian case. In this case, high frequency shock waves are seen at very small values of the Reynolds number and large values of κ. To avoid these shock waves, we present results for a parameter set different from that used for simulations with the Newtonian model. Figure 24 shows the result of a simulation with Re =10 4. As before, we find that Reynolds number does not affect the frequency of the oscillations provided that it is sufficiently small. We also find that, as for the Newtonian case, decreasing the compressibility increases the frequency of the oscillations. 2.4 Conclusions In this chapter, we have studied the behavior of extrusion when a multivalued, pressure dependent model for wall slip is used. The main features of spurt flow and multiplicity can be modeled using this relation. The steady state flow curves using Newtonian, shear thinning and viscoelastic models predict a critical shear stress τ c2 for the onset of spurt

71 P b Q e Figure 23: Non-oscillatory flow of a Newtonian fluid with Λ = 250, Q p = and other parameters as in figure 21: Pressure drop and exit flow rate vs. time. time

72 Pb Q e time Q e P b Figure 24: Oscillatory flow of a UCM fluid (C =10 6, Λ=10, Re =10 4, κ =10 5, De = De =10, λ 1 = λ 2 =0.01, Q p =0.081): (a) Barrel pressure and exit flow rate vs. time (b) Plot of Q e vs. P b superimposed on the steady state flow curve.

73 53 flow that decreases with increase in Λ. In all cases, we see a loss in multiplicity at high Λ, although the slip relation itself gives a multivalued slip velocity for a given shear stress at all pressures in the range involved. Time-dependent simulations taking fluid compressibility into account show that attempting to operate on the decreasing branch of the flow curve gives rise to oscillations in the exit flow rate and pressure drop, which is in accord with the results of previous researchers using multivalued slip models. The frequency of these oscillations is virtually independent of the Reynolds number, provided that it is small enough. Decreasing compressibility increases the frequency of the oscillations. No oscillations are observed if the flow curve does not show multiplicity. Although there is little experimental data in the literature for experiments conducted at high Λ values at constant pressure, the behavior we observe is similar to that reported in Kalika and Denn (1987) for a constant piston speed experiment. The main conclusion of this work is that multiplicity in the slip model does not guarantee multiplicity in the flow curve.

74 54 Chapter 3 The filament stretching instability In this chapter, we discuss the instability that occurs during the elongation of a polymer filament. Here, an initially axisymmetric filament undergoes a buckling instability producing a non-axisymmetric shape. We use the idea of stress localization to model the process as the stretching of an elastic membrane enclosing a passive, but incompressible fluid. We show that this simple model exhibits an instability that is similar to that seen in the stretching of a polymer filament. 3.1 Introduction Flows with significant elongational components are common in industrial applications such as fiber spinning, as well as in rheometry, during the measurement of elongational viscosity. The most common types of elongational flow instabilities are draw resonance, necking, capillary breakup, and elastic filament breakup, Detailed reviews of which may be found in Petrie and Denn (1976) and more recently by Larson (1992), so we content Most of the material in this chapter has been published in Kumar and Graham (2000)

75 55 ourselves with a brief overview of these instabilities. We then summarize the observations in the literature that are related to elastic filament breakup, and postulate a new mechanism by which this instability could occur. Fiber spinning is defined (Petrie and Denn, 1976) as a process whereby a filament or sheet is extruded from a die and drawn down in cross sectional area by being taken up at a velocity greater than the extrusion velocity. The draw ratio D r,isdefined as the ratio of the take-up velocity to the extrusion velocity. This also equals the ratio of the initial to drawn fiber area. Draw resonance is described by Christensen (1962) as a periodic variation in fiber diameter, which occurs at constant take-up speed, when small variations in the diameter of fiber at the take-up spool produce oscillations in the fiber tension. The analysis of Pearson and Matovich (1969) for a Newtonian fluid with negligible inertia, gravity, and surface tension showed that the steady state solution is unstable to timeperiodic disturbances when D r exceeds A later analysis by Pearson and Shah (1972) showed that inertia acts to stabilize the flow, while surface tension is destabilizing. The corresponding stability analysis for viscoelastic fluids has been performed by Zeichner (1973), Fisher and Denn (1975, 1976), and Chang and Denn (1980). An important parameter in this case is the Deborah number, defined as De = λv(0)/l, where λ is the relaxation time of the fluid, v(0) is the extrusion velocity, and L is the length of the fiber. The analysis of Fisher and Denn (1976) for a UCM fluid shows that elasticity has a stabilizing influence. The analysis also shows that there is no steady solution for certain values of De, an effect related to the unbounded growth of normal stress predicted by the UCM equation for elongational flow, at certain Deborah numbers. Nonlinear analyses carried out for Newtonian fluids (Kase, 1974; Fisher and Denn, 1975) and for power law

76 56 and UCM fluids (Fisher and Denn, 1976) show that finite amplitude effects are stabilizing. Finally, fluids that thin in elongation are found to be less stable to draw resonance than Newtonian fluids, while elongation thickening fluids are more stable (Pearson and Shah, 1974; Fisher and Denn, 1976). When the fiber is extruded at a constant tension rather than a constant take-up speed, Fisher and Denn (1976) found that the flow is stable to draw resonance. However, small local indentations can grow as they are convected down the length of the fiber. This phenomenon is known as necking. Necking occurs because the constant force applied along the fiber length translates to a greater elongational stress in the neck as compared to adjoining regions in the fiber. Thus, the neck thins faster than the neighboring regions. It follows that elongation thinning, which speeds up the reduction in area, is destabilizing, while elongation thickening leads to increased stability. They also showed that if the residence time of the neck in the fiber is long enough, necking can lead to fiber breakage. The phenomenon of capillary breakup was first analyzed by Rayleigh (1879). He showed that a jet of Newtonian liquid with nonzero surface tension is unstable to axisymmetric disturbances whose wavelength is greater than the circumference of the jet. For fiber spinning, the capillary breakup instability augments necking, but the additional effect can be shown to be negligible (Larson, 1992). For UCM fluids, the analysis of Goldin et al. (1969) showed that in the linear limit, disturbances have higher growth rates than for Newtonian fluids. However, nonlinear effects quickly become significant, and the lubrication and finite element analysis of Bousfield et al. (1986) which takes these into account shows that the growth rates slow down significantly, so that the net effect is a stabilization compared to Newtonian fluids. Once again, it is the elongation thickening property of the UCM fluid that leads to increased stability, by resisting the thinning and

77 57 breakup of threads. While the capillary breakup instability is not of significance in fiber spinning, it has been suggested (Larson, 1992) that the measurement of growth rates of the disturbances may be used to estimate the surface tension of viscoelastic fluids. The next instability that we discuss is the breakup of a viscoelastic liquid bridge in elongation. Unlike the instabilities we have discussed so far, this instability is purely elastic, in the sense that it is not seen in corresponding experiments with Newtonian fluids with similar viscosity. The elongation of a liquid bridge confined between two endplates is used in rheometry to measure the elongational viscosity of a fluid (Tirtaatmadja and Sridhar, 1993; McKinley et al., 1996). A schematic of the setup is shown in figure 25. In ideal uniaxial elongational flow, a liquid column of initial length L 0 is extended at an exponential rate so that its length at time t is given by L t = L 0 exp( ɛ 0 t), where ɛ 0 is the strain rate. The radius of the filament remains uniform throughout its length. The elongational viscosity is then defined as η (τ zz τ rr ) / ɛ 0, (65) where τ zz and τ rr are respectively the axial and radial normal stresses. This stress difference, and hence η, may be calculated from the force required to produce the elongation. A measure of the displacement is given by the Hencky strain ɛ, given by log e (L t /L 0 ). The Deborah number is given by De = λ ɛ 0, where λ is the relaxation time of the fluid. A filament stretching rheometer, such as the one shown in figure 25 attempts to reproduce uniaxial elongation by separating the endplates at an exponential rate. The actual kinematics differ from this due to the presence of the two endplates. While conducting experiments with polyisobutylene based Boger fluids in a filament stretching rheometer, McKinley et al. (1996) found that the liquid bridge partially decohered from one of the endplates at high strain rates (corresponding to De > 1). The instability began as a

78 58 L 0 polymer solution L t Figure 25: Schematic of a filament stretching rheometer. The setup on the left shows the undeformed state of the liquid bridge. non-axisymmetric deformation near the endplates at a certain critical strain, with a well defined azimuthal wavenumber of 4. Upon further elongation, the column broke into fibrils, which themselves bifurcated into smaller fibrils at larger elongations. The critical strain for onset of the instability was found to decrease with increasing Deborah number. The dependence on the initial aspect ratio (defined as L 0 /R 0, where R 0 is the initial filament radius) was weaker, with the critical strain increasing slightly at larger aspect ratios. They report that Newtonian liquid columns of similar viscosity do not display the instability, and in fact break up due to the capillary instability at comparable strains. McKinley et al. (1996) propose that the mechanism is similar to the one seen in the Saffman-Taylor instability (Saffman and Taylor, 1958), which occurs when a less viscous fluid attempts to displace a more viscous one. In this mechanism, a small perturbation at the interface between the two fluids causes the less viscous fluid to find itself in a region of lower pressure, hence amplifying the perturbation. The Boger fluids used in

79 59 the experiment of McKinley et al. (1996) are strongly strain hardening. This means that, at large strains, the central portions of the liquid bridge which are highly stretched resist further decrease in diameter. This in turn means that a large portion of the filament is being extended as a cylinder of almost constant radius, with the additional volume of fluid being supplied by the rapidly depleting pool of liquid near the two endplates. At large strains, the thickness of this pool is small compared to its radius, so that the flow of liquid in this region is radially inward. This indicates that the pressure at the center of the pool must be lower than at the edge. If the free surface near this region is perturbed, the surrounding medium (in this case, air at atmospheric pressure) finds itself in a region of lower pressure, which causes the interface to deform further. This instability is relatively new, so not much work has been done in terms of modeling it. Spiegelberg and McKinley (1998) performed a numerical simulation of the stretching of a filament of Oldroyd-B fluid using a commercial software program based on the finite element method. They reported that the initial flow inhomogeneity caused the formation of stress boundary layers near the free surface of the filament at large strains. For Hencky strains ɛ>4, they found that it became increasingly difficult to model the rapidly draining region near the endplates. Significantly, the elastic instability sets in close to ɛ =4for large Deborah numbers. Rasmussen and Hassager (1999) performed an analysis of the stretching of a UCM filament. They considered two cases: the first was was the elongation of a purely cylindrical filament, and the second was the elongation a cylindrical filament with a small uniform non-axisymmetric perturbation superimposed on it. For the non-axisymmetric initial state, they found that while the central region and the region near the endplates remained almost circular, the deviation from axisymmetry in the region just above the

80 60 endplates grew as the filament was extended. They also found that stress boundary layers were formed near the free surface for the axisymmetric initial condition. For nonaxisymmetric initial conditions, the stress boundary layer was relieved in regions which bulged out as compared to the axisymmetric state, while it was more pronounced in the regions that had retracted. Care must be taken when interpreting their results, however, since the initial condition was non-axisymmetric, in contrast to the axisymmetric base state in the experiments. Also, numerical difficulties restricted their analysis to Hencky strains less than 2.5 for Deborah numbers larger than 1, so their simulations could not capture the behavior of the filament at large elongations and high Deborah numbers seen in the experiments by Spiegelberg and McKinley (1998). While the mechanism suggested by McKinley et al. (1996) is plausible, the importance of fluid elasticity, as evidenced by the fact that the instability is not observed at small Deborah numbers or in Newtonian fluids, indicates that a purely elastic mechanism may be at work. The formation of stress boundary layers near the free surface as seen in the simulations of Spiegelberg and McKinley (1998) and Rasmussen and Hassager (1999) suggests that the physics of the instability may be captured by examining a small region near the interface. Since the flow in the region near the endplates is radially inward, compressive hoop stresses develop in the fluid (Kumar and Graham, 2000). Further, at high Deborah numbers, the Oldroyd-B fluid behaves like an elastic solid. Hence, we expect the region near the free surface to behave like a membrane subjected to a compressive hoop stress. We conduct a preliminary investigation of our proposed mechanism by modeling the region of the liquid bridge near the endplates as an elastic membrane enclosing a passive incompressible fluid. The membrane is then stretched with the radii at the two ends kept

81 61 fixed. The incompressible fluid then exerts a force of constraint on the membrane, which acts to keep the enclosed volume fixed. The constraint forces cause compressive hoop stresses to develop in the membrane, and thus we expect the initial axisymmetric shape to bifurcate into non-axisymmetric modes at sufficiently large strains. In the subsequent section, we demonstrate the build up of stress near the surface by modeling an ideal planar elongation experiment. This serves to justify the use of our membrane analogy. We next present the formulation of the membrane model for a reference configuration obtained by chopping away the region near the apex of a cone. This shape approximates the geometry near near the endplates. We conclude the chapter with a discussion of the important results and conclusions of our analysis. 3.2 Planar elongation: a model problem In this section, we examine a model problem that captures the main features of the flow field near the central region in a filament stretching experiment. Figure 26 shows a portion of the central region in the planar elongation of a UCM fluid. The flow field is Hamiltonian, with the stream function ψ = ɛxy. Thus, the velocity components are given by v x = ψ y = ɛx v y = ψ = ɛy. (66) x The origin is a stagnation point, which means that polymer molecules which are carried by the flow near the origin will experience large elongations. Thus, on physical grounds, we would expect high stresses to develop in the neighborhood of the origin.

82 62 y=0 y x x=0 y=-l Figure 26: Schematic of a planar elongation setup and flow field. We can get a quantitative picture of the stress buildup by following the motion of a fluid element initially located at (x 0, l) (which we assume is a point of zero stress) as it moves along a streamline. The evolution of the stress tensor τ is described by the UCM equation as τ + λ ( D τ Dt ) {τ v} {τ v}t = η 0 γ, (67) where v is the velocity vector, γ =( v +( v) t ), λ is the relaxation time, η is the viscosity and D/Dt denotes the material derivative / t+v. We make these equations dimensionless by scaling the length with l, velocity with l ɛ, time with ɛ, and stress with the shear modulus η/λ. With these scalings, equation 67 is written as ( ) 1 D ˆτ We ˆτ + {ˆτ ˆv} {ˆτ ˆv}t = D ˆt ˆ γ, (68) with the Weissenberg number We is given by λ ɛ. Here, the hats above the variables are used to denote the scaled versions of the respective quantities, and will be dropped in the rest of this section for convenience.

83 63 For the flow described above, the UCM equation gives dτ xx dt ( 2 1 ) 2=0, (69) We for the evolution of τ xx on a particular streamline. The fluid element is chosen so that it is initially located at (x 0, 1), where x 0 (0, 1), with the stress τ xx (t =0)=0. The fluid element is carried along with the flow according to dx dt dy dt = x, (70) = y. (71) Equations (69) to (71) may be readily solved to yield x(t) = x 0 exp(t), (72) y(t) = exp( t), (73) ( ( 2 We τ xx (t) = 1+exp 2 1 ) ) t. (74) 2 We 1 We Equation 73 may be used to eliminate t from equation 74 to yield ( ( 1 τ xx (y) = 2 We ) 2 1/We 1). (75) 2 We 1 y Equations 74 and 75 yield the well known result that the axial stress in planar elongation of a UCM fluid blows up at We =0.5. It is also evident from equation 75 that τ xx is independent of x for the initial condition of zero stress at y = 1. In the limit We, equation 75 gives τ xx =1/y 2 1, which indicates that at high Weissenberg numbers, a stress boundary layer exists near the free surface y =0. This serves as justification for using the membrane approximation for large We, where we assume that the physics of the instability can be captured by modeling a small region near the interface.

84 64 At first sight, it seems somewhat surprising that the limiting value of τ xx, or equivalently, the thickness of the boundary layer, is independent of We. This is resolved by observing that, in general, the Weissenberg number governs the length l, at which the normal stresses are close to zero. This in turn means that We would affect the length scale for y, and hence the limiting value of τ xx. 3.3 Elongation of a truncated cone In this section, we apply the theory of membrane elasticity to determine the evolution of the shape of a truncated cone under axial elongation. This geometry is chosen because it approximates the reservoir region near the endplates in a filament stretching experiment. The cone encloses an incompressible fluid, so that its volume remains constant under elongation. Thus, the liquid exerts a force on the membrane to keep the volume fixed, but otherwise plays no role in the formulation. This force exerted by the liquid is a force of constraint which does not contribute to the total energy of the system (Goldstein, 1980). Our goal is to demonstrate that compressive hoop stresses develop in this system, resulting in a symmetry breaking bifurcation to a non-axisymmetric shape Problem Formulation The mathematical formulation uses aspects of tensor analysis, differential geometry, and finite elasticity. An overview of finite elasticity, to the extent needed to formulate this problem, is presented in Appendix A. In keeping with the Einstein notation, we will use

85 65 superscripts to denote the components of vectors or tensors that transform contravariantly, and subscripts for those that transform covariantly. Scalars and constants are invariant under transformation, and subscripts used in conjunction with them merely serve to distinguish between different quantities. Consider the portion of the cone shown in figure 27. In its undeformed (rest) configuration, that the radius varies linearly from r 1 to r 2 over a length L. Thus, at any axial location x 3 (with x 3 <L), the radius of the cone is given by r(x 3 )=r 1 sx 3, (76) where s =(r 1 r 2 )/L. In the limit of s =0, this reduces to a cylindrical membrane. ) The volume enclosed by the membrane is given by V 0 = π (r 21 L r 1 sl 2 + s23 L3. The membrane is subjected to a deformation at constant enclosed volume. This can be achieved by extending either end or both ends, but it is convenient from our point of view to think of the left end (the end near the origin) as being held fixed, while the right end is extended by an amount l. We begin by defining the Gaussian surface coordinates for the reference configuration. Let v 1 be the distance along a circular cross section of the cone measured from the intersection of the cone with the x 2 =0plane. Thus, at any axial location, the azimuthal angle (measured anti-clockwise) is given by θ = v 1 /r(x 3 ). The second surface coordinate, v 2 is defined to be the distance of a point from the intersection of the cone with the x 3 =0plane, measured along its intersection with the x 1 x 3 plane (see figure 27). Thus, the x 3 coordinate of a point whose second surface coordinate is v 2 is given by x 3 = v 2 cos(φ), where tan(φ) =(r 1 r 2 )/L. Hence we have 0 v 1 < 2 πr(x 3 ), and 0 v 2 L sec(φ). The position vector in Cartesian coordinates of a point whose

86 66 x 1 L v 2 r 1 v 1 r2 x 3 x 2 Figure 27: Coordinate system for the truncated cone. surface coordinates are (v 1,v 2 ) is given by a = r(v 2 )cos(θ) e 1 + r(v 2 )sin(θ) e 2 + v 2 cos(φ) e 3. (77) We scale all lengths by L and write a = r (v 2 )cos(θ) e 1 + r(v 2 )sin(θ) e 2 + v 2 cos(φ) e 3. (78) In future, we will drop the asterisks for convenience. We now consider an axisymmetric deformation such that the position vector of a point whose surface coordinates in the undeformed configuration are (v 1,v 2 ) is given by A = λ 1 (v 2 ) r(v 2 )cos(θ) e 1 + λ 1 (v 2 ) r(v 2 )sin(θ) e 2 + λ 2 (v 2 ) e 3. (79)

87 67 The boundary conditions on the stretches are: λ 1 (0) = 1, λ 1 (sec(φ)) = 1, λ 2 (0) = 0, λ 2 (sec(φ)) = 1 + l. (80) Given the position vectors a and A, we can compute the covariant and contravariant components of the surface metric tensors in the reference and deformed states, as described in section A.1.1. Let us denote by a αβ the covariant components, and by a αβ the contravariant components of the surface metric tensor in the reference state. The corresponding components in the deformed state are given by A αβ and A αβ respectively. The strain invariant I 1 is then given by I 1 = a αβ A αβ + a/a. (81) Thus, the dimensionless strain energy stored in a neo-hookean form is given by the equation E = 1 2 π (r1 sz) 0 0 (I 1 3) dv 1 dz, (82) where z = v 2 cos(φ), and the neo-hookean coefficient C 1, and a factor of cos φ have been absorbed into E to make it dimensionless. Note that the integral is over the undeformed state of the membrane. The next step in the formulation is to compute the volume enclosed by the membrane. One way to do this is to assume that the cone is a solid body, so that the Cartesian coordinates (x 1,x 2,x 3 ) of a point enclosed by the membrane are given in terms of v 1 and

88 68 v 2 by x 1 = λ 1 (v 2 ) r cos(v 1 /r(v 2 )), x 2 = λ 1 (v 2 ) r sin(v 1 /r(v 2 )), x 3 = λ 2 (v 2 ), (83) where 0 <r<r 1 sv 2 cos(φ). The Cartesian volume element dv = dx 1 dx 2 dx 3 is related to the differential surface coordinates by dx 1 dx 2 dx 3 = J dr dv 1 dv 2, (84) where J is the determinant of the Jacobian matrix of the transformation. Since J is not a function of r, we can perform the integration over r. The volume enclosed by the deformed membrane is given by V l = 1 2π(r1 sz) 0 Hence, the dimensionless strain energy function is given by (r 1 sz) λ 2 1 λ 2,z dv 1 dz. (85) F = E +Λ(V l V 0 ), (86) where E is given by equation 82 and V by equation 85, and Λ is a Lagrange multiplier. For a given value of l, λ 1 and λ 2 take values so that the first variation of F is zero, i.e., λ 1 and λ 2 are solutions to the Euler-Lagrange equations corresponding to the variational principle δf =0(Greenberg, 1978) Method of solution and stability analysis The standard method of solving variational problems is by means of the Ritz method. Here, we expand λ 1 and λ 2 as polynomial series which satisfy the boundary conditions

89 69 and solve for the coefficients such that F is extremized. We choose the following functional form: N λ 1 = 1+(z 2 1) a i T i (η), i=0 N λ 2 = (1+l) z + b i T i (η), (87) where η =2z 1and T i (η) is the Chebyshev polynomial of i th order with argument η. Substitution of these expressions into equation 86 gives F in terms of a i, b i and Λ. The variational statement δf =0then corresponds to the requirement that the partial derivatives of F with respect to each of the a i, b i and Λ vanish. This gives us a set of 2 N +3nonlinear equations for a i, b i and Λ which we can solve by Newton iteration. Having obtained the axisymmetric solution, we now check for bifurcations to nonaxisymmetric states. To do this, we assume that the position vector in the deformed state can be written as A = (λ 1 (v 2 )+ˆλ 1 (v 1,v 2 )) r(v 2 )cos(θ + ˆλ 3 (v 1,v 2 )/r(v 2 )) e 1 + (λ 1 (v 2 )+ˆλ 1 (v 1,v 2 )) r(v 2 )sin(θ + ˆλ 3 (v 1,v 2 )/r(v 2 )) e 2 + (λ 2 (v 2 )+ˆλ 2 (v 1,v 2 )) e 3. (88) The perturbations ˆλ 1, ˆλ 2 and ˆλ 3 are set to be 0 at the two ends of the membrane. In addition, we impose a periodicity condition in the azimuthal direction. We can now perform an analysis similar to the one described in the previous section to determine the constrained strain energy function for non-axisymmetric deformations F a = E a +Λ a (V l,a V 0 ), (89) on the same lines as equation 86. Note that F a reduces to F if we take ˆλ 1, ˆλ 2 and ˆλ 3 to be zero (the axisymmetric shape). In the linear limit, the azimuthal modes decouple, so we i=0

90 70 can examine each mode separately. We therefore choose the following functional form for the perturbations: N ˆλ 1 = (z 2 1) â i T i (η) cos(nθ), i=0 ( N ˆλ 2 = (z 2 1) ˆbi T i (η) cos(nθ)+ i=0 ( N ˆλ 3 = (z 2 1) ê i T i (η) cos(nθ)+ N i=0 N i=0 i=0 ) ˆd i T i (η) sin(nθ), ) ˆf i T i (η) sin(nθ), (90) where θ = v 2 /(r 1 sz), and n is the azimuthal mode number. The form of ˆλ 1 is chosen so as to fix the phase of the perturbation. Substituting equations 87 and 90 into equation 89 gives us F a in terms of a i, b i, â i, ˆb i, ˆd i, ê i, ˆfi, and Λ. The requirement that the partial derivatives of F a with respect to these variables be zero gives us a set of 7 N +8equations in these variables. One solution to this set of equations corresponds to the axisymmetric solution calculated above, i.e., a i, b i and Λ corresponding to the axisymmetric solution, and â i, ˆb i, ˆd i, ê i, ˆf i all equal to zero. A bifurcation to a different solution occurs when one or more eigenvalues of the Hessian matrix of F a, evaluated at the axisymmetric solution, crosses the origin with non-zero slope. 3.4 Results and Discussion The axisymmetric shape is obtained using a Newton iteration. In general, 9 Chebyshev polynomials provide enough accuracy to capture the shape. Since the equations are highly nonlinear, we need a good initial guess, so we start at l =0where the exact solution is

91 71 given by a i =0, b i =0, and Λ=0, and then do a continuation in l, with the initial guess at each point computed from the tangent at the previous point. We use a Broyden update method (Broyden, 1965) to avoid having to recompute the Hessian between successive iterations of the Newton method at a given value of l. We use two different methods to check our results. First, for s =0, the truncated cone is a cylinder, and the evolution of the shape in the absence of the volume constraint was computed by Yang and Feng (1970). Our results for the corresponding case are in agreement with theirs. We also check that equation 179, which is in the nature of a momentum conservation equation for the membrane, is satisfied both in the base state, and also in linearized form by the bifurcating solution. We first consider the specific case of r 1 =0.5and s =0.45, and examine its stability with respect to a mode 3 disturbance. Figures 28(a) (c) show the development of the three dimensional structure of the cone as we increase l from 0 to the bifurcation point, l =0.5. It is clear from these figures that much of the deformation is concentrated in the central portion of the filament, and the right end of the filament stretches into a cylindrical portion of nearly constant radius. This is what we desire if we wish to match this region in some way to the constant thickness central region in a filament stretching experiment. Denoting the elongation l at which the bifurcation occurs by l c,wefind that the cone undergoes a bifurcation to the n =3mode at l c =0.50. The structure of the bifurcating solution is shown in figure 28(d). This is obtained by adding a small multiple of the eigenvector corresponding to the bifurcating eigenvalue and adding it to the base eigenvector. As mentioned in the introduction, the instability results because portions of the membrane are in compression. To verify this, we plot the amplitude of the perturbation radial stretch ˆλ 1 and the hoop stress τ θθ (calculated from equation 163) in figure 29. We

92 72 (a) (b) (c) (d) Figure 28: Evolution of the shape of a truncated cone under elongation: (a) undeformed configuration (b) axisymmetric configuration at l = 0.2 (c) axisymmetric configuration at l = l c =0.5 (d) post bifurcation non-axisymmetric shape at l = l c =0.5. The figures on the right track the change in a cross section originally at a distance of 0.75 units from the left edge. In (d), the perturbation has been exaggerated for clarity.

93 τ θθ ^λ z Figure 29: Spatial profile of the hoop stress, τ θθ, and the amplitude of the bifurcating solution ˆλ 1. observe that the perturbation has its maximum amplitude close to the point where the hoop stress has its largest negative value, i.e., near the point of largest compressive stress. Table 2 shows the values of l c at which the n =3mode bifurcates for various values of r 1 and s. While the dependence on s is non-monotonic, we see that for a fixed value of s, increasing r 1 pushes l c to higher values. r 1 s l c Table 2: Variation of the bifurcation point for the n =3mode of the truncated cone with r 1 and s. So far, we have confined our discussion to the behavior of the n =3mode. While computing values of l c for different modes is a computationally intensive task, we can

94 74 get an idea of the dependence of n by looking at the eigenvalue spectrum of different modes for the l c of the n =3mode. This computation indicates that modes with higher wavenumbers bifurcate sooner, an observation consistent with the fact that even very small increases in length result in compressive hoop stresses in parts of the membrane. Thus, the elongation of this geometric shape at fixed volume is always unstable to nonaxisymmetric perturbations, with a critical wavenumber n. Atfirst sight, this seems unphysical, but a similar situation exists in the Saffman-Taylor problem (Saffman and Taylor, 1958) in the absence of surface tension. What is absent in the present formulation is a mechanism to damp out large wavenumber disturbances. In a viscoelastic filament, two such mechanisms would be surface tension and bending moments arising from nonzero boundary layer thickness. In fact, an approximate analysis of the radial inflow of a thin layer of an Oldroyd-B fluid in a washer shaped domain with free surface boundary conditions at the top and bottom (Kumar and Graham, 2000), which models the flow the polymer solution just above the end plates, shows that disturbances of large wavelength are the most unstable, thus providing evidence for the stabilizing role of surface tension. Another example of this is in the Saffman-Taylor problem, in which when surface tension is taken into account, the fastest growing disturbance has a finite wavenumber. 3.5 Conclusions In this chapter, we have proposed a new mechanism to explain the purely elastic instability seen in filament stretching at large Weissenberg numbers. In the first part of this chapter, we used the simple model of a planar elongation experiment to show how large stresses build up in free surface flows. Based on this, and upon evidence in the literature

95 75 (Rasmussen and Hassager, 1999; Spiegelberg and McKinley, 1998) on the formation of stress boundary layers near the free surface at large Weissenberg numbers, we postulated that the physics of the instability can be captured by modeling the region near the interface as an elastic membrane enclosing an incompressible fluid. We showed that such a membrane is unstable in elongation to non-axisymmetric disturbances. The instability is related to the formation of compressive hoop stresses in the membrane, and these are largest close to the point where the perturbation has the greatest magnitude. Our calculations show that disturbances with n are most unstable, but we expect that the inclusion of selection mechanisms like surface tension and a non-zero bending moment which impose an energy penalty on large wavenumber disturbances will yield a finite critical wavenumber. Validation of the mechanism proposed here would require numerical simulations using a viscoelastic constitutive equation at large Deborah number and up to large Hencky strains. For the mechanism proposed here to be effective, the simulations would need to show stress boundary layers with regions of compressive hoop stress close to the portions of the filament that have buckled.

96 76 Chapter 4 Stabilization of Dean flow instability In this chapter, we will discuss the stabilization of the purely elastic instability in the Dean flow geometry. Instabilities in Dean flow, and the closely related circular Couette flow geometry are interesting because they serve as models for more complex flows in polymer coating operations. Therefore, we will motivate the discussion by providing an overview of instabilities in coating flows. We will then describe the main features of the elastic instability in Dean flow and present a means of suppressing this instability. 4.1 Instabilities in coating flows Coating may be defined as the process of replacing air with a new material on the substrate (Cohen and Gutoff, 1992). Coatings play an important part in everyday life, being Most of the material in this chapter has been published in Ramanan et al. (1999)

97 77 used in the production of common products such as photographic film, paper, and magnetic media used for data storage. The most visible part of the industry is the manufacture of paints and protective films for automobiles, houses, and other structures. In the United States, this segment of the industry alone has an annual revenue in of about $17 billion (The Freedonia Group, 2000). When the less visible part of the industry, which includes such areas as print and publishing and printed circuit boards for electronics for is taken into account, the value exceeds $300 billion. There are several coating methods currently in use. The most common single layer coating methods are rod coating, forward and reverse roll coating, blade coating, air knife coating, gravure coating, slot coating, and extrusion coating. Multiple layer coatings processes include slide and curtain coating. Schematics of some of these processes are shown in figure 30. The choice of coating method to use for a given application is determined by several factors: these include the type of substrate (porous or non-porous), coating speed, the thickness and accuracy desired, the viscosity and viscoelasticity of the coating fluid, and the number of layers to be deposited. Booth (Booth, 1958a,b) was the first to publish a guide to choose the right coating method based on the criteria mentioned above. More recently, a simpler guide has been proposed by Cohen (1992). Throughput in coating operations is generally limited by interfacial instabilities which give rise to surface distortions on the coating (Strenger et al., 1997). Perhaps the most well known of these interfacial instabilities is the so called ribbing instability in forward and reverse roll coating (Saffman and Taylor, 1958; Pearson, 1960; Pitts and Greiller, 1961). A photograph of this instability in forward roll coating is shown in figure 31. The photograph clearly shows the spatially periodic patterns on the surface of the rollers,

98 78 (a) (c) (d) (f) (b) (e) Figure 30: Some commonly used coating industrial coating processes: (a) Dip coating and rod coating. (b) Blade coating and air knife coating. (c) Gravure coating. (d) Reverse roll coating. (e) Extrusion coating. (f) Slide coating and curtain coating (Cohen, 1992)

99 79 which translate to surface distortions on the coating. Since the flow in the gap in between the rollers is complicated, most of the earlier analyses of the ribbing instability used simplified models of the flow (Pitts and Greiller, 1961; Savage, 1977a,b; Gokhale, 1981, 1983b,a; Savage, 1984; Benkreira et al., 1982). With the advent of more powerful computers, it became possible to perform proper asympototic and fully numerical calculations using finite element methods, and to analyze its stability for Newtonian liquids (Ruschak, 1985; Coyle et al., 1986, 1990b). As a result, the critical conditions for the onset of ribbing for Newtonian liquids are now very well predicted by theory (Coyle, 1992). These results are typically reported in terms of the capillary number, which is a measure of the ratio of viscous to surface tension effects. Figure 31: Photograph of ribbing instability in forward roll coating (Coyle et al., 1990b). Many industrial coating processes use polymeric fluids, and this can have a significant effect on the instability. Bauman et al. (1982) showed that adding as small an amount as 10 ppm of high molecular weight polyacrylamide can reduce the critical capillary number for instability by a factor of 2 to 5 in forward roll coating. Slot coating (Ning et al., 1996) and reverse roll coating (Coyle et al., 1990a) are also significantly destabilized when polymeric fluids are used. Strong destabilization has also been observed recently in free

100 80 surface flows in eccentric cylinder geometries (Grillet et al., 1999) with dilute solutions of polyisobutylene in polybutene/kerosene mixtures. Soules et al. (1988) and Fernando and Glass (1988) performed a series of experiments which demonstrated that the earlier onset of ribbing correlates well with the extensional viscosity of the polymer. It is clear from the observations listed above that the addition of polymer has a marked effect on the stability of coating flows in industrial use. Therefore, it is important to understand the mechanisms by which polymers can cause flow instabilities. Rather than attempt the complex and computationally demanding task of analyzing coating flows, we look instead for simpler geometries where some of these mechanisms are manifested. Since we are interested in instabilities caused by elastic effects alone, we restrict our attention to flows where inertial effects are negligible. Examples of these include the flow of polymer solutions in cone-and-plate and plate-and-plate geometries (Larson, 1988; Byars et al., 1994), the circular Couette geometry (Larson et al., 1990), and the Dean and Couette-Dean geometries (Joo and Shaqfeh, 1991, 1994). In the latter three geometries, the fluid flows in the gap between two concentric cylinders. In circular Couette flow, the flow is driven by the motion of one of the cylinders, while in Dean flow, a pressure drop is applied in the azimuthal direction. In Couette-Dean flow, a combination of the two methods is used to drive the flow. Apart from these, recirculation flows, such as flow in a lid-driven cavity (McKinley et al., 1996) also exhibit elastic instabilities. In each of these flows, polymer molecules are stretched along the curved streamlines in much the manner as in the rod climbing effect discussed in chapter 1. This extra tension in the streamlines, or the so called hoop stress, is what causes the instability. It is obvious from an examination of figure 30 that curved streamlines exist in many coating flows. In addition, recent flow visualizations (figure 10) have shown the presence of recirculation

101 81 regions in blade coating. Thus, it is plausible that the instabilities seen in these flows are driven by mechanisms similar to those that drive the instability in simpler geometries, and that any stabilization mechanism that suppresses such instabilities in the simpler geometries such as Couette and Dean flows carries over to the more complex coating flows as well. 4.2 Elastic instability in Dean flow Before discussing the elastic instability in Dean flow, we first present an overview of the instability in the related geometry of circular Couette flow. Larson et al. (1990) performed the first theoretical and experimental analyses of elaticity-driven instabilities arising in circular Couette flow. Their theoretical analysis, using an Oldroyd-B fluid, showed that this viscoelastic Taylor-Couette (VETC) instability occured even at negligible Reynolds number. The criterion for the instability to occur is that ɛ 1/2 We θ = O(1), where We θ is the azimuthal Weissenberg number (ratio between material and flow time scales) and ɛ is the gap width, non-dimensionalized with respect to the radius of the cylinder. The scaling reflects how large We θ must be for the azimuthal normal (hoop) stress to contribute to the leading order radial momentum balance. For comparison, the well known inertial Taylor-Couette instability occurs when ɛ 1/2 Re is O(1), where Re is the Reynolds number. The mechanism of destabilization proposed by Larson et al. (1990) and later refined by Joo and Shaqfeh (1994) to include non-axisymmetric modes, was based on the coupling of stress perturbation to the base state velocity gradient to produce an azimuthal normal stress that drives radial and transverse motions leading to the formation of cells. The equations governing perturbations to the basic Couette flow are

102 82 O(2) symmetric, i.e., they are invariant under both reflections and translations. Theory (Golubitsky et al., 1985) indicates that bifurcations in such systems are either pitchfork or degenerate Hopf (i.e., the Jacobian matrix has two pairs of complex conjugate eigenvalues crossing the imaginary axis at the onset of the instability). In agreement with this prediction, Larson et al. (1990) found that the instability took the form of a degenerate Hopf bifurcation. While Larson et al. (1990) only considered axisymmetric modes, it was later found (Joo and Shaqfeh, 1994; Sureshkumar et al., 1994) that the most unstable disturbance is in fact a non-axisymmetric mode, with a wavenumber between 1 and 3. Further, Renardy et al. (1996) conducted a nonlinear analysis to study mode interactions arising from the introduction of inertia and thus the classical Taylor-Couette instability into the system. Computations (Northey et al., 1992; Avgousti and Beris, 1993; Avgousti et al., 1993) and experiments (Shaqfeh et al., 1992) have shown that finite gap effects tend to stabilize the flow. Recently, Al-Mubaiyedh et al. (2000) have shown that non-isothermal effects can give rise to an entirely new mode of instability at very low values of the Weissenberg number. In contrast to the isothermal case, this instability takes the form of a stationary, axisymmetric bifurcation. Al-Mubaiyedh et al. (2000) also investigated the effect of fluid rheology, in particular, accounting for multiple relaxation times, on the viscoelastic circular Couette instability. They found that the critical Weissenberg number asymptotes to a value that is half that predicted by a single mode constitutive equation, but that the mechanism of the instability and structure of the bifurcated modes remain the same. While the analyses discussed above have focused on the behavior close to the bifurcation point, i.e., the linear and weakly nonlinear regimes, experiments (Groisman and Steinberg, 1997; Baumert and Muller, 1999) have shown that a rich variety of dynamical phenomena can occur far away from the bifurcation point. In

103 83 chapter 5, we describe how a fully nonlinear analysis can capture some aspects of these observations. In the context of Dean flow, Joo and Shaqfeh (1991) performed a theoretical investigation for an Oldroyd-B fluid to show that such an elastic instability did indeed occur. Joo and Shaqfeh (1992b) generalized this to Couette-Dean flow. In line with the fact that this instability is purely elastic, they found that the most unstable fluid was one where there was no contribution to the extra stress tensor from the Newtonian solvent, and that the onset of the instability was delayed when solvent viscosity was present. Later, Joo and Shaqfeh (1992a) investigated the effect of inertia on Dean and circular Couette flows. They found that Dean flow was destabilized by inertial effects, while circular Couette flow was stabilized if the flow was driven by the rotation of the outer cylinder, and destabilized if it was driven by the inner cylinder rotation, consistent with the mechanism of the inertial instability. Joo and Shaqfeh (1994) presented experimental confirmation of the elastic instability in Dean flow, and also performed an experimental and theoretical investigation of the effects of finite gap width on the instability. As in circular Couette flow, they found both theoretically and experimentally that finite gap effects were stabilizing. The perturbation equations in Dean flow are also O(2) symmetric, and in this case, Joo and Shaqfeh (1991) found that the axisymmetric mode bifurcates as a pitchfork. They performed an energy analysis and determined that the mechanism of the instability involved the coupling of base state hoop stresses with radial velocity perturbations. In a later publication (Joo and Shaqfeh, 1994), they found experimental confirmation that the primary bifurcation in Dean flow was a stationary wave, and performed a theoretical analysis to show that non-axisymmetric modes are always more stable than axisymmetric ones.

104 84 The mechanism of the instability may be understood from figure 32. Polymer molecules are modeled as pairs of beads connected by springs. Molecules near the outer cylinder are more highly stretched than those closer to the center. An inwardly directed radial velocity perturbation convects highly stretched molecules towards the center, and thus increases the local hoop stress. This has the effect of reinforcing the radial velocity perturbation, hence driving the secondary flow. A simpler way of understanding this mechanism is to imagine that the azimuthal flow stretches polymer molecules so that they act like a stretched cylindrical membrane. This membrane in turn exerts an inward compressive force on the fluid, which causes fluid columns to buckle in the axial direction, causing the instability. v a c d b Figure 32: Mechanism of the elastic instability in Dean flow. Knowledge of the mechanism of the flow can be used to develop methods to stabilize it. A recent study by Graham (1998) has shown that the addition of a relatively weak axial flow results in significant stabilization of the isothermal VETC instability. The axial flow increases the critical Weissenberg number for onset of the instability and moves the most unstable mode to longer wavelengths. The stabilization is due to the additional axial normal stress generated by the axial flow, which suppresses radial displacements. Scaling analyses and numerical simulation showed that non-axisymmetric disturbances are

105 85 strongly suppressed. A weakly nonlinear analysis was performed to determine whether the bifurcation was subcritical or supercritical. In the narrow gap limit, it was found that the primary bifurcation in circular Couette flow is subcritical. The bifurcation remains subcritical if an axial Couette flow is added, while an axial Poiseuille flow changes the bifurcation to a supercritical one. Given that the effect of an axial flow is to suppress radial velocity perturbations, it seems reasonable to suppose that the same mechanism would stabilize the Dean flow instability as well. In the context of the simple instability mechanism stated above, axial flow has the effect of stretching the polymer membrane in the axial direction. This creates a restoring force that acts against radial buckling, as shown in figure 33.. In the first part of this work we extend the analysis of Graham (1998) to demonstrate that, as expected, the addition of axial flow stabilizes the viscoelastic Dean flow instability. In particular, we show that at large axial shear rates, the critical value of ɛ 1/2 We θ scales linearly with the axial shear rate, thus confirming that a relatively weak axial flow results in dramatic stabilization. This is similar to the behavior in circular Couette flow. We also perform a weakly nonlinear analysis for Dean flow to examine the effect of adding axial flow on the criticality of the bifurcation. Having established that the addition of a steady axial flow does stabilize viscoelastic Dean flow, we next examine the effect of adding a time-periodic axial flow. There are two reasons to study this. Firstly, it is more practical to superimpose an oscillatory axial motion in an industrial polymer processing operation than it is to superimpose steady axial motion. Secondly, viscoelastic fluids have characteristic relaxation time scales, and these might be expected to interact with the time scale of the parametric forcing to produce interesting dynamical phenomena. Note that the axial normal stress depends

106 86 τ rz, τ zz τ rz, τ zz τ θθ /r Figure 33: Illustration of the mechanism by which additional axial stresses generated by a superimposed axial flow stabilize the viscoelastic Dean flow instability on the square of the axial shear rate in the base flow. Thus, the average of this stress over one period of the oscillation is non-zero. Superimposing an oscillatory flow on the circular Couette flow has been shown to stabilize the inertial Taylor-Couette instability both experimentally (Weisberg et al., 1997) as well as theoretically (Hu and Kelly, 1995; Marques and Lopez, 1997). No corresponding analysis is available for Newtonian Dean flow. The organization of this chapter is as follows. We first begin by formulating the problem for an Oldroyd-B fluid in the narrow gap approximation in section 4.2. Section 4.3 describes the details of the linear stability analysis used in the study. To examine the effect of adding a steady axial flow, we use a standard linear stability analysis. However, adding an oscillatory axial flow results in time-periodic coefficients in the stability equations. In this case, the stability analysis is performed using Floquet theory. Rather than construct the entire monodromy matrix, we use the Arnoldi technique to determine its dominant eigenvalues. Finally, in section 4.4, we present results from our numerical analysis and compare them to predictions from the asymptotics.

107 Formulation We consider the isothermal flow of an Oldroyd-B fluid in the annular region between two infinitely long circular cylinders. Consideration of isothermal flow ensures that only the purely elastic mechanism of destabilization operates. The annulus is assumed to be open, i.e., no constraint is imposed on the net axial flow rate. The fluid has a relaxation time λ; the polymer and solvent contributions to the viscosity are η p and η s respectively, and the ratio η s /η p is denoted by S. The primary flow is produced by imposing a constant pressure drop in the azimuthal direction, K θ = P/ θ. A schematic of this geometry is shown in figure 34. R 2 Κ θ R 1 Figure 34: Dean flow geometry, shown with superimposed Poiseuille flow. The equations governing the flow are the momentum, constitutive and continuity equations. In dimensionless form, they are given by τ p + We θ S 2 v =0, (91)

108 88 ( τ τ + We θ t + v τ ( τ v +(τ v) t) ) ( = We ) θ v + v t, (92) v =0, (93) where v is the velocity, p is the pressure and τ is the polymer stress tensor. No-slip boundary conditions are imposed at the two cylinders. The Weissenberg number is defined in terms of the pressure drop as where ɛ is the dimensionless gap width, We θ = K θλɛ 2 η p, (94) ɛ = R 2 R 1 R 2. (95) The Newtonian fluid is recovered in the limit S, with an appropriate rescaling of pressure. We measure length in terms of distance from the inner cylinder, scaled by the gap width, i.e., r = r R 1 R 2 R 1, (96) where r is the dimensional radial position, and r is the dimensionless radial coordinate, scaled and shifted so that r =0is the inner cylinder and r =1is the outer cylinder. Stress is scaled by η p /λ, and velocity is scaled by K θ ɛ 2 R 2 /2 η p. Time is made dimensionless by the ratio of the gap width, ɛr 2, to the velocity scale. In terms of the dimensionless variables, the flow domain is given by {(r, θ, z) : 0 r 1, 0 θ<2π, <z< }. Axial Couette flow is imposed by moving the inner cylinder with the velocity V cos(ωt), where V is the amplitude of the modulation and ω is the frequency. The scaling used for

109 89 ω is the inverse of that used for time, so that ωt is O(1). Setting ω =0is tantamount to imposing steady axial Couette flow. Later, we will see that ω 0 is a singular limit. The imposition of axial flow introduces an axial Weissenberg number, We z,defined as We z = λv (1 ɛ). (97) R 2 ɛ When the axial flow is imposed by means of a constant axial pressure drop, P z, we can similarly define another Weissenberg number For convenience, we will use the following acronyms: We z = P zλɛr 2 2 η p. (98) DAC: Dean flow with steady Axial Couette flow superimposed (ω =0). DMAC: Dean flow with Modulated Axial Couette flow (ω 0). DAP: Dean flow with steady Axial Poiseuille flow. We only consider the case where the axial pressure drop is steady. For an Oldroyd-B fluid, it is possible to obtain exact analytical expressions for the base state velocities v and polymer stresses τ in Dean flow subjected to superimposed time-periodic or steady axial flow. The narrow gap limit of these expressions (i.e. the leading term in a Taylor series about ɛ =0) are shown in Appendix B. We are interested in the stability of these steady and periodic solutions to infinitesimal perturbations. We define a vector of perturbations u =( τ rr, τ rθ, τ rz, τ θθ, τ θz, τ zz, ṽ r, ṽ θ, ṽ z, p), where, for example, τ θθ = τ θθ τ θθ. It was shown by Joo and Shaqfeh (1991) that the elastic instability occurs when We θ is O(ɛ 1/2 ). We enforce this by defining a scaled Weissenberg number Wp = ɛ 1/2 We θ, which is O(1) in the limit ɛ 0. We now scale the

110 90 perturbation stresses and velocities by requiring that all terms in the Oldroyd-B equations appear in the perturbation equations for steady axial flow. This may be obtained by imposing the following scalings on the perturbation velocities and stresses: τ rr = O(1), τ rθ = O(ɛ 1/2 ), τ rz = O(1), τ θθ = O(ɛ 1 ), τ θz = O(ɛ 1/2 ), τ zz = O(1), ṽ r = O(ɛ 1/2 ), ṽ θ = O(1), ṽ z = O(ɛ 1/2 ), p = O(1) and We z = O(1). Terms involving the azimuthal wavenumber n do not appear in the governing equations at leading order. To get around this, we must either consider terms of O(ɛ 1/2 ) relative to the leading order terms in the equations (as done by Joo and Shaqfeh (1994)) or equivalently, define a new azimuthal wavenumber ñ by n =ñɛ 1/2, where ñ is O(1) (as done by Graham (1998)). We choose the latter approach. Thus, in our formulation, ñ is not necessarily an integer, although n must be. The scaling on the stresses and velocities is identical to that on the base state stresses and velocities, and also to the scaling obtained by Graham (1998) for circular Couette flow with imposed axial flow. For time-periodic axial flow, a balance of terms is obtained by rescaling time and frequency so that ω = ω 1 ɛ 1/2 and t = t 1 ɛ 1/2 where ω 1 and t 1 are O(1). With this scaling, the dimensionless relaxation time of the polymer is Wp. This is the low frequency regime that includes all possible terms in the narrow gap approximation. Setting ω 1 =0in this regime, we recover the equations of DAC flow. We also investigate the effect of imposing a high frequency oscillation by scaling ω to be an O(1) quantity i.e. we specify ω 1 = O(ɛ 1/2 ). Dimensionally, this corresponds to a forcing frequency on the order of the azimuthal shear rate. In this regime the stability characteristics depend on the order of magnitude of the axial shear rate. In section 4.5 we show that the relative magnitude of We z with respect to ω 1 determines the appropriate balance of terms that affects the stability of the system. Asymptotic and numerical results are presented for both regimes.

111 Stability and Numerical Analysis Linear Analysis and Floquet Theory To analyze the stability of the base flow, we expand the perturbation vector as u = ˆδφ(r, t)e iαz + O(ˆδ 2 )+c.c., (99) where c.c. stands for complex conjugate and α is the axial wavenumber which is a parameter in the problem. We first consider the imposition of steady axial flow. In this case, the vector φ(r, t) is expanded out as φ = ξ(r)e i(nθ αct), (100) where ξ(r) =(ˆτ rr (r), ˆτ rθ (r), ˆτ rz (r), ˆτ θθ (r), ˆτ θz (r), ˆτ zz (r), ˆv r (r), ˆv θ (r), ˆv z (r)), and c = ĉ r + i ĉ i is the growth rate. This expression is added to the base state stresses and velocities, and the result substituted into equations 91 to 93. The equations may then be expanded as a series in ˆδ, and at O(ˆδ 0 ) we recover the equations governing the base state stresses and velocities. At O(ˆδ), we get the equations that govern the evolution of perturbations to the base state. We may write these succinctly as iαceu= L(α, We θ,we z, ñ)u, (101) where L is the linearization of the governing equations about the base state, and E is a diagonal matrix with 1 on the diagonal entries corresponding to the Oldroyd-B equations and zeros elsewhere. The nonzero components of E are shown in Appendix B for ñ =0. Equation 101 is in the form of a generalized eigenvalue problem for the growth rate c. If the other parameters are given, we can numerically solve for the eigenvalue c. For stability, ĉ i must be less than zero.

112 92 The numerical technique we use is a Chebyshev collocation method (Canuto et al., 1988). We use a primitive variable formulation of the conservation, constitutive and continuity equations, i.e., our unknowns are the six stress components, the three components of the velocity, and the pressure. Since the conservation equations only contain derivatives of the pressure, it is necessary to define pressure on a staggered grid (Canuto et al., 1988) to avoid spurious modes. In the formulation we use, the velocity components and stresses are defined at the N +1Gauss-Lobatto points (for an N th order collocation), which are r k =cos ( ) πk,k=0,...,n, (102) N while the pressure is defined at the N Chebyshev-Gauss points, which are given by ( ) π(k +1/2) r k =cos,k=0,...,n 1, (103) N so that pressure is defined at one less point than the other variables. In evaluating the conservation equations, pressure is interpolated onto the Gauss-Lobatto points from the staggered grid, while the continuity equation is evaluated on the staggered grid by interpolating the velocity components from the Gauss-Lobatto points. We find that it is sufficient to use N =32for accurate calculation of c and spatial resolution of the eigenvectors. We now move on to the stability analysis for the time-periodic case. In this case, the vector φ(r, t) is expanded as φ(r, t) =(ˆτ rr (r, t), ˆτ rθ (r, t), ˆτ rz (r, t), ˆτ θθ (r, t), ˆτ θz (r, t), ˆτ zz (r, t), ˆv r (r, t), ˆv θ (r, t), ˆv z (r, t), ˆp(r, t)). (104)

113 93 We will only consider axisymmetric perturbations because, as will be shown in section 4.5, it is the axisymmetric mode which is most unstable whether or not axial flow is present. The equations governing evolution of the perturbations are given by E u = A(t)u. (105) Here, A(t) is a matrix with time-periodic coefficients such that A(t) = A(t + T ), where T =2π/ω is the period of the forcing function. The nonzero components of A are shown in Appendix B. The problem lends itself to analysis by Floquet theory (Iooss and Joseph, 1989). The solution u(t ) at t = T, given the initial vector u(0), is u(t )=Φ(T )u(0), (106) where Φ(T ) is the monodromy matrix, whose eigenvalues β, known as Floquet multipliers, determine the stability of the system. Stable and unstable behavior is indicated by β < 1 and β > 1 respectively. The Floquet exponent σ is defined by the relation β =exp(σt). (107) Suppose Ψ is an eigenvector of Φ(T ) corresponding to the Floquet exponent σ. Then, it can be shown (Iooss and Joseph, 1989) that the solution w(t) to Ė = A(t)w with w(0) = Ψ has a T periodic component ζ(t) given by ζ(t) =e σt w(t), (108) such that ζ(t) =ζ(t + T ). (109) Thus, the ζ(t) corresponding to the dominant Floquet multiplier gives us information on the spatial structure and time evolution of the disturbance.

114 94 The standard way to calculate Φ(T ) is to do it column by column, integrating equation 105 with u(0) = e k, where e k is the k th column of the unit matrix of same dimension as E or A(t). Given the size of our problem, this is a computationally expensive task. However, we only need the dominant eigenvalues (those with the largest modulus) of Φ(T ), and we can conveniently obtain these using the Arnoldi method (Arnoldi, 1951). We use a primitive variable formulation as in the steady case, with pressure defined on a staggered grid. We use the public domain code ARPACK (Lehoucq et al., 1997) to carry out the Arnoldi calculations. The advantage of the Arnoldi method lies in the fact that we do not need to explicitly construct Φ(T ) in its entirety; we need only determine the action of the matrix on a vector, q Φ(T )p. In our problem q is the solution vector obtained by integrating the time evolution equations for a given initial vector p. The choice of the initial vector p is arbitrary, with the only requirement being that it satisfy the algebraic components of equation 105. In our case, this is ensured by choosing an arbitrary stress profile and solving the linear momentum and continuity equations for the corresponding velocities and pressure. To compute the periodic component ζ(t), we calculate the disturbance vector w(t) by integrating the time evolution equation 105 with initial conditions given by the eigenvector of the monodromy matrix obtained from ARPACK. This choice of starting value ensures that the periodicity condition given by equation 109 is not violated. The integration of the time-dependent viscoelastic equations is performed using the EVSS decomposition (Rajagopalan et al., 1990) of the total stress into elastic and viscous components, coupled with a fully implicit first-order time stepping procedure. The time integration and eigenvalue routines were benchmarked against known eigenvalues for the

115 95 steady case (ω =0) from the DAC results presented later in this chapter. Most calculations are performed with a time step t = T/100, while at very low frequencies we employ increased temporal resolution of t = T/1000. This high resolution is necessary to ensure the accuracy of the eigenvalue computation, which is checked by monitoring the periodicity of ζ(t) Weakly nonlinear analysis Following the methodology of Graham (1998) which in turn follows Iooss and Joseph (1989), we conduct a weakly nonlinear analysis to determine the criticality of the bifurcation in DAC and DAP flows. In the presence of either type of axial flow, the loss of stability takes the form of a Hopf bifurcation, while in the absence of axial flow the bifurcation is a pitchfork. In either case, the vectors take the form of complex conjugates, and since we are only interested in real valued solutions, the analysis described below works whether or not an axial flow is present. The nonlinear solutions are constructed by expanding all variables as a power series in the amplitude ˆδ, substituting in the governing equations, and applying the narrow gap approximation at each order in ˆδ. We let µ = Wp Wp c, ˆω 0 = αc r, and s =ˆωt. Thus, the solution takes the form u(z, s, ˆδ) µ(ˆδ) = ˆδ k k! k=1 u k (z, s) µ k, (110) ˆω(ˆδ) ˆω 0 ˆω k where u is taken to be the solution at the Chebyshev points. The linear operator L can

116 96 be written as L(µ) =L 0 + ˆδµ 1 L + ˆδ 2 2 µ 2L +... (111) Further, we define the following inner products: a(z, s),b(z, s) α 2π/α 2π a(z, s) 4π b(z, s) ds dz, (112) and [a(z, s),b(z, s)] 9(N+1)+N 1 l=0 a l (z, s),b l (z, s). (113) At O(ˆδ), we recover the linear stability problem, which has the real valued solution u 1 = z + z. Note that u is time-dependent in the absence of axial flow, and a traveling wave otherwise. The eigenvectors are normalized so that [z, z] =1, where z is the solution to the adjoint problem. A solvability condition at O(ˆδ 2 ) gives 2 i ˆω 1 [Eu 1, z ]+2µ 1 [L u 1, z ]= 2[N 2 (u), z ]. (114) The structure of the solutions at first order implies that the right hand side of this equation vanishes. Upon separating the real and imaginary parts, the coefficient matrix of the resulting 2 2 real system has, in general, a nonvanishing determinant, and hence admits only the trivial solution for µ 1 and ˆω 1. Substituting these values into the O(ˆδ 2 ) problem, we see that the particular solution at that order can be written as u 2 = u 20 + u 22 e 2i(αz+s) + ū 22 e 2i(αz+s). (115) Here, u 20 is given by L 0,0 u 20 = 2 N 2 (u), 1, (116)

117 97 where L 0,0 is obtained by replacing α by 0 in L 0. The vector u 22 is given by ( 2iˆω 0 E + L 0,2α )u 22 = 2 N 2 (u), 1e 2i(αz+s), (117) where 1 is a column vector with 1 in all its entries, and L 0,2α is obtained by replacing α with 2α in L 0. Unlike in Graham (1998), the present formulation retains pressure in the momentum equations. Hence, neither of the above equations is singular and the solution is obtained by LU decomposition. A solvability condition at O(ˆδ 3 ) gives the single complex equation (or the 2 2 real system) iˆω 2 [Eu 1, z ]+µ 2 [L u 1, z ]= 2[N 3 (u), z ], (118) from which µ 2 and ˆω 2 can be obtained. 4.5 Results and Discussion Scaling Analysis As discussed in the introduction, the elastic instability is caused by an unstable stratification of the hoop stress, τ θθ. Therefore, if this term is absent from the momentum balance in some asymptotic limit, the flow is stable. It is instructive to perform an asymptotic analysis to determine how large Wp must be for the hoop stress to enter into the momentum balance at leading order. This can also be used to check the results from the numerical analysis. We first consider the imposition of steady axial flow, with We z 1. The scalings for both DAC and DAP flows are identical, and the effect of O(1) solvent viscosity does not affect them. We first consider the results for axisymmetric perturbations. For these, the

118 98 scaling regimes and asymptotics are identical to the VETC flow scaling regimes discussed in Graham (1998), so we merely restate them here. The method of determining these scalings is to assume that Wp scales as We m z, and then use the constitutive equations to determine the stress scalings. These are then substituted into the momentum balance, and the requirement that τ θθ appear at leading order in the radial momentum balance is used to solve for m. Depending on the value of α, three scaling regimes exist for axisymmetric disturbances: αwe z 1 : Wp = O(α 1/2 We 3/2 z ), (119) We z 1, αwe z = O(1) : Wp = O(We z ), (120) We z 1, αwe z 1 : Wp = O(1/α). (121) Clearly, it is the second scaling which has the lowest Wp of the three regimes. This means that for We z 1, the most unstable wavenumber will be O(1/We z ), and the critical Wp will scale linearly with We z. For non-axisymmetric modes, we first consider ñ = O(1), and α = O(1). For this case, the dominant balance reveals that m =1,or Wp = O(We z ). This is the only regime for ñ = O(1), and indicates that for a given ñ, Wp scales linearly with We z, as in the axisymmetric case. Note that the scaling for nonaxisymmetric modes differs from the corresponding scaling in VETC flow determined by Graham (1998), where Wp must be O(We 2 z ) for non-axisymmetric disturbances to become unstable. Disturbances with n 1 are very strongly suppressed. For example, when n = O(ɛ 1 ), Wp has to be O(ɛ 1 ) to keep τ θθ in the momentum balance. The linear scaling of Wp with We z has an important practical consequence. Recalling that We θ = Wp ɛ 1/2, we see that the ratio We z /We θ, which is a measure of the magnitude of axial flow relative to the azimuthal flow, is O(ɛ 1/2 ). Since ɛ 1, this means

119 99 that a small amount of axial flow provides significant stabilization. Although we have assumed that We z 1 in the asymptotic analysis above, numerical computations indicate that these results are valid down to about We z =1, yet another instance of the power of asymptotics in supplying information about qualitative behavior. We now present scaling results for the imposition of time-periodic axial flow. Our aim here is to determine how the imposition of a time-periodic axial flow changes the stability characteristics of the system. We have already observed that the stabilization is due to the stress τ zz induced by the axial flow. From Appendix B, we see that the expression for τ zz has a We 2 z term multiplied by a factor that is the sum of a constant term and a term that depends on the frequency and time. From our analysis above, we know that the constant term has a stabilizing influence. We seek to determine the effect of the frequency and time dependent terms on the stability characteristics. We find that, depending on the frequency, this term can either stabilize or destabilize the flow. We first consider the high frequency regime, ω 1 = O(ɛ 1/2 ), with time rescaled appropriately to reflect the shorter time scales. Further, we also retain the scalings on the axial and azimuthal Weissenberg numbers We z = Wp = O(1) and restrict α to be O(1). In this regime, the axial strain rate is large, but the strain amplitude is small (O(ɛ 1/2 )), and we find that the leading order evolution equation for τ zz does not contain the We 2 z contribution that has been established to be the source of high normal axial stresses that contribute to the stabilization in the flow. The We 2 z term comes from the time-averaged non-zero base state axial stress, which tends to zero with increasing ω and fixed We z. Thus, we expect decreased stabilization in this regime. Further, the ω 1 = O(ɛ 1/2 ) regime also lets us apply the method of averaging, a rigorous asymptotic technique (Sanders and

120 100 Verhulst, 1985), to the time dependent equations. Briefly, the averaging method transforms the system ẋ =ˆɛf(x,t,ˆɛ) with f(x,t,ˆɛ) =f(x,t+ T,ˆɛ), (122) to an autonomous system ẏ =ˆɛ 1 T T 0 f(y,t,0) dt =ˆɛ f(y). (123) In other words, the averaging procedure replaces the time-dependent coefficients with their averages over the period of oscillation. This procedure is valid for arbitrary amplitude of oscillation as long as ˆɛ 1. It can be shown that in our problem the equations governing the stress perturbations in this regime are of the form shown in equation 122 with ˆɛ = ɛ 1/2, ɛ being the dimensionless gap width defined by equation 95 in section 4.3. Importantly, we note the absence of the contribution due to the zero frequency terms in τ zz in the leading order equations for τ rz and τ zz. The remaining coefficients of We z are periodic with zero mean and drop out in the averaging procedure, leaving a system that is independent of We z and ω 1. Hence in the ω 1 = O(ɛ 1/2 ) and We z = O(1) regime, the system is reduced to the steady Dean flow limit with no axial motion. Our numerical computations also indicate that in the limit of large frequencies, we recover the steady Dean flow results. In the second high frequency regime of interest we investigate the effect of O(1) axial deformation on the dynamics of the system. In this regime we have ω 1 = O(ɛ 1/2 ) and We z = O(ɛ 1/2 ), which corresponds to high frequency and large deformation. By considering time scales of the order of the relaxation time of the fluid, we discover the presence of stress boundary layers near the cylinders of size O(1/We z ) for O(1) wavenumbers and of size O(1/αWe z ) for O(ɛ 1/2 ) wavenumbers. The existence of boundary

121 101 layers at high shear rates is consistent with the analysis of Renardy (1997) and Graham (1998). The asymptotic balances reveal that the destabilizing azimuthal stress, τ θθ, drops out of the radial momentum balance, thus resulting in stabilization. Again, computational results confirm this prediction Numerical Results We begin by presenting results for the steady case. As mentioned in the introduction, the basic Dean flow profile is invariant under translations and reflections, i.e., it has O(2) symmetry. Symmetry breaking bifurcations in such systems take the form of pitchforks or degenerate Hopf bifurcations (Golubitsky et al., 1985). In agreement with this prediction, Joo and Shaqfeh (1991) found that the primary bifurcation in Dean flow was a pitchfork. When axial flow is added, the reflection symmetry is lost, and the flow is SO(2) symmetric, and theory (Iooss and Joseph, 1989) predicts that symmetry breaking takes the form of Hopf bifurcations. In line with the predictions above, we find that the primary bifurcation in pure Dean flow is a pitchfork, and when axial flow is added, symmetry breaking takes the form of a Hopf bifurcation. Figure 35 shows the results of a linear stability analysis for DAC flow when S =0. The curves are plots of the critical value of Wp (denoted by Wp c ) versus the wavenumber α for given We z. For each curve, there is a global minimum value of Wp c, which we denote by Wp c,min, with the corresponding wavenumber denoted by α min. Thus Wp c,min denotes the smallest value of Wp at which the base flow becomes unstable, and the destabilizing disturbance has a wavenumber α min. The figure clearly shows that Wp c,min increases with We z. A similar plot is shown for DAP flow in figure 36. Here too, the effect is increased stability at high We z. Note also that in both DAC and DAP flows,

122 102 α min decreases with increasing We z. Figure 37 shows a plot of Wp c,min versus We z for both DAC and DAP flows. In accordance with the asymptotic analysis, Wp c,min increases linearly with We z. Note also that in both these flows, Wp c,min increases monotonically as We z is increased, in contrast to the initial destabilization observed in circular Couette flow (Graham, 1998). Solvent viscosity has a stabilizing effect in both DAC and DAP flows, as indicated in figure 38 which plots Wp c,min versus We z for S =0and S = We z =0 We z =1 We z =2 We z =3 Wp α Figure 35: Neutral stability curves for DAC flow (S=0). In each case, the position of Wp c,min is denoted by a. As predicted by the asymptotic analysis, non-axisymmetric modes are similarly stabilized. Solvent viscosity has a stabilizing effect on non-axisymmetric modes as well, and we only show the results for S =0. Plots of Wp c,min versus ñ for different We z are shown in figure 39 for DAC flow and figure 40 for DAP flow. We see increased stabilization as ñ increases and also as We z increases, in accord with the asymptotic analysis. Figure 41 shows a plot of Wp c,min versus We z for ñ =1. The linear scaling predicted by the asymptotics is evident.

123 We z = 0 We z = 1 We z = 2 We z = Wp c α Figure 36: Neutral stability curves for DAP flow (S=0). In each case, the position of Wp c,min is denoted by a Wp c,min 5.0 DAP flow DAC flow We z Figure 37: Plot of Wp c,min vs. We z for DAC and DAP flows (S=0). Note the linear scaling at high We z.

124 Axial Couette (S=0) Axial Couette (S=10) Axial Poiseuille (S=0) Axial Poiseuille (S=10) 20.0 Wp c,min We z Figure 38: Plots of Wp c,min vs. We z for S =0and S =10, displaying the stabilizing influence of solvent viscosity We z =0 We z =0.5 We z =1.0 We z =1.5 Wp c,min ~ n Figure 39: Plot of Wp c,min vs. ñ for different values of We z for DAC flow (S =0).

125 We z =0 We z =0.5 We z =1.0 We z =2.0 Wp c,min ~ n Figure 40: Plot of Wp c,min vs. ñ for different values of We z for DAP flow (S =0) Wp c,min We z Figure 41: Plot of Wp c,min vs. We z for DAC flow with ñ =1.0 (S =0). Note the linear scaling for We z > 1.

126 106 Figure 42 shows a portion of the neutral stability curve for pure Dean flow. We point out that there is a change in the slope of the curve at α =21.9. At this point, four eigenvalues with zero real parts are simultaneously neutrally stable. This is a codimension-2 Takens-Bogdanov bifurcation point (Takens, 1974; Bogdanov, 1975; Knobloch and Proctor, 1981; Guckenheimer and Knobloch, 1983). Hereafter, we refer to this value of α as α tb. For wavenumbers greater than α tb, the bifurcation is a Hopf, with a four dimensional center manifold, similar to the one in circular Couette flow. As we decrease α towards α tb, the period of the Hopf bifurcation increases, reaching infinity at α tb, where the form of the bifurcation changes to a pitchfork. From a physical point of view, this bifurcation point is expected, since large wavenumber disturbances are localized in a small region near the outer cylinder, and hence experience a base velocity profile that is locally linear, similar to the profile in circular Couette flow. Thus, we expect that for large wavenumbers, the destabilization mechanism, and hence the nature of the bifurcation, will be similar to the one in circular Couette flow. Experimental results by Genieser (1997) on the flow of viscoelastic fluids through a planar contraction display instabilities that occur as transitions from steady flow to either steady or oscillatory flow, depending on contraction ratio. The oscillatory flows have very low frequency, so it may be that these observations are another manifestation of Takens-Bogdanov bifurcation and concomitant change in destabilization mechanism found here. The unfolding of a Takens-Bogdanov bifurcation in an O(2) symmetric system was performed by Dangelmayr and Knobloch (1987). They showed that, depending on the value of the bifurcation parameters, observable patterns include a nontrivial steady state, traveling waves, standing waves and modulated waves. The results of the weakly nonlinear analysis at Wp c,min and α min are summarized in

127 Hopf (oscillatory) 7.5 Wp c pitchfork (steady) Takens Bogdanov point α Figure 42: Neutral stability curve of pure Dean flow at high α. The Takens-Bogdanov bifurcation point is indicated by a. Tables 3 and 4 for DAC flow and Tables 5 and 6 for DAP flow. Negative values of µ 2 indicate subcritical behavior while positive values imply supercriticality. We note that in the absence of solvent viscosity (S = 0), the bifurcation changes from subcritical to supercritical at We z 0.05 while at finite solvent viscosity the change occurs at We z 1. Results for higher We z show that for S 0the bifurcations revert back to being subcritical. Clearly, by varying the extent of axial motion we may not only change the position of the bifurcation point but also ensure that we stay in a regime that admits stable solutions. Similar behavior arises for the DAP flow, where there is a change in the sign of µ 2 at at We z 0.25 for S =0, and We z 1.0 when S =1. Here, however, nonzero S does not change the criticality of the dominant unstable mode at higher values of We z. We note that a change in the criticality of bifurcation has also been observed in

128 108 We z α min Wp c,min µ 2 ˆω Table 3: µ 2 and ˆω 2 for DAC flow (S =0). We z α min Wp c,min µ 2 ˆω Table 4: µ 2 and ˆω 2 for DAC flow (S =1). the weakly nonlinear analysis of cone and plate flow of an Oldroyd-B fluid (Olagunju, 1997). In the final portion of this section, we present results for the imposition of timeperiodic axial flow. As in the steady case, solvent viscosity has no qualitative effect on the nature of the results, and we confine our discussion to the fluid with zero solvent viscosity. Addition of solvent viscosity simply results in increased stabilization relative to S =0, but does not otherwise change the nature of the results. Also, our analysis for the imposition of steady axial flow indicates that non-axisymmetric modes are always more stable than axisymmetric ones, so we will only consider axisymmetric perturbations here. The addition of time-periodic axial flow introduces a new parameter ω 1, which increases the dimension of the parameter space. Scanning the entire regime of We z, α and

129 109 We z α min Wp c,min µ 2 ˆω Table 5: µ 2 and ˆω 2 for DAP flow (S =0). We z α min Wp c,min µ 2 ˆω Table 6: µ 2 and ˆω 2 for DAP flow (S =1). ω 1 is a computationally demanding task. Since our primary goal is to determine the effect of the periodic axial flow on the stability of pure Dean flow, we limit ourselves to a smaller parameter space. For the most part, we choose Wp =4.06, which corresponds to Wp c,min for pure Dean flow. For all the values of α, We z and ω 1 that we considered, we always found increased stabilization relative to pure Dean flow. Thus, the effect of time-periodic axial flow is always stabilizing. This can be seen quite dramatically in figure 43, which shows the decay of the Euclidean norm of the destabilizing perturbation hoop stress when timeperiodic axial flow is imposed. The forcing chosen was We z =1and ω 1 =0.5. The initial condition was chosen to be the eigenvector for pure Dean flow with Wp =4.06 and α =6.6, which corresponds to the most unstable disturbance in pure Dean flow. A time sequence of density plots of the perturbation hoop stress, τ θθ, corresponding to these

130 110 parameters is shows in figure 44. By t =12T, the perturbation has essentially decayed to the point where the base flow is recovered. The complete stabilization found in Dean flow is in contrast to the results reported by Ramanan et al. (1999) for the imposition of time-periodic axial flow in the circular Couette case. While they find that axial flow results in increased stability for most cases, they also find that the system shows instability (synchronous resonance) for certain values of ω 1 and We z. 2.0 t/t t/t Figure 43: Decay of the perturbation hoop stress τ θθ when axial flow is imposed. Parameters are: We z =1.0, ω 1 =0.5, Wp =4.06, S =0. As mentioned in section 4.4, stability requires that the magnitude of the Floquet multiplier be less than one. We show a plot of the magnitude of the Floquet multiplier, β versus the frequency ω 1 for different wavenumbers in figure 45. The other initial conditions were We z =0.5, and Wp =4.06. In each case, the magnitude of the Floquet multiplier is less than 1, indicating stabilization relative to pure Dean flow. Figure 46 shows the effect of increasing We z. The parameters chosen were Wp =4.06 and α =6.6. As in the steady case, increasing We z increases stabilization, reflected here in the smaller

131 111 t=0 t=12t t=t/4 t=9t t=3t/4 t=6t t=t t=3t Figure 44: Time sequence of density plots of the perturbation hoop stress τ θθ when axial flow is imposed. The parameters are identical to figure 43, so that without axial flow, the flow is neutrally stable. Each frame shows a z r cross-section of the geometry.

132 112 magnitude of the Floquet multipliers for larger axial forcing. The figure also shows that for large ω 1, the Floquet multiplier tends asymptotically to unity. This is in agreement with the scaling analysis for high frequency discussed in the preceding section, which indicates that for ω 1 1, the O(We 2 z ) contribution to τ zz in the base flow has a zero average over a single period of the forcing, thus reducing the problem to zero axial forcing. For the values of Wp and α we chose in this simulation, the base flow is marginally stable, and hence β α=2.0 α=4.0 α=6.6 α= β ω 1 Figure 45: Plot of the magnitude of the Floquet multiplier β vs. ω 1 for different values of α. (We z =0.5, Wp =4.06, S =0). In the limit ω 1 0, we might expect the results from our simulations with timeperiodic forcing to reduce to those with steady axial forcing. It turns out however, that this is not the case, i.e., the limit ω 1 0 is singular. This has also been observed in the stabilization of Newtonian circular Couette flow (Marques and Lopez, 1997), as well as in the linear stability analysis of other time-dependent flows (Rosenblat, 1968; von Kerczek and Davis, 1974). Davis and Rosenblat (1977) analyzed a damped Mathieu-Hill equation

133 β We z =0.5 We z =1.0 We z = ω 1 Figure 46: Plot of the magnitude of the Floquet multiplier β vs. ω 1 for different values of We z.(wp =4.06, α =6.6, S =0). β asymptotes to 1 at large ω 1 in agreement with the asymptotic prediction. with externally imposed modulation and showed that at low frequencies the eigenfunctions have large temporal increases in magnitude within the period of oscillation; these become unbounded as ω 1 0. Thus, while the Floquet analysis may predict overall stable behavior for a suitable choice of parameters at low frequencies, the long periods of oscillation allow for large transient increases in the response. Since the perturbations in the system are now no longer infinitesimally small as is required for linear stability analysis, one would have to resort to a nonlinear theory to be able to adequately represent the solution of the system. This type of behavior is seen in DMAC flow at small ω 1, and is shown in figure 47, where the Euclidean norm of the periodic component of the hoop stress is plotted against time (normalized by the period of the oscillation) for ω 1 =0.01. The hoop stress is normalized with respect to its minimum value. We see in the figure that there are periods where the stress increases close to 500 times its minimum value in

134 114 the cycle. In contrast, a similar plot for ω 1 =1.0 (figure 48) does not show this behavior, with the stress remaining O(1) relative to its minimum value in the cycle. Similar behavior has been observed in the context of periodic axial flow applied to circular Couette flow (Ramanan et al., 1999) ~ τ θθ t/t Figure 47: Plot of the magnitude of the periodic component of the hoop stress τ θθ over a cycle of the forcing for ω 1 1 (We z =1.0, Wp =4.06, α =6.6, ω 1 =0.01). Note the large increases in magnitude. The results presented above for periodic axial forcing reflect the stability properties of the forced system relative to the critical Wp of pure Dean flow. We conclude this section with a sample calculation showing how the minimum critical value of Wp is shifted by the presence of axial oscillations. For DMAC flow, simulations for S =0,ω 1 =0.8,We z = 2.0 yield Wp c,min =4.37 and α min =6.0. This is greater than Wp c,min for pure Dean flow (4.06), thus indicating increased stabilization. For comparison, the value of Wp c,min for We z =2in steady axial flow would be 7.39 for DAC flow.

135 ~ τ θθ ω 1 Figure 48: Plot of the magnitude of the periodic component of the hoop stress τ θθ over a cycle of the forcing for ω 1 = O(1) (We z =1.0, Wp =4.06, α =6.6, ω 1 =1.0). The magnitude remains O(1) over the entire cycle. 4.6 Conclusions In this chapter, we showed that the elastic instability in isothermal Dean flow could be delayed by the addition of steady axial flow either in Couette or Poiseuille form. Both axisymmetric and non-axisymmetric disturbances are suppressed. The stabilization is a result of the additional axial normal stress resulting from the axial flow. We conducted a weakly nonlinear analysis to determine the criticality of the bifurcation in Dean flow with and without axial flow. Our results indicate that the bifurcation is subcritical for pure Dean flow, and the subsequent nature of the bifurcation depends on We z and S. We also showed that time-periodic axial Couette flow can be used to stabilize Dean flow. Finally, we report a codimension-2 Takens-Bogdanov bifurcation point at an axial wavenumber of 21.9 for Dean flow without axial forcing. This bifurcation point represents a change in the mechanism of the instability.

136 116 Chapter 5 Localized solutions in viscoelastic shear flows In the previous chapter, we discussed how the primary instability in Dean flow can be suppressed by the addition of an axial flow that is small in magnitude when compared to the primary flow. The tools that we used were asymptotic analysis, linear stability analysis, and weakly nonlinear analysis. These tools are useful in giving us information about the dynamics close to the point where the flow first loses stability. As recent experimental work (Groisman and Steinberg, 1997, 1998; Baumert and Muller, 1999; Groisman and Steinberg, 2000) shows, the dynamics far from the bifurcation point can be very complex. Of particular interest to us are the observations of stationary, long wavelength structures in circular Couette flow by Groisman and Steinberg (1997), Groisman and Steinberg (1998), and Baumert and Muller (1999). This chapter describes how a fully nonlinear Most of the material in this chapter has been published in Kumar and Graham (2000a) and submitted for publication in J. Fluid Mech.

137 117 analysis can capture the main features of these localized solutions in circular Couette flow. 5.1 Introduction Spatially localized structures are common in pattern forming physical systems (Cross and Hohenberg, 1993). Such patterns are interesting and important because they are an indication of significant nonlinear effects, and their interaction with other patterns may give information on spatiotemporal behavior. Examples of oscillatory localized structures can be found in binary liquid mixtures (Moses et al., 1987; Heinrichs et al., 1987; Kolodner et al., 1988), parametrically excited surface waves (Wu et al., 1984), elastic media (Wu et al., 1987), granular media (Umbanhowar et al., 1996; Lioubashevski et al., 1996; Fineberg and Lioubashevski, 1998), and colloidal suspensions (Lioubashevski et al., 1999). Recently, stationary, two dimensional finite amplitude localized states have been computed in Newtonian plane Couette flow (Cherhabili and Ehrenstein, 1995, 1997). These solutions are isolated from the base Couette flow branch and were computed by numerical continuation of traveling wave solutions in plane Poiseuille flow. Although unstable, these may be related to coherent structures observed in turbulent plane Couette flow. In flows of viscoelastic liquids, long wavelength structures were first observed by Beavers and Joseph (1974) in a circular Couette device. These structures, termed tall Taylor cells, are primarily inertia driven patterns (Taylor vortices) modified by elasticity. Similar patterns have been computed by Lange and Eckhardt (2000). In contrast, the structures seen by Groisman and Steinberg (1997), Groisman and Steinberg (1998),

138 118 and Baumert and Muller (1999) are driven purely by elasticity, since the Reynolds number is negligibly small in their experiments. There are three interesting aspects to their observations: (1) isothermal linear stability analysis in this geometry never predicts stationary bifurcations, (2) these vortex pairs, dubbed diwhirls by Groisman and Steinberg (1997), and flame patterns by Baumert and Muller (1999) and are very localized, i.e., there does not seem to be a selected axial wavelength for these patterns, and (3) the transition back to the base Couette flow is hysteretic, i.e., the shear rate at which the Couette flow base state is recovered is much lower than the onset point at which it loses stability. Here, we seek answers to the following questions, motivated by these observations: (1) Do isolated branches of stationary solutions exist in a simple model for a viscoelastic fluid? (2) Are such solutions, if they exist, localized in space? (3) Can the results from the computations be used to postulate a self sustaining mechanism for these structures? We address these questions by fully nonlinear computations of the branching behavior of an isothermal inertialess Finitely Extensible Nonlinear Elastic (FENE) dumbbell fluid in the circular Couette geometry. Our computations show that an isolated branch of stationary solutions does indeed exist in the circular Couette geometry. In common with the experimentally observed patterns (which, adopting the nomenclature of Groisman and Steinberg (1997), we term diwhirls), they are long wavelength solutions, exhibit significant asymmetry between radial inflow and outflow, and show hysteresis. In addition, these solutions persist at arbitrarily large wavelengths: some of the solutions we have computed have an axial wavelength that is more than a hundred times larger than the gap width. We also use the results from our computations to propose a self-sustaining mechanism for these patterns. Along with the circular Couette flow base state, these structures

139 119 may form the building blocks for complex spatiotemporal dynamics in the flow of elastic liquids, such as the recently observed phenomenon of elastic turbulence (Groisman and Steinberg, 2000). In addition, they may be linked to localized defects seen in polymer processing operations and possibly to the strongly nonlinear and long-wave features observed in free surface flows (Grillet et al., 1999). In the previous chapter, we discussed the mechanisms by which elastic instabilities arise in Dean and circular Couette flows due to the inward radial force associated with tensile stresses along curved streamlines. As mentioned earlier, non-isothermal effects can give rise to stationary, axisymmetric bifurcations. However, the shear rates at which these bifurcations occur are an order of magnitude lower that those at which the diwhirls are observed. Therefore, the non-isothermal mechanism appears to have limited relevance for the diwhirls, but seems to explain the very weak stationary vortices seen experimentally by Baumert and Muller (1995, 1997). Hereafter we consider isothermal flow. The absence of a stationary bifurcation from the circular Couette flow base state means that any branch of stationary solutions that exists in this flow must be isolated from the base state flow, i.e., there can be no direct path from the base state flow to this branch of solutions. One way of accessing such an isolated branch is to use a technique known as homotopy. The idea in this technique is to start with a problem different from the original one, but whose solution has the desired properties. For example, the modified problem may be easier to solve than the original one. After the solution to the modified problem is computed, it is tracked as the problem is morphed to the original one. Recently, homotopy has been used to find isolated solutions in plane Couette flow (Waleffe, 1998). For our problem, we seek a flow whose base state has a stationary bifurcation, and which can be easily morphed to circular Couette flow. Clearly, Dean flow is a very good

140 120 candidate to satisfy this criterion, since the flow geometry is identical to that of circular Couette flow, and there is a stationary bifurcation from the base state for a wide range of parameters. Starting from Dean flow, we can approach circular Couette flow in a smooth way by progressively decreasing the pressure drop, while simultaneously increasing the rotation speed of one of the cylinders. The linear stability characteristics of viscoelastic Couette-Dean flow were studied by Joo and Shaqfeh (1992b). As we might expect, this flow is unstable to a stationary axisymmetric mode when the pressure gradient is the dominant driving force, whereas a non-axisymmetric oscillatory mode is the most dangerous when cylinder rotation dominates. Work on nonlinear analysis in viscoelastic circular Couette and Dean flows has concentrated on regimes close to the bifurcation point - there have been no extensive computational studies of fully nonlinear behavior in these flows. Renardy et al. (1996) conducted a nonlinear analysis to study mode interactions arising from the introduction of inertia into the system. Graham (1998) performed a weakly nonlinear analysis to determine the criticality of the bifurcation in circular Couette flow in the narrow gap limit upon addition of axial flow. Later, Ramanan et al. (1999) extended this analysis to Dean flow. Khayat (1999) used a low dimensional model in an attempt to determine the dynamical behavior in purely elastic and inertio-elastic circular Couette flow. It should be noted, however, that stress localization (a striking example of which will be seen below) is common in viscoelastic flows, and it is questionable whether a simple low dimensional model, which is essentially a low-resolution Galerkin projection, can adequately capture such behavior. The strategy we adopt to search for isolated branches of stationary solutions in circular Couette flow is a fully nonlinear analysis of the governing equations. We use a

141 121 numerical continuation procedure (Seydel, 1994) to trace out stationary nontrivial solutions bifurcating from the trivial branch in Dean or Couette-Dean flow and see if these solutions persist as a parameter is varied smoothly to change the flow from Dean or Couette-Dean to pure circular Couette flow. Any such stationary solutions that persist in the limit of circular Couette flow have to be part of an isolated branch since there is no stationary bifurcation from the base state isothermal circular Couette flow. In the remainder of the chapter, we report our procedure and results as follows. Section 5.2 contains a discussion of the geometry, governing equations, and scalings that are used in the computations. In section 5.3, we present a discussion of the discretization scheme and the numerical method that we use to solve the sparse linear systems arising in the Newton iterations during the continuation process. This section includes discussion on a preconditioner that we have found to be especially useful. In section 5.4, we discuss the results of continuation in the various parameters, mechanism of the diwhirl solutions we compute, their stability with respect to time dependent axisymmetric and non-axisymmetric disturbances, and present a quantitative comparison of our computed diwhirls with experimental data (Groisman and Steinberg, 1998). Finally, we conclude in section 5.5 with a summary of our main findings. 5.2 Formulation We consider the flow of an inertialess polymer solution between two concentric cylinders (figure 49). The inner cylinder has radius R 1 and the outer cylinder has radius R 2.The fluid has a relaxation time λ; the polymer and solvent contributions to the viscosity are

142 122 R 2 K θ Ω R 1 Figure 49: Geometry of Couette-Dean flow in an annulus denoted respectively by η p and η s, with the ratio η s /η p denoted by S. The solution viscosity η t, is given by the sum of the solvent and polymer viscosities, η s + η p. The flow is created by a combination of the motion of the inner cylinder at a velocity ΩR 1 and by the application of an azimuthal pressure gradient K θ = P/ θ. The equations governing the flow are the dimensionless momentum and continuity equations τ p + We θ S 2 v =0, (124) v =0, (125) where v is the velocity, p is the pressure and τ is the polymer stress tensor. The polymer molecules are modeled as dumbbells connected by finitely extensible springs. Approximate constitutive equations for this model include the FENE-P equation (Bird et al., 1987b) and the FENE-CR equation (Chilcott and Rallison, 1988), which were described

143 123 in chapter 1. In dimensionless form, they are We θ ( QQ t ) + v QQ { QQ v} t { QQ v} ( ) QQ + (1 tr( QQ )/b) I (1 c r tr( QQ )/b) =0, (126) where QQ is the ensemble average of the polymer conformation tensor, b is a dimensionless measure of the maximum extensibility of the dumbbells (the dimensionless form of Q 0 in chapter 1), We θ is the Weissenberg number, which is the product of the polymer relaxation time and a characteristic shear rate, and c r is a parameter which takes the value 1 for the FENE-CR model and 0 for the FENE-P model. Values of c r between 0 and 1 do not correspond to any standard constitutive equation; this parameter merely serves as a convenient way of performing numerical continuation between the FENE-P and FENE- CR equations. We have already discussed the behavior of the FENE-P and FENE-CR models in shear and extension in chapter 1. Here, we simply mention that the FENE- P model has been found to better approximate the behavior of the kinetic theory based FENE model in steady shear and elongational flows than the FENE-CR model (Herrchen and Öttinger, 1997). Given the differences between the FENE-P and FENE-CR models even in simple flows, we would expect them to exhibit different behavior in complex flows as well, and our computations confirm this. For both models, Q 2 0 and the components of QQ are scaled by kt/h, where k is Boltzmann's constant, T is the temperature, and H is the spring constant. Distance is scaled by the gap width, and time by the inverse of a characteristic shear rate. Since the FENE-P model does not yield an analytical solution for the base state in Couette- Dean flow, we take the characteristic shear rate to be the shear rate at the outer cylinder for an Oldroyd-B fluid i.e., 1/b = 0) flowing through the same geometry. The

144 124 velocity scale is chosen to be the product of the time scale and the gap width. The polymer stress, scaled by the shear modulus, is obtained from QQ using the relation τ = QQ /(1 tr( QQ )/b) I/(1 c r tr( QQ )/b). Other parameters of importance are the dimensionless gap width ɛ =(R 2 R 1 )/R 2, and δ, which measures the relative importance of the pressure gradient as the driving force for the flow, given by δ = K θ ɛ 2 R 2 /(2 η t ) (1 ɛ)r 2 Ω K θ ɛ 2 R 2 /(2 η t ), (127) so that δ = 0 is circular Couette flow and δ = 1 is Dean flow. Explicit forms for the scalings used are presented in Appendix C. The velocity satisfies no slip boundary conditions on the walls of the cylinder. 5.3 Discretization and solution methods Equations 124, 125 and 126 form a set of partial differential equations for the three components of the velocity, the pressure, and the six components of QQ. We look for steady, axisymmetric solutions that are periodic in the axial direction with a dimensionless period (scaled by the gap width) of L, so each variable only depends on two spatial directions, the radial direction r (shifted and scaled so that r =0is the inner cylinder and r =1the outer cylinder), and the axial direction z. In performing the numerical discretization, we can take advantage of certain symmetry properties of the solutions we seek. In particular, we take the radial and azimuthal velocities to be reflection symmetric about the plane z = L/2, and the axial velocity to be reflection anti-symmetric. This implies that QQ rr, QQ rθ, QQ θθ, QQ zz, and p are reflection symmetric, while QQ rz and QQ θz are reflection anti-symmetric. Thus, the computational domain is Γ ={0 r 1, L/2 z L}, which is half the size of the physical domain.

145 125 Considerable care needs to be exercised when choosing a discretization scheme. Since we are looking for localized solutions, our primary consideration is to choose a discretization scheme that can place a high concentration of points in regions of strong velocity and stress localization. In our work, we experimented with two discretization schemes. The first scheme that we used was a global spectral method. Spectral methods enjoy the very desirable property of exponential convergence as long as all the features of the solution are captured. For our problem, we used Chebyshev discretization in both directions. In the axial direction, we implemented the symmetry conditions by choosing odd Chebyshev interpolants for the reflection anti-symmetric components, and even Chebyshev interpolants for the reflection symmetric components. Chebyshev interpolants work better than Fourier interpolants in the periodic direction because they have an uneven distribution of collocation points. Regions at the two ends of the domain have a higher concentration of collocation points, and we can get an improvement in convergence compared to Fourier method by making these regions coincide with the areas of stress localization. The early computations were performed with this global method. However, we found that as the degree of localization of the solutions increased, very high order polynomials were needed, which resulted in unacceptable increases in memory requirements, and presented difficulties in solving the linear systems associated with the continuation scheme outlined below due to the poor condition number of the matrices. Our experience with the spectral method indicated that the appropriate discretization scheme would permit efficient local concentration of points. To this end, we attempted a domain decomposition Chebyshev collocation scheme (Canuto et al., 1988) in which we split the computational domain into four conforming rectangular sub-domains and used Chebyshev collocation in both directions in each of the domains. The interface conditions

146 126 for each variable were continuity of the value of the variable and its normal derivative between the domains. Since the governing equation for QQ is hyperbolic, continituity of the normal derivative is not technically required for its components. However, we found that the solutions on the non-trivial branch would not converge unless this condition was imposed. Even when these conditions were imposed, the performance of this scheme was worse than the global spectral method. One reason could be that the interface conditions on the components of QQ impose stronger continuity requirements than needed, thus inhibiting convergence. Also, there is a degree of arbitrariness in the direction of the normal derivative at the corner point common to all four domains. The next scheme that we tried was the spectral element method (Patera, 1984). This method, which we describe below, allows efficient local refinement by subdividing the domain into several sub-domains or elements. Within each element, the solution components are approximated by tensor products of high order orthogonal polynomials. As long as the solution components within each element are well resolved, this scheme preserves the exponential convergence properties of the global spectral method. As will be evident from the description below, interface and corner points are treated naturally within this formulation: the governing equations at these points are the sum of contributions from all elements that border them. We found that this method was superior to both the global spectral method as well as the domain decomposition collocation scheme in terms of resolving the stress and velocity localization. In addition, we found that the linear systems associated with the continuation scheme presented later in this section are also better conditioned and more sparse for a spectral element scheme than for a global spectral method. Finally, the computational domain is rectangular in shape, which makes this method relatively easy to implement.

147 127 In the spectral element formulation, we apply Galerkin weighting on the conservation and continuity equations (i.e., the test and weight functions are the same), and streamline-upwind/petrov-galerkin weighting (Brooks and Hughes, 1982; Marchal and Crochet, 1987) on the constitutive equations. The formulation, which is based on the weak form of the governing equations, is given by ( p I + τ + We θ S v) : u dγ = u ( pi + τ + We θ S v) nds, u U, Γ Γ Γ (128) q vdγ =0, q Q, (129) ( ( ) QQ We θ + v QQ { QQ v} t { QQ v} + Γ t ) QQ (1 tr( QQ )/b) I ( w + c v w) dγ =0, w W, (1 c r tr( QQ )/b) (130) where U H 1 (Γ), the space of functions whose first derivatives are square integrable over Γ, W, Q L 2 (Γ), the space of functions which are square integrable over Γ, and c = h /V where h is a characteristic length scale of an element, and V is a characteristic velocity. We take h to be the square root of the area of the element, and V to be the average of the magnitude of the velocity at the four corners of each element. We take u, q, and w to be the same as the interpolating functions for the velocity, pressure and polymer stress components. These functions must be chosen to satisfy the symmetry conditions discussed above. In each element, the variables are approximated by tensor products of Lagrange polynomials defined on the Gauss-Lobatto-Legendre (GLL) grid. We take the symmetry properties into account by treating the elements bordering the axial edges in a different way

148 r Figure 50: A spectral element mesh 2.5 with 16 3 axial 3.5 and 16 radial 4 elements 4.5 with fifth order z polynomials in each direction in each domain. Note the dense concentration of points near r =1and z = L/2. The high resolution is necessary to capture the intense stress localization in these regions. than the interior elements. Since the GLL grid is defined in the domain { 1 ξ 1}, we map each interior element to { 1 ξ r 1, 1 ξ z 1}. We map the elements bordering the left edge (z = L/2) to the range { 1 ξ r 1, 0 ξ z 1}, and use even axial interpolants for the reflection symmetric components and odd interpolants for the reflection anti-symmetric components. Similarly, we map the elements bordering the right edge (z = L) to { 1 ξ r 1, 1 ξ z 0} and use even or odd interpolants as appropriate. A sample spectral element mesh is shown in figure 50. In order to avoid spurious pressure modes, the relative approximation orders of velocity and pressure need to satisfy the Ladyzheskaya, Babuska and Brezzi condition. We ensure this by using interpolants for the pressure that are based on a grid that has two

149 129 fewer points in each direction than the velocity grid in each element. We choose the approximation orders for the components of QQ to be the same as for the velocity. We perform the integrations in equations 128 to 130 using Gauss-Legendre quadrature on the GLL grid in each element and construct the final system by direct stiffness summation. This procedure reduces the system of nonlinear partial differential equations to a system of nonlinear algebraic equations for the nodal values of the variables on the appropriate GLL grid in each element. These equations can be written in compact form as E y t = f(y,we θ,s,ɛ,b,l,δ). (131) The matrix E has zeros in the rows corresponding to the momentum and continuity equations. Steady states correspond to solving f(y,we θ,s,ɛ,b,l,δ)=0. (132) Solutions to equation 132 are tracked using a numerical continuation procedure, the starting point for which is the base state Oldroyd-B solution. This solution is used as an initial guess for the FENE-P or FENE-CR base state solution and refined using a Newton iteration. We calculate subsequent points along the branch using a pseudo-arclength continuation algorithm (Seydel, 1994), which we briefly describe here. Let us denote the set of the values of the variables at the collocation points by the vector y, and the continuation parameter by µ. Here, µ could be We θ, b,orl. In pseudo-arclength continuation, we consider both y and µ to be functions of a step length parameter s. Thus, we can write the set of discretized equations in the compact form f(y(s),µ(s)) = 0. (133)

150 130 Given a point (y 0,µ 0 ) on the solution branch, the idea is to find the next point (y 1,µ 1 ) such that, apart from satisfying the governing equations, it obeys an additional constraint N(y,µ)=ẏ 0 (y y 0 )+ µ 0 (µ µ 0 ) s =0, (134) where (ẏ 0, µ 0 ) is the unit tangent at the point (y 0,µ 0 ), and s is a specified step length. This equation requires that the next computed solution point lie a distance s from the current point, in the direction of the tangent to the solution curve. At each step, we use a Newton iteration to solve the augmented set of equations f(y(s),µ(s)) = 0, (135) N(y(s),µ(s)) = 0. The Jacobian matrix of this system is given by J = and is not singular at a turning point. f y f µ ẏ 0 µ 0 (136) While tracing a solution branch, we check for stationary bifurcations using a test function method (Seydel, 1994). The test function is a scalar function that changes sign at a stationary bifurcation point, and is relatively inexpensive to compute. Suppose that (y 0,µ 0 ) is a stationary bifurcation point. Then, it follows that f y (y 0,µ 0 )h = 0, (137) where h is the eigenvector corresponding to the zero eigenvalue of f y (y 0,µ 0 ). Suppose now that we are at a point (ȳ, µ) different from the bifurcation point. Then equation 137 with (y 0,µ 0 ) replaced by (ȳ, µ) has no nontrivial solution for h. However, we can get

151 131 a solution to an equation that resembles this closely. We arbitrarily choose two indices l and k, and require that h k =1. We do this by replacing the l th row in equation 137 by the equation e t k h =1, where e k is the column vector with a 1 in the k th position and zeros elsewhere. After this substitution, equation 137 becomes J lk h = e l, (138) where J lk is the matrix obtained after performing the indicated substitutions in f y.ifwe are exactly at the bifurcation point, then h is simply the eigenvector corresponding to the zero eigenvalue, normalized so that its k th component is 1. If equation 138 is solved close to a bifurcation point, then h is a good approximation to the eigenvector corresponding to the zero eigenvalue. In particular, the scalar function t lk = e t lf y (y,µ)h (139) is zero at a bifurcation point, and changes sign as a bifurcation point is crossed. We use t lk with l = k to check for stationary bifurcations. If a stationary bifurcation is detected, as for example when the trivial branch in Dean flow becomes unstable, we need to begin tracking the new branch. To compute a first approximation to a point on the new branch, we use the fact that h closely approximates the eigenvector corresponding to the zero eigenvalue, and write z = ȳ + δ 0 h, (140) for some small value of δ 0, as an approximation to a point on the new branch. However, if we perform a Newton iteration starting with z as the initial guess, we will likely converge back to the old branch. Instead, we perform a Newton iteration on the augmented system

152 132 of equations f(z,µ) z k z = 0 (141) to solve for a point (z,µ ) on the new branch. In equation 141, we are simply specifying the value of a solution component on the new branch and solving for the point (z,µ ) where this holds. The method is especially effective when switching from the base solution to the nontrivial branch because we can choose k to be a component of v r or v z, both of which are zero in the base state. The question of determining stability with respect to oscillatory disturbances can be divided into two parts. For axisymmetric disturbances, we need to find the eigenvalues ω of the generalized eigenvalue problem f y q = ωeq. (142) If any of the eigenvalues have positive real parts, the solution is unstable with respect to disturbances that have the same symmetry properties that y does and have wavelengths L such that L/L is an integer, otherwise it is stable with respect to such disturbances. We do not attempt here the much more demanding task of determining stability with respect to disturbances of arbitrary wavelength. Since we are only interested in determining stability, we do not need to find the entire spectrum of eigenvalues. It is only necessary to check if any of the ω have positive real parts. To do this, we use an Arnoldi scheme, as implemented in the public domain software package ARPACK (Lehoucq et al., 1997), to calculate a few such eigenvalues. Since ARPACK does not have a built in option to calculate a specified number of eigenvalues with positive real parts, we use a spectral transformation suggested by Christodolou and Scriven (1988). Using this transformation,

153 133 we find the eigenvalues κ of the matrix P =(E f y ) 1 (E + f y ). These eigenvalues are related to the the eigenvalues ω in equation 142 by means of the transformation κ i = 1+ω i 1 ω i. (143) The eigenvectors of P are identical to those of equation 142. This transformation maps the eigenvalues in the left half of the complex plane to the interior of the unit circle. Thus, the eigenvalues of equation 142 with positive real parts map on to the eigenvalues of P with the largest magnitude, and are easily found by ARPACK. We should mention here that the Arnoldi scheme constructs a Krylov subspace by the successive action of P on a vector. As is evident from the definition of P, the construction of each such vector requires the solution of a linear system. Thus, the eigenvalue computation is an expensive process, and we only perform it for a few points. The process for determining stability with respect to non-axisymmetric modes is somewhat more complicated. We write the solution vector φ as φ(r, z, θ) = φ(r, z)+ɛ φ(r, z) exp(ct+ inθ), (144) where ɛ φ is a small perturbation, c is the growth rate, and n is the azimuthal wavenumber of the perturbation, assumed to be an integer. Substituting this in the governing equations, and retaining terms at O(ɛ ) gives a complex generalized problem for the growth rate c: J ũ = c E ũ. (145) As in the axisymmetric case, we can reduce this to a regular eigenvalue problem using the spectral transformation K =(E J) 1 (E + J). At this point, it is clear that every step of the procedure involves the solution of a

154 134 system of linear equations. Let us denote the generic equation we solve as Ax= b. (146) It is well worth expending effort to make the solution process efficient. Firstly, A is a sparse matrix, so considerable savings in memory result from just storing the nonzero entries together with integer pointer arrays that store information about the coordinates of each stored entry. The sparse nature of A also means that a properly implemented iterative method could be more efficient at solving equation 146 than a direct scheme. The iterative scheme we use is GMRES (Saad and Schultz, 1986). However, GMRES will converge only if A is well conditioned, and this is generally far from being true for the systems we solve. Therefore, we solve a preconditioned system MAx= Mb, (147) where M is an approximation to A 1. In the remainder of this section, we present a discussion on some of the preconditioners that we have used. The first preconditioner that we describe is the incomplete LU decomposition preconditioner with zero fill level, or ILU(0) for short. This preconditioner is a sparse version of the full LU decomposition algorithm, with sparsity being preserved by only keeping those entries in the L and U matrices where the corresponding positions in A are nonzero. This technique gives no consideration at all to the size of the entries that are dropped, so we would not expect this to be a good preconditioner. However, our experience has been that this preconditioner is surprisingly robust, and works reasonably well up to b = 1500, where b is the finite extensibility parameter. At this value of b, we need about twenty five thousand degrees of freedom to resolve the components of QQ, and solving these linear systems takes about 450 GMRES iterations. At larger values of b, however, ILU(0)

155 135 fails due to numerical instability: the dropping of entries causes the buildup of very large or very small numbers in the LU decomposition, which makes it impossible to complete it. Pivoting actually makes the situation worse by giving rise to zero pivots. Although we can complete the decomposition by replacing these with small numbers, the resulting matrix is useless as a preconditioner. The discussion above suggests that a preconditioner that drops entries based on numerical size would be more effective than ILU(0). A variant of ILU which takes this into account is threshold ILU, or ILUT (Saad, 1996). In this method, entries in the incomplete LU decomposition are dropped based on tolerances applied at two different stages. The algorithm is shown below, where we denote the dimension of the matrix by N and use the symbols l and u to represent the entries in the L and U matrices respectively. In each entry, the first subscript represents the row, and the second represents the column. The algorithm used a one-dimensional work array of length N, which we denote by w below. 1. Do i=1, N 2. w:=a i, (Copy the nonzero entries of row i into w) 3. Do k=1, i 1 and when w k 0 4. w k := w k /u kk 5. Apply a dropping rule to w k 6. If w k 0then 7. w j = w j w k u kj 8. EndIf

156 EndDo 10. Apply a dropping rule to row w 11. l ij = w j for j =1,..., i u ij = w j for j = i,...,n 13. w := EndDo The decomposition is sparse because of the dropping rules applied in steps 5 and 10. The usual choice in step 5 is to keep entries which are larger than some fraction (tol) of the norm of the row in the original matrix. In step 10, it is usual to keep the p entries which have the largest magnitude in order to control the amount of fill in. In using ILUT, we found that the algorithm would generate zero pivots unless p was made unacceptably large. Again, we can complete the decomposition by replacing these pivots with small numbers, but the resulting preconditioner is completely ineffective. We found that this problem could be overcome by performing ILUT on a modified version of A. We construct this matrix, denoted by Ã, by setting to zero in A the entries corresponding to the velocities and pressure in the rows corresponding to the constitutive equation, and the entries corresponding to the components of QQ in the rows corresponding to the momentum and continuity equations. We then construct an ILUT decomposition of à and apply this as a preconditioner for A. Wefind that this is a very effective technique, and the largest linear systems that we have solved (O(60000) unknowns) converge to a relative accuracy of 10 6 in about 350 GMRES iterations, with the preconditioner having about three times the number of nonzero entries that A does. Smaller systems converge

157 137 faster and fewer nonzero entries can be kept in the preconditioner. Henceforth, we shall refer to this preconditioner as ILUT. Figure 51 shows a comparison between ILU(0) and ILUT for a sample problem, which clearly demonstrates the superiority of the ILUT preconditioner over ILU(0) ILUT * ILU(0) Log 10 (residual) iterations Figure 51: Comparison of ILUT and ILU(0) preconditioners. The test problem was the calculation of the unit tangent for a point on the nontrivial branch in Dean flow. The matrix A had a dimension of In using ILUT to solve the complex generalized eigenvalue problem indicated by equation 145, we need to modify the problem so that it only involves real numbers. Recall that the spectral transformation requires the solution of the linear system (E J) x = b (148) for each iteration of ARPACK. Instead, we rewrite equation 145 as a real valued problem c E 0 0 E u r u i = J r J i J i J r u r u i, (149)

158 138 where the subscripts r and i denote respectively the real and imaginary parts of the vector or matrix. The spectral transformation now requires the solution of linear systems involving the matrix S = (E J) r (E J) i (E J) i (E J) r, which is real. To precondition this matrix, we note that the matrix (E J) i has far fewer entries than (E J) r. Therefore, as a first approximation, we can neglect it in computing the preconditioner. We construct the preconditioner by performing ILUT on (E J) r. If M represents this decomposition, we precondition S using the matrix M M Results and discussion Stationary bifurcations from the Dean and Couette-Dean flows Rather than explore large volumes of parameter space, we restrict our attention to values close those used in the experiments by Groisman and Steinberg (1997). Specifically, we fix the value of S at 1.2, and except when we examine the effect of varying the gap width, set ɛ =0.2. For most of our work, we use the FENE-P model which has been shown to be a better approximation to the exact FENE model (Herrchen and Öttinger, 1997), at least in simple flows. In order to track stationary nontrivial branches, it is first necessary to find out from where they bifurcate. Therefore, the logical starting point of our investigation is the

159 139 linear stability diagram for Dean flow. Figure 52 shows such a diagram computed using the FENE-P model. Takens-Bogdanov points, where the bifurcation switches from a stationary mode to an oscillatory mode, are marked as TB. Unlike in the Oldroyd-B model, where one such point is seen only at very small wavelengths (Ramanan et al., 1999), we see that, as the polymer becomes stiffer (i.e., b decreases), these points are shifted to larger wavelengths. We also see that for a sufficiently small value of b, there is no stationary bifurcation from the base state at all. From the point of view of numerical simulation, we would prefer to work with as small a value of b as possible, since we would expect stress boundary layers to be less sharp for smaller values of b, which in turn makes the computations easier. However, decreasing b tends to lower the elastic character of the fluid and suppresses elastic instabilities. Before moving on to the nontrivial solutions, we present some linear stability results for Dean flow of the FENE-CR model in figure 53. Note that, unlike in the FENE-P model, stationary bifurcations are seen even at low values of b. Thus, the FENE-CR model predicts linear stability behavior that is qualitatively different from that predicted by the FENE-P model. We will have more to say on the differences between the two models in section For now, we simply concentrate on the FENE-P model The branch structure of viscoelastic Couette-Dean flows We begin the nonlinear analysis by tracking the bifurcating branch of stationary solutions in Dean flow at b = 700 and L =1.05. Since this value of b is small, numerical continuation is not difficult and a crude numerical scheme suffices. We performed these calculations using a global Chebyshev collocation scheme in both the radial and axial directions. At L =1.05, a pair of complex conjugate eigenvalues crosses the imaginary

160 TB TB b=600 b=700 b=800 b= TB We θ 25.0 Unstable 20.0 Stable α Figure 52: Linear stability curves at δ =1(Dean flow) computed using the FENE-P model. The points marked TB are Takens-Bogdanov points. The lines correspond to points where the base state flow loses stability to stationary axisymmetric perturbations b=500 b=600 b=700 We θ π/L Figure 53: Linear stability curves at δ =1(Dean flow) computed using the FENE-CR model. As in figure 52, only stationary axisymmetric perturbations are considered. Note the complete absence of non-stationary bifurcations.

161 141 axis at We θ = Upon further increasing We θ, the two unstable eigenvalues coalesce and form a pair of unstable real eigenvalues which then move in opposite directions. The smaller one of these re-crosses the imaginary axis at We θ = We track in We θ the stationary branch bifurcating as result of this crossing. The result is shown in figure 54, where we plot the solution amplitude, measured by the quantity v r = ( Nr ) N 1/2 z v r,ij 2, (150) i=0 j=0 as a function of We θ, where N r +1and N z +1are the number of Chebyshev collocation points used in the radial and axial directions respectively. We see that the bifurcation is mildly supercritical, but quickly turns back and shows a marked hysteretic character. The turning point at We θ =22.34 is much lower than the value of where the base solution loses stability. When we pick a point on this branch and continue it down in δ, wefind however, that it does not extend all the way to δ =0. For instance, picking We θ =23.35 on the upper branch in figure 54 and tracking it in δ,wefind that the branch turns back at δ =0.69. We have tried this for other points as well, but in all cases, they turn back well before reaching δ =0. Therefore, at least for this value of b and L, there is no direct path from nontrivial solutions in Dean flow to those (if any) in circular Couette flow. Given the apparent absence of a direct route from δ = 1 to δ = 0, we focussed on smaller values of δ and larger values of b and L. As b increases, the solutions become more localized, and the global scheme we originally used is inefficient. Hence, we switched to the spectral element/supg method described in section 5.3. At δ =0.576, b = 1830, and L =2.71, a stationary bifurcation occurs at We θ = We tracked the bifurcating branch at this value of We θ up to L =3.07 and then

162 v r Hopf point We θ Figure 54: Continuation in We θ of a stationary solution in Dean flow. The parameter values are L = 1.05, b = 700, ɛ = 0.20, and S = 1.2. At the Hopf point, a pair of complex conjugate eigenvalues become unstable. These collide and form two real eigenvalues, one of which re-crosses the imaginary axis at We θ = 29.57, where the stationary branch originates. The solution amplitude used here differs from that used in subsequent figures and is defined in equation 150.

163 Multiple steady states in circular Couette flow v r δ=0 Continuation to L= δ Figure 55: A path to stationary solutions in circular Couette flow. The parameters are We θ =25.15, L =3.07, b = 1830, ɛ =0.2, and S =1.2. down in δ. This path is shown in figure 55. The velocity norm used in this and all subsequent figures is defined as ( 1/2 v r = vr dγ) 2. (151) Γ As figure 55 shows, we found that this branch persists all the way to δ =0. This computation demonstrates that an isolated branch of nontrivial solutions does indeed exist in the circular Couette geometry. This is the first time that stationary nontrivial solutions have been computed in zero Reynolds number circular Couette flow Nontrivial stationary solutions in Couette-Dean flow - Diwhirls We now proceed to discuss the effect of changing parameters on the stationary solution at δ =0. The first parameter we focus on is the wavelength. The results of continuing

164 144 our solution at We θ =25.15 in L are shown in figure 56. The upper branch, being a stronger flow, is much harder to track than the lower branch, and the end point of this branch represents the largest value of L at which we could obtain converged solutions on the upper branch for this value of We θ. The lower branch presents fewer problems and we were able to track it with relative ease. The key observation from figure 56 is that as L increases, both the lower and upper branches become flat, suggesting that the spatial patterns are becoming independent of the size of the computational domain, i.e., they are becoming localized. Examination of the solution components confirmed that this was indeed the case, with the localization occurring in the region near z = L/2. Since the components of the solution show little or no axial variation far away from z = L/2, we can simply use their values at the collocation points for a lower value of L as an initial guess for the solution at a larger value of L, while increasing the axial extent of the domains bordering the edges (i.e., z = L). This method of remeshing captures the localization effectively and avoids the necessity of computing solutions at intermediate values of L. Using this technique, we were able to get converged solutions on the lower branch for wavelengths that are in excess of 100 times the width of the gap between the cylinders. We have also used this technique to compute such long wavelength solutions on the upper branch at lower values of We θ. We show the results of one such computation in figure 57. This figure shows the streamfunction contours and a density plot of QQ θθ at L = and We θ = For clarity, we only show the center and edges of the domain. The streamfunction contours are strongly localized near the center of the flow cell, which is a region of very strong inflow. Away from the core is a region of weak outflow, and even further away, the solution is pure circular Couette flow. The QQ θθ field shows an even stronger localization. It is the necessity of capturing this

165 v r L Figure 56: Results from continuing the stationary circular Couette flow solutions in L. The parameters are We θ =25.15, b = 1830, ɛ =0.2, and S =1.2. The gaps in the lower branch correspond to places where we changed the mesh. Note the flatness of the branches as L increases. We have computed extensions of the upper branch at lower values of We θ. strong localization that requires the use of a numerical method that permits efficient local refinement. The streamlines at the core are remarkably similar to those in figure 10 on page 2457 of Groisman and Steinberg (1998). Henceforth, we will call our solutions diwhirls as well. Figure 58 shows the results of continuation in We θ for solutions at three different values of L. We see that all three curves show turning points in We θ, i.e., there is a lower limit in We θ below which the diwhirls are not seen. Note that the curves at L =9.11 and 4.74 are close together, and are both well separated from the curve at L =3.07. This further highlights the independence of the solutions on L for large enough L. At L =3.07, the critical We θ at which the base circular Couette flow solution loses stability to an axisymmetric time-dependent mode is 29.65, which is significantly higher than the

166 146 Outer cylinder 1 Inner cylinder Outer cylinder Inner cylinder L= Figure 57: Density plot of QQ θθ (white is large stretch, black small) and contour plot of the streamfunction at L = (We θ =24.29, b = 1830, S =1.2, and ɛ =0.2). For clarity, most of the flow domain is not shown. Note the very strong localization of QQ θθ near the center. The maximum value of QQ θθ at the core is 1589 which gives τ θθ = Compared to this, the maximum value of QQ θθ in the circular Couette base state is 706, which gives τ θθ = Away from the core of the diwhirl, the structure is pure circular Couette flow. The streamlines show striking similarity to those in figure 10 of Groisman and Steinberg (1998). This point was generated by stretching the point at the corresponding We θ on the upper branch of the curve for L =9.11 in figure 58.

167 v r L=9.11 L=4.74 L= We θ Figure 58: Diwhirl solution amplitudes as functions of We θ and L. Note that the curves at L =9.11 and L =4.74 are very close together, while both curves are well separated from the curve at L =3.07 (b = 1830, S =1.2, and ɛ =0.2). location of the turning point, located at The critical We θ for the other values of L are even higher. Therefore, the overall bifurcation structure shows hysteresis. In figure 59, we show a plot of the location of the turning point in Weissenberg number (We θ,c ) as a function of the wavelength. The most interesting feature in figure 59 is the flatness of the curve at large L, indicating yet again that for large L, the characteristics of the solution are independent of the wavelength. Another interesting feature in figure 59 is that the curve shows a minimum, i.e., the diwhirl patterns exhibit wavelength selection. This minimum, which occurs at a Weissenberg number of approximately 23.3, is therefore the lowest Weissenberg number at which the FENE-P model with the chosen parameters predicts diwhirls to occur. More important than the absolute value is the relative position of the turning point and linear stability limits. The base state circular Couette flow is unstable with respect to axisymmetric disturbances above We θ =20.37.

168 148 This means that all the solutions that we compute lie above the linear stability limit of circular Couette flow. In contrast, Groisman and Steinberg (1998) observe diwhirls at Weissenberg numbers as low as 10, well below the linear stability limit of the base flow. One reason for this discrepancy could be the approximate nature of the FENE-P model, which does not take into account the internal degrees of freedom of a real long chain polymer. Yet another could be that our numerical simulations have not been able to access a sufficiently high value of b. Infigure 60, we plot the position of the turning point in We θ as a function for b for L =4.74, which is close to the minimum in figure 59. Also plotted in the figure is the minimum critical value of We θ at which the base state circular Couette flow loses stability with respect to axisymmetric perturbations. This figure shows that as b increases, the position of the turning point shifts to lower Weissenberg numbers at a faster rate than the shift in the minimum of the linear stability curve. This result is not unexpected, because the polymer molecules are much more highly stretched in the core of the diwhirl than in the base state. Therefore, we would expect the nonlinearity of the FENE-P spring law to have a greater effect on the diwhirls than on the base state Couette flow. It is conceivable to suppose, based on the results shown in figure 59, that the two curves would cross at larger values of b (which we are not able to access due to limitations in the numerical scheme) and that the diwhirls would come into existence below the linear stability limit of the base flow. We point out here that Baumert and Muller (1999) and Groisman and Steinberg (1997) performed their experiments with very high molecular weight polymers, for which the values of b are likely to be much higher than we have been able to access in our simulations. We now present a rough quantitative comparison of our patterns with those from the experimental observations of Groisman and Steinberg (1998). In figure 9 of their

169 We θ,c 24.0 diwhirls 23.5 no diwhirls L Figure 59: Plot of the location of the turning point, We θ,c, versus L at S =1.2 and ɛ =0.2. Note the flatness of the curve at large L. paper, Groisman and Steinberg present the radial velocity profile as a function of z at a constant radial position near the middle of the gap, where v r has maximum amplitude. To compare our results with this figure, we chose a point on the upper branch of the curve for L =4.74 with We θ =23.50 in figure 58 of this work. We then converted our radial velocity into dimensional units by using values for the physical parameters from Groisman and Steinberg (1998). Specifically, the values we used were λ =1s and R 2 =41mm. For ɛ =0.2 used in our computations, this gives a gap width of 8.2 mm, slightly higher than the 7 mm gap used in the experiments. In figure 61(a), we present a profile of the radial velocity as a function of z for r =0.6, where the radial velocity is maximum. The peak inflow velocity we find is 5.9 mm/sec, which should be compared to the value of 3.8 mm/sec that Groisman and Steinberg (1998) show in their figure 9, which we reproduce here as figure 61(b). The qualitative and quantitative

170 diwhirl onset 24.0 We θ,c linear onset 20.0 Figure 60: Plot of the position of the linear stability limit in circular Couette flow with respect to axisymmetric 19.0 disturbances and the turning point in We θ for the diwhirls as a function of b. The parameters are S =1.2and ɛ =0.2. The computations for the diwhirls b were performed at L =4.74, which is close to the minimum in figure 59.

171 151 similarity between these two values is remarkable, more so when we consider how simple the FENE-P model is, and that the Weissenberg number used in figure 61(a) is roughly twice that at which Groisman and Steinberg (1998) report their results, which means a larger radial velocity should be expected. Furthermore, the wiggles at the shoulders of the peak in figure 61(a) are not numerical artifacts; similar features in figure 61(b) V r, mm/s (a) z, d (b) Figure 61: (a) The axial variation of v r at r =0.6 for L =4.74 and We θ =23.50 on the upper branch. (b) Figure 9 on page 2457 of Groisman and Steinberg (1998), shown here for purposes of comparison. We have shifted the axial coordinate so that the symmetry axis of the computed diwhirl in (a) is at z =0, to make comparison with (b) easier The dimensionless gap width or curvature ɛ plays a critical role in generating elastic instabilities. Based on the generic mechanism of elastic instabilities, we expect a decrease in curvature to have a stabilizing effect, i.e., keeping other parameters fixed, we would expect the diwhirl pattern to vanish at small enough values of ɛ. Infigure 62, we show the dependence of the diwhirl solution amplitudes on ɛ. In agreement with expectations, we observe turning points as ɛ decreases. The role of streamline curvature will become clear in section where we propose a mechanism for the diwhirls. As mentioned in section 5.1, an important reason for attempting to numerically simulate experimentally observed flow patterns is to determine whether a constitutive equation

172 v r L=4.74 L= ε Figure 62: Variation of solution amplitudes with ɛ. Here, We θ =25.15, and the other parameters as in figure 58. can model complex flows of viscoelastic liquids. Both the FENE-P and the FENE-CR equations are derived by applying closures to the evolution equation for QQ for a dilute solution of noninteracting dumbbells connected by nonlinear springs. We have already seen that the linear stability curves predicted by the FENE-P and FENE-CR models show significant differences, and we have seen that the FENE-P model equation has stationary solutions in circular Couette flow, which indicates that it is able to capture, at least qualitatively, the mechanism behind the diwhirls. A natural question to ask is whether the FENE-CR model can do so as well. To this end, we perform a continuation of the diwhirl solutions in the parameter c r, starting from the FENE-P solutions (c r =0). If the solutions persist at c r =1, then we will have obtained solitary solutions for the FENE- CR model in circular Couette flow. Figure 63 shows the results of these computations. Both values of L that we chose exhibit turning points at small values of c r, indicating that these solutions do not exist for the FENE-CR model, at least for the parameter values

173 v r L=3.57 L= c r Figure 63: Diwhirl solution amplitudes as a function of the parameter c r for two different wavelengths. We θ =25.15 and the other parameters are as in figure 58. The existence of turning points demonstrates that these solutions cannot be extended to the FENE-CR model. that we have chosen. Note, though, that in the limit of b, both the FENE-P and FENE-CR models are equivalent to the Oldroyd-B model. Hence, at sufficiently large values of b, the FENE-CR model should yield these localized solutions as well. What these computations show is that the parameter values at which the solutions come into existence depends very strongly on the details of the model being used. The work of Al-Mubaiyedh et al. (2000) has shown that non-isothermal effects introduce a new mode of instability in circular Couette flow. While this mode of instability does not appear to be relevant for the diwhirls, it is still of interest to quantify the effect of viscous dissipation at the core of the diwhirl, where we would expect dissipation to be largest. In dimensionless form the viscous dissipation, scaled by the product of the shear

174 154 modulus G and the characteristic shear rate γ s, is given by the relation ˆΦ = ( τ + We θ S( v + v t ) ) : v. (152) Here ˆΦ represents the work done by viscous dissipation per unit time, per unit volume of fluid. In figure 64, we show a density plot of ˆΦ for a point on the upper branch at L =4.74. For the sake of clarity, we have only shown a twentieth of the wavelength, centered around the symmetry plane. As expected, ˆΦ has its largest amplitude in the core. r z Figure 64: Intensity plot of the dimensionless viscous dissipation for We θ =23.73 on the upper branch at L =4.74. Light areas represent areas of large viscous dissipation and dark areas represent regions of low viscous dissipation. The horizontal axis is stretched by a factor of two relative to the vertical axis for clarity. To get an estimate of the magnitude of the viscous dissipation, we can compute the work done on a fluid element as it traverses the core of the diwhirl, and use that to calculate its temperature rise in the absence of heat loss to the neighboring fluid. This can be

175 155 calculated using E v = t1 t 0 ˆΦdt. (153) Since the diwhirl is axisymmetric and v z =0at the symmetry plane, we can replace dt by dr/v r and write E v = r1 Once E v is known, we can calculate the temperature rise using r 0 ˆΦ v r dr. (154) T = G γ s(r 2 R 1 )E v ρc p, (155) where ρ is the mass density of the polymer solution, C p its specific heat capacity at constant pressure on a mass basis, and the product G γ s (R 2 R 1 ) is required to convert E v to dimensional form. We performed this integral using Simpson's rule for the data in figure 155 choosing (arbitrarily) r 0 =0.98 and r 1 =0.2. From the data in Groisman and Steinberg (1998), we use a solvent viscosity of 0.1 Pa s, and a relaxation time of 1s.Asafirst (and rough) approximation, we assume the heat capacity and density of the sugar syrup to be the same as that of water ( Jkg 1 C 1 and 1000 kg m 3 respectively). This yields a temperature rise of about 10 4 C in one cycle. For a PIB/PB Boger fluid such as the one used in the experiments by Baumert and Muller (1999), however, the temperature rise would be much higher. For instance, using the data for their medium viscosity Boger fluid (η s =6.5Pas, λ =0.23 s, ρ = 880 kg m 3, and C p Jkg 1 C 1 ) yields a temperature rise of 0.27 C if the geometrical parameters are taken to be the same as those used by Groisman and Steinberg (1998). Given the low magnitude of the temperature rise in the PAA Boger fluids, it would appear that

176 156 the experiments of Baumert and Muller (1999) are more likely to be subject to nonisothermal effects than those of Groisman and Steinberg (1998). However, since similar structures are seen in both fluids, this would seem to indicate that non-isothermal effects do not significantly influence either the diwhirls or the flame patterns Self-sustaining mechanism Since there is no stationary bifurcation in circular Couette flow, the diwhirl solutions that we have computed are part of an isolated branch that does not connect directly to the base state. Therefore, the sustaining mechanism for these patterns must be inherently nonlinear. Following their experiments, Groisman and Steinberg (1998) proposed one such mechanism. They argued that the difference in symmetry between inflow and outflow results in the elastic forces performing net positive work on the fluid. While this argument shows that finite amplitude stationary structures that exhibit significant asymmetry between inflow and outflow are physically plausible, it does not explain the mechanism by such structures sustain themselves. Having the detailed velocity and stress fields available to us from our computations, we propose a more complete mechanism. Figure 65 shows a vector plot of v at the axial centerline of the vortex. The azimuthal velocity field has a parabolic structure near the outer cylinder, similar to the velocity field in the outer half of the channel in Dean flow. This velocity field results in an unstable stratification of azimuthal normal stress (Joo and Shaqfeh, 1992b). We therefore propose the following fully nonlinear mechanism for the diwhirls: a finite amplitude perturbation near the outer cylinder results in a locally parabolic velocity profile. This in turn creates an unstable stratification of hoop stress (visible in figure 65(b) for r < 0.99), as in Dean flow, which drives inward radial motion. As the fluid moves inward, it accelerates azimuthally due to

177 r r 0.97 θ (a) 0.97 θ (b) Figure 65: (a) Vector plot of v near the outer cylinder at the center of the diwhirl structure (oblique arrows) and the base state (straight arrows). The length of the arrows is proportional to the magnitude of the velocity. The axial velocity is identically zero in the base state, and is zero by symmetry at the center of the diwhirl. (b) Principal stress directions at the same location as for (a). The Couette flow stress is not shown because it is very small in comparison. This figure shows how fluid elements at larger radii are pulled down and forward sustaining the increase in v θ. the base state velocity gradient. The azimuthal tension in polymer chains drags the fluid at larger r forward and down (figure 65(b)). This maintains the increase in v θ and results in a self-sustaining mechanism (figure 66). Thus we see that, in common with other elastic instabilities, the mechanism is based on the inward radial force associated with tensile stresses along curved streamlines. This mechanism shows that we have come a full circle. We began our search for stationary, finite amplitude solutions in circular Couette flow by starting from Dean flow. We now find that these solutions sustain themselves by a mechanism similar to the one resulting in the primary instability in Dean flow.

178 158 local increase in v θ near outer cylinder tension in the streamlines pulls fluid near the outer outer cylinder forward and down unstable stratification of τ θθ, generates inward radial motion Figure 66: Nonlinear self-sustaining mechanism for the diwhirl patterns Stability fluid moves azimuthally due to base state shear We now address the stability of the diwhirl patterns with respect to axisymmetric and non-axisymmetric perturbations. We determine this by finding the eigenvalues ω in equation 142 for axisymmetric perturbations or equation 145 for non-axisymmetric perturbations. These computations are expensive, so we only perform them for a few points along the upper branch for L =4.74 in figure 58. Here, we report the results from one such computation, performed at We θ = To resolve this point, we needed 16 radial elements and 14 axial elements using fifth order polynomials in both directions in each element, which is a slightly coarser mesh than the one shown in figure 50, and results in a system with unknowns for axisymmetric disturbances, and double that number for non-axisymmetric perturbations. The computation requires the storage of two sparse matrices, the preconditioner for one linear system, and the Krylov basis. This was well above the storage capacity available to us on a single computer, so we stored the Krylov basis and one of the matrices on one machine and the second matrix and preconditioner

179 159 on another, and used Message Passing Interface (MPI) subroutines to communicate between the two machines. We used a 600 vector basis (ARPACK would not converge if significantly fewer vectors were used) and asked for the most unstable eigenvalues. For axisymmetric disturbances, we found two pairs of complex conjugate eigenvalues that had positive real parts. The eigenvectors corresponding to one pair had an irregular grid scale structure, suggesting that they are part of the continuous spectrum of eigenvalues (Graham, 1998; Wilson et al., 1999; Renardy, 2000). These modes are expected to be stable, but since the eigenvectors are nonintegrable (Graham, 1998), they will not converge exponentially in a spectral element scheme and can display spurious instability. The structure of the other two eigenvectors is shown in figure 67. They show that although the branch is unstable, the destabilizing disturbance has significant amplitude only near the ends of the domain, where the flow is essentially circular Couette flow. The base circular Couette flow has a minimum critical We θ of with respect to axisymmetric disturbances, and so is linearly unstable at We θ = Hence, it is not surprising that the portion of flow pattern where the flow is essentially circular Couette would be susceptible to destabilizing disturbances. What is interesting, however, is that the core of the pattern, where the diwhirl lies, is entirely unaffected. This shows that the diwhirl pattern is dynamically distinct from the oscillatory finite wavelength axisymmetric pattern arising from the linear instability of circular Couette flow. For non-axisymmetric disturbances, the stability picture is slightly more complicated. We performed computations with n =1and n =2and found three pairs of unstable complex conjugate eigenvalues. Their growth rates are summarized in table 7, while their structures are shown in figure 68 and 69. For n =1, we see that there are two modes that have their largest amplitude close to the core of the diwhirl (figures 68 (a) and

180 160 (b)), while the third (figure 68 (c)) has a large amplitude away from the core. The third mode is directly related to the linear instability of circular Couette flow with respect to non-axisymmetric disturbances with n =1. For n =2, the picture is slightly different. There is still one mode (figure 68(a)) that is largely concentrated outside the core, and which therefore seems related to the linear instability of circular Couette flow. The mode in figure 68 (b) is largely concentrated at the core, and shows similarities to figures 68 (a) and (b). In addition, there is non-localized mode (figure 69(c)) that is absent for n =1. n =1 n = ± i ± i ± i ± i ± i ± i Table 7: Growth rates for the unstable nonaxisymmetric modes at We θ =23.87, L = 4.74, S =1.2, and b = Dean flow revisited In the previous sections, we have demonstrated that stationary, localized solutions of very large wavelength exist in circular Couette flow. Here, we return to Dean flow to investigate whether such solitary solutions are possible there. We do this by tracking long wavelength steady states that bifurcate from the base state in Dean flow. To make comparison with the circular Couette flow results easier, we choose b = 1830, S =1.2, and ɛ = 0.2, which are the same parameter values that we used for circular Couette flow. In figure 70, we show the bifurcation diagram for L = This diagram is similar to figure 54, in that there is a large subcritical region, and the nontrivial branch exhibits a turning point, i.e., there is a critical value of the Weissenberg number below which the nontrivial solution does not exist. In figure 71, we show an density plot of

181 161 r (a) z r (b) z r (c) Figure 67: Density plot of v r showing the (a) real and (b) imaginary parts of the destabilizing axisymmetric disturbance, and (c) the streamlines of the base diwhirl. Note that the core of the diwhirl is entirely unaffected by the disturbance. The parameters are We θ =23.87, S =1.2, ɛ =0.20, b = 1830, and L =4.74. z