Numerical Simulation Of Stratified Flows. And Droplet Deformation In 2D Shear Flow. Of Newtonian And Viscoelastic Fluids

Transcription

1 Simulation Of Stratified Flows And Droplet Deformation In 2D Shear Flow Of Newtonian And Viscoelastic Fluids A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AT VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS By T IRIVAN HU CHIN YOKA Supervised by: YURIKO & MICHAEL RENARDY Committee: Jong Kim, Tao Lin, Shu-Ming Sun DEPARTMENT OF MATHEMATICS VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY BLACKSBURG, VIRGINIA Keywords: Stratified Flows, Droplet Deformation, Viscoelastic fluids November 15, 24

2 Simulation Of Stratified Flows And Droplet Deformation In 2D Shear Flow Of Newtonian And Viscoelastic Fluids. By Tirivanhu Chinyoka Abstract Analysis of multi-layer fluid flow systems or, in general, flows with interfaces often leads to mathematical expressions and equations too complicated for pencil and paper hence numerical computation is almost always necessary. In this dissertation, we develop a numerical code for tracking deformable interfaces. In particular this code is a viscoelastic version of the volume of fluid algorithm developed in [11]. The code uses the piecewise linear interface calculation method to reconstruct the interface and the continuous surface force formulation to model interfacial tension forces. Our numerical algorithm is primarily designed to simulate the flow of (i) superposed fluids (herein referred to as fluid-fluid systems) and (ii) the drop in a flow problem (droplet-matrix systems) in 2D shear flows of viscoelastic fluids. However by taking the viscoelastic parameters to be zeros, we in fact can consider cases were either or both of the phases in the fluid-fluid or droplet-matrix system will be assumed viscoelastic or Newtonian. The extra stresses governing viscoelasticity will herein be treated with the Oldroyd-B constitutive equations. The part our work dealing with two-layer flows is in the same spirit as among others that of Renardy et. al [29], who investigated the Poiseuille flow counterpart. As mentioned earlier, this part can also be thought of as a natural extension of the work of Li et. al, [11], to the viscoelastic regime. Our subsequent work on deformable drops is closely connected to the experimental investigation of Guido et. al [6], and the numerical works of Sheth et. al. [32], Pillapakkam et. al [18], and Renardy et. al [28], all of whom considered the drop in a flow problem in various contexts. As in [11] we employ the volume

3 of fluid scheme with a semi-implicit Stokes solver (enabling computations at low Reynolds numbers) in our numerical algorithm. In the first part, the code is validated against linear theory for the superposed shear flow of well documented fluid-fluid systems. validation in the second part will mostly be against the results of the four major worked cited earlier. iii

4 Contents List of Figures vi List of Tables vii Acknowledgements viii Dedication ix Declaration x 1 Introduction Interfacial Instabilities Rayleigh-Taylor Instabilities Saffman-Taylor Instabilities Viscous Instabilities Surface Tension Gradient Instabilities Elastic Instabilities Scope Governing Equations 8 iv

5 2.1 Dimensionless Governing Equations Stability Analysis Bifurcation to traveling wave solution Mean Flow Component Implementation Volume Of Fluid (VOF) Method Piecewise Interface Calculation (PLIC) Spartial Discretization and MAC Method Temporal Discretization, Projection Method Semi-Implicit Stokes Solver Multigrid Method Modeling interfacial tension force (CSF) Oldroyd-B Constitutive Equation Validation I Calibrating The Code Velocities and Extra-stresses Time derivatives Growth rates of amplitudes and velocities Growth Rates at the Interface Waveforms Effects of cell shape Effects of Surface Tension Growth rates away from Interface v

6 4.2.6 Maximum growth along a horizontal line Growth rates of Extra-stresses Growth at the interface Growth away from Interface Maximum growth along a horizontal line Harmonic averaging Growth rates of amplitudes and velocities Growth at the Interface Maximum growth along a horizontal line Growth rates of Extra-stresses Growth at the interface Maximum growth along a horizontal line Validation II Growth rates of amplitudes and velocities Growth at the Interface Waveforms Droplet Deformation Problem formulation and assumptions Validation of Results Small deformation case Large deformation case Temporal and spatial convergence vi

7 7.3.1 Newtonian drops in viscoelastic fluids Purely Newtonian case and effects of inertia vii

8 List of Figures 2.1 Flow Schematics MAC cell Residual data for velocities and extra-stresses Residual data for time derivatives log(h), log(v max ), log( v 2 ). respectively Wave forms at (a) t =.1, (b) t =.1, (c) t =.3 and (d) t = Effects of cell shape for the case α = R(σ) vs. Tension, T Effects of surface tension Spectrum of eigenvalues for α = (a) R(σ)= (b) R(σ)= log(v) & log(u) in the Newtonian fluid log(v) & log(u) in the Viscoelastic fluid log(v max ) & log(u max ) along a horizontal line in Newtonian fluid log(v max ) & log(u max ) along a horizontal line in Viscoelastic fluid log(ampl) & log(t ijmax ) log(t ij ) in the viscoelastic fluid viii

9 4.16 log(t ijmax ) in the Viscoelastic fluid log(h), log(v max ), log( v 2 ). respectively log(v max ) & log(u max ) along a horizontal line in Newtonian fluid log(v max ) & log(u max ) along a horizontal line in Viscoelastic fluid log(ampl) & log(t ijmax ) log(t ijmax ) in the Viscoelastic fluid log(h), log(v max ), log( v 2 ). respectively max(amplitude) vs time and wave shapes Wave forms at (a) t =.1, (b) t =.1, (c) t =.3 and (d) t = 1 where R 1 = 186, t = 1 3, t max = 1 and h = Computational domain Temporal evolution of D, when Ca = Temporal evolution of D when Ca = Deformed drops at t = Deformed drops at t = Deformed drops at t = Contour plots of first normal stress at difference t = Temporal evolution of D when Ca =.24 and zero initial stresses Evolution of D when Ca = Temporal evolution of D when Ca = Elongation of drops at t = 1 when Ca = 6, R =.3, De = ix

10 7.12 Effect of mesh size on numerical breakup of drops, results shown for a Newtonian drop in a viscoelastic fluid at t = 1 with R =.3, De = 8.&Ca = Convergence with mesh resolution, for a Newtonian drop in a viscoelastic fluid at t = 1 with R =.3, De = 8. & Ca = 6, Red= , Blue= Green= Steady state shapes with spatial (a) & spatio-temporal (b) refinement Transient deformation with temporal refinement velocity vector plots for Newtonian drop in a viscoelastic fluid where Ca = 6, R =.3, De = 8 at (a) t = 2, (b) t = 4, (c) t = 5, & (d) t = 6, velocity vector plots for Newtonian drop in a viscoelastic fluid where Ca = 6, R =.3 & De = 4, t = 2, De = 4, t = 6, De = 2, t = 2, & De = 2, t = 6, velocity vectors showing linear streamwise profiles at left and right hand side edges, Ca = 6, R =.3, De = Surface and contour plots of first normal stress difference at t = 2 where Ca = 6, R =.3, De = 8 in (a) & (b) and De = 4 in (c) & (d) Surface and contour plots of first normal stress difference at t = 2 where Ca = 6, R =.3, De = 2 in (a) & (b), De =.5 in (c) & De =.5 in (d) Surface and contour plots of first normal stress difference at t = 2 where Ca = 6, R =.3, De =.1 in (a) & (b) and De =.1 in (c) & (d) x

11 7.22 Surface and contour plots of pressure at t = 2 where Ca = 6, R =.3, De = 8 in (a), (b) and De = 4 in (c), (d) Surface and contour plots of pressure at t = 2 where Ca = 6, R =.3, and De = Contour plots of pressure at t = 2 where Ca = 6, R =.3, De =.1 in (a) and De = in (b) Evolution of drop deformation in time (a) Ca =.2, m = 1, (b) Ca =.2, m = 1, (c) Ca =.4, m = 1 & (d) Ca =.4, m = Drop shapes at t = 3 (a) Ca =.2, m = 1, (b) Ca =.2, m = 1, (c) Ca =.4, m = 1 & (d) Ca =.4, m = xi

12 List of Tables 4.1 Properties of fluids: Conditions of experiments: Deformation parameter (D) and Angle (φ max ) for four drop-matrix systems at dimensionless t = 3, Ca = D and φ max at dimensionless t = 1, Ca = D and φ max at t = 3, Ca =.24 & zero initial stresses D & φ max at t = 3, Ca = D and φ max at t = 1, Ca = xii

13 Acknowledgements This work would not have been possible without the tremendous help and immense input from two brilliant Mathematicians: my advisors, Professors Yuriko & Michael Renardy, my acknowledgements should start with them. By extension I would also like to acknowledge the help, direct or indirect of Professors Damir Khismatullin and Jie Li and the Virginia Tech Mathematics department. I should also take this opportunity to confer a big thank you to two of the finest people who ever graced this world and who naturally brought me this far, my parents Tizirai Kunashe and Gelly Gumbo. My siblings Tiani and Pilia also deserve special mention for bearing all the burden during my absence, especially following the passing on of our wonderful sister Piripina Hahani way before her time, we will forever miss her. Lastly, I extend a hand to the one person I would least think of thanking, the one who has unfolded himself to a higher level, without much hope from the sidelines......self! tiri xiii

14 Dedication To those who by the simplicity of their deeds, make our days complicated, without forethought but with unconcious attention. tiri xiv

15 Declaration No portion of the work referred to in this dissertation has been submitted in support of an application for another degree or qualification of this or any other university or other institution of learning. xv

16 Chapter 1 Introduction In recent years, the stability problem of multi-fluid flows has attracted a considerable amount of research interest among scientists from the fields of aeronautics, applied mathematics, bio-medical and chemical engineering and physics to name but a few. This can be readily attributed to the ever increasing number of industrial applications of such problems for example in lubricated pipelining, manufacture of co-extrusion polymers, [14], photographic film development and deicing aircraft wings [17]. 1.1 Interfacial Instabilities The earliest significant theory of instability of flow of superposed fluids can be traced back to Yih [37] with his work on long wave instability of two-layer flows (Couette & Plane Poiseuille) with viscosity stratification. The years following the initial publications of Yih saw a ballooning of interest and literature in the field of multi-fluid flow stability. A concise account can be found say in the book by Joseph and Renardy [14]. We are here simply going to try to just give a brief overview of the parallels that can be drawn from most of these works. 1

17 We should here point out the interesting works of earlier researchers before Yih for example G. I. Taylor who made the observation that (provided gravity is set aside) if two fluids, separated by a flat horizontal interface are accelerated perpendicular to this interface, then there will be an instability of the interface if the acceleration is directed out of the lighter fluid into the heavier one [34], [13] Rayleigh-Taylor Instabilities The observation by Taylor is of course a generalization of what are generally termed Rayleigh-Taylor instabilities in which under gravitational acceleration the set-up with a heavy fluid lying above a light fluid is unstable. This can also be generalized the the case of multi-layer flows with density stratification. Microgravity environments, which simulate the suppression of gravity effects have been achieved through either density matching or by conducting the relevant experiments in space laboratories aboard space shuttles. An interesting case however is the case of thermocapillary instabilities under microgravity, in which there is well-documented evidence of disagreements between results from experiments conducted in space and theoretical results [39]. The workers in [39] cite Coriolis forces resulting from the orbiting motion of space shuttles as a possible source of the discrepancies. Rayleigh-Taylor instabilities have also been investigated in, [21], with viscous effects in [36], and numerically in [35] Saffman-Taylor Instabilities When one fluid displaces another from a porous solid, say a Hele-Shaw cell, displacements of the interface usually grow into finger-like structures of the penetrating fluid in the compliant fluid. This is an example of what are referred to as Saffman-Taylor instabilities. Since however, most of the important flows in this case (over porous solids) do not behave according to the usual Navier-Stokes 2

18 equations, and are instead governed by Darcy s law we will therefore not dwell much on these except to just refer to the (quite interesting as we shall see in the next paragraph) conclusions of Saffman-Taylor, i.e. the displacement of a more viscous fluid by a less viscous one in a porous solid may be unstable whereas the reverse situation is stable, see for example [13] for a complete analysis Viscous Instabilities Curiously, when the Navier-Stokes equations do apply, say in industrial scale pipeline transport of liquids, we usually come to completely opposite conclusions to those just stated. In this case, the displacement of a more viscous fluid (e.g. oil) by a less viscous fluid (e.g. water) leads to a stable arrangement where the less viscous fluid migrates to the walls and acts as a lubricant for the more viscous fluid [14]. In general for two-layer flows with equal densities and subject to long-wave disturbances, such long wave instabilities can be stabilized via the so-called thin-layer effect (i.e. putting the less viscous fluid in a thin layer) [14], [1]. Short wave instabilities for such flows (which may surprisingly be due to the viscosity difference and not damped out by it!) will be stabilized by surface tension [9], [14]. Li, Renardy and Renardy [11], investigated the viscous counterpart of the simulation of an inviscid Kelvin-Helmholtz instability. Their paper looks at periodic disturbances in two-layer Couette flow with an emphasis on the effects of viscosity stratification, and its role in fingering instabilities. Ganpule and Khomami [4] showed the mechanism of instability of short and long waves to viscosity stratification in both plane Poiseuille and plane Couette flow by performing a rigorous viscous energy analysis. For short waves, they showed that the interfacial contribution to the viscous energy term was the mechanism of instability whereas for long wave disturbances, it is the Reynolds stress term of the disturbance energy equation. 3

19 These kinds of discussions are also typical of most stability studies via asymptotic analysis, in which the instabilities are depicted exclusively as either short or long wave in nature. However, using a rigorous linear analysis, [21] showed the existence of unstable regions not described by asymptotic analyses of short or long waves Surface Tension Gradient Instabilities Even though we just noted that inclusion of surface tension can stabilize short wave disturbances, it should be noted that surface tension (or interfacial tension as we should correctly refer to it for multi-layer flows, surface tension being for the case when a liquid is in equilibrium contact with its vapor) is a thermodynamic property which depends on both temperature and composition (concentration of surfactants). Differences in temperature and/or composition in the tangential direction leads to inhomogeneity of surface tension and thus a tangential interfacial tension gradient which may produce flow or change an existing one and this may produce instabilities. Surface tension gradient driven convection is also known as Marangoni driven convection after the Italian physicist or also thermocapillarity convection in case it arises only from temperature differences. If on the other hand the temperatures and/or composition gradients are perpendicular to the interface, then only if such gradients are strong enough can they lead to instability otherwise since they do not produce any inhomogeneities of the interfacial tension, one would not expect any interfacial tension related instabilities. For a thorough treatment of the effects of interfacial tension inhomogeneities on multifluid flow stability we refer the reader to the recent book [16]. In our current work we will assume that the temperature is constant, the composition is fixed and hence the interfacial tension is constant. 4

20 1.1.5 Elastic Instabilities Our work also deals with viscoelastic fluids hence it is worth reviewing some of the recent work that has been done by other researchers focusing on the effects of elasticity on stability of two-layer flows. Y. Renardy in [22] showed that for short wave disturbances, the order of magnitude of elasticity stratification in determining growth rates of such disturbances is one less than the stabilizing effect of surface tension and one more that the destabilizing effects of density and viscosity stratification, and hence under suitable choices of the surface tension, densities and viscosities, elasticity stratification may stabilize or destabilize the flow. Based on this work, Joseph and Renardy [14] showed that under long wave disturbances, elasticity stratification may also stabilize or destabilize the flow. Again by performing an energy analysis for long wave as well as short wave purely elastic instabilities on both plane Poiseuille and Couette flows Ganpule and Khomami [4] showed the mechanism of instability to be associated with the coupling between the jump in the base flow normal stresses across the interface and the perturbation velocity field. The experimental, computational and analytical works of importance which we did not present here include the work of Khomami, Renardy, Su & Clark [29], and all the other numerous publications, far too many to mention, say of K. Chen, V. Coward, D. Joseph, D. Kothe, M. Renardy, Y. Renardy, S. Zaleski, and their co-workers. 1.2 Scope This Dissertation is mainly concerned with developing a numerical scheme for investigating the effects of elasticity stratification on the superposed shear flow of two liquids and in droplet deformation. This numerical code uses a volume of fluid (VOF) scheme similar to that of [11] and [5]. The VOF method is a 5

21 fixed mesh approach that allows for accurate interface advection and handles changes in interface topology [11]. The interface is reconstructed using using a piecewise linear interface calculation (PLIC) method where the interface is assume to be linear in each computational cell [19]. Other algorithms that have been proposed for interface reconstruction are the SLIC and least squares methods [19]. To handle the boundary conditions at the interface, we use the continuous surface force (CSF) technique where the interfacial tension forces are incorporated as body forces per unit volume in the momentum equations [1], [11]. There are different variants of the VOF method and other approaches of tracking the interface between two fluids, for a complete description of how these work and their relative efficiencies, we refer the reader to [11] and the references there in, mostly to the same authors. The part of our code that will not be found in [11] handles the calculation of the extra-stress tensor using a semi-implicit scheme for the Giesekus equation. This is our viscoelastic contribution. The dissertation is organized in two parts named Parts I & II respectively. Part I was originally designed to focus on the numerical validation of the fluid-fluid systems but for chronological reasons, we found it necessary to also include the (i) detailed outlines of the equations governing the motion of the flow system including details of the Giesekus model which governs the behavior of the extra stresses in the viscoelastic phase and (ii) development of the numerical code for the simulation of the two flow systems of concern. Part II is mostly devoted to numerical validations of the droplet-matrix system. 6

22 PART I TWO LAYER FLOWS

23 Chapter 2 Governing Equations We analyze the linear stability of two-layer shear flow of immiscible fluids, in which at least one of the fluids is viscoelastic. The flow geometry is sketched below with the lower fluid denoted Fluid 1 and the upper fluid represented by Fluid 2. Figure 2.1: Flow Schematics Unless otherwise stated, we will herein use the subscript j to represent 1 or 2, with the understanding that each of these indices describe quantities characteristic to the corresponding fluid, e.g. ρ 2 represents the density of Fluid 2 etc. Densities will 8

24 be denoted ρ j, solvent viscosities η sj, polymeric viscosities η pj, total viscosities µ j = η sj + η pj and relaxation times λ j. We also define β j = η sj /µ j, the viscosity ratio m = µ 1 /µ 2 and density ratio r = ρ 1 /ρ 2. The lower and upper walls are located at z = and z = l respectively, where the asterisks are used for dimensional variables. In the basic flow; The upper plate moves with velocity (U p, ) and the bottom plate is at rest. Fluid 1 occupies z l 1 and Fluid 2 occupies l 1 z l. The interfacial velocity is (U (l 1), ) and we denote U (l 1) by U i. 2.1 Dimensionless Governing Equations The velocity, distance, time and pressure are made dimensionless with respect to U i, l, l /U i, and ρ 1 U 2 i the same as the pressure. respectively. The extra stress components are scaled Reynolds and Weissenberg numbers in fluid j are denoted R j = U i l ρ j /µ j and W j = U i λ j /l respectively, where the requirement, mr 1 = rr 2, should be satisfied. For Couette (and Poiseuille) flows, there are 13 dimensionless parameters: a Reynolds number, say R 1, a Weissenberg number W 1, the undisturbed depth l 1 of fluid 1, a surface tension parameter T =(surface tension coefficient S )/(µ 2 U i ), a Froude number F given by F 2 = U 2 i /gl where g is the gravitational acceleration constant, a dimensionless pressure gradient G = G l /(ρ 1 U 2 i ), the viscosity ratio m, a density ratio r, the ratio of relaxation times w = λ 1 /λ 2 = W 1 /W 2, β 1, β 2, κ 1 and κ 2. The physically relevant range is κ j (1 β j )µ j /(µ 1 R 1 ) <.5, [7]. The dimensionless equation of motion is u t + u u = ρ 1 ρ j ( T p) + F + β j R j 2 u, (2.1) 9

25 where F represents body forces (which includes gravity and interfacial tension forces) and the total stress tensor is τ = pi + T + ( β j R 1 )( µ j µ 1 ) 1 2 where T is the extra stress tensor. constitutive relation ( uj + u ) i, (2.2) x i x j The Giesekus model has the differential T t + (u )T ( u)t T( u)t + κt 2 + T W j = G j ( u + ( u) T ), (2.3) where the elastic modulus at time t =, G is G j = µ j(1 β j ) µ 1 R 1 W j, and κ is the Giesekus non-linear parameter. Taking κ in the Giesekus model leads to the Oldroyd-B model, which is what we are exclusively going to consider as governing the viscoelastic phase in this work. At the interface, the velocity and tangential stress are continuous, the jump in the normal stress is balanced by surface tension and curvature, and the kinematic free surface condition holds. For the combined Couette-Poiseuille flow under the Oldroyd-B model, the dimensionless basic velocity (U(z),) is the same as the Newtonian case: GR 1 z 2 /2 + c 1 z, z l 1, U(z) = rgr 2 (z 1) 2 /2 + c 2 (z 1) + U p, l 1 z 1, where (2.4) c 1 = (1 + GR 1 l1/2)/l 2 1, l 2 = 1 l 1, c 2 = m( GR 1 + c 1 ), and the upper plate speed is U p = 1 + ml 2 GR 1ml 2. l 1 2 The basic pressure field P is also the same as the Newtonian case and satisfies dp/dx = G and dp dz = 1/F 2, z l 1, 1/(rF 2 ), l 1 z 1. 1 (2.5)

26 We note that in our shear flow case, we will take G. The basic extra stress tensor has the form T = C 1(z) C 2 (z), (2.6) C 2 (z) where C 1 (z) = 2(1 β j ) ( µj µ 1 R 1 ) W j [U (z)] 2, C 2 (z) = (1 β j ) ( µj µ 1 R 1 ) U (z). (2.7) The basic shear stress condition is [C 2 +(β j µ j )U /(R 1 µ 1 )] = at z = l 1. Solutions that are perturbations of the above basic flow are sought. The perturbations to the velocity, pressure and interface position are denoted by (u, v), p and h, respectively. The perturbation to the extra stress tensor is The equations of motion in each fluid yield T 11 T 12. (2.8) T 12 T 22 u t + U u x + vu + ρ 1 p ρ j x ρ ( 1 T11 ρ j x + T ) 12 β ( j 2 ) u z R j x + 2 u 2 z 2 = u u x v u v t + U v x + ρ 1 ρ j p z ρ 1 ρ j z, (2.9) ( T12 x + T ) 22 β j ( 2 v z R j x + 2 v 2 z ) 2 = u v x v v z, (2.1) The constitutive equations for the Oldroyd-B fluid (κ 1 = κ 2 = ) yield the following coupled equations for the extra stress components: ( T11 T 11 + W j + U T ) 11 t x + u vc 1 2C 1 x 2C u 2 z 2T 12U ( ) 2µj u (1 β j ) µ 1 R 1 x ( u = 2W j x T 11 + u ) ( z T 12 W j u T 11 x + v T ) 11, (2.11) z 11

27 ( T12 T 12 + W j + U T ) 12 t x + v vc 2 C 1 x T 22U ( ) ( µj v (1 β j ) µ 1 R 1 x + u ) z ( v = W j x T 11 + u ) ( z T 22 W j u T 12 x + v T ) 12, (2.12) z ( T22 T 22 + W j + U T ) ( ) 22 t x 2C v 2µj v 2 (1 β j ) x µ 1 R 1 x ( v = 2W j x T 12 + v ) ( z T 22 W j u T 22 x + v T ) 22. (2.13) z Continuity is u x + v z =. (2.14) The boundary conditions are u = v = at z =, 1. The conditions at the interface are posed at z = l 1 + h(x, t). The unknown h(x, t) is assumed to be small; the interfacial conditions are expanded as Taylor series about z = l 1, and retaining only the linear order terms. The shear stress conditions are [t τ n] = where the unit tangent vectors are t = (1, h x )/ 1 + h 2 x, the normal is n = ( h x, 1)/ 1 + h 2 x where [x] denotes x(fluid 1) - x(fluid 2). The normal stress condition is [n τ n] = T h xx, (2.15) mr 1 [1 + h 2 1/2 x] where the dimensionless interfacial tension parameter is T = S /(µ 2 U i ). Continuity of velocity yields h[u ] + [u] = h[ u z ], [v ] = h[ v ]. (2.16) z 12

28 Continuity of shear stress yields [T 12 ] h x [C 1 ] + [ β j R 1 µ j µ 1 (u z + v x )] =. (2.17) Here, [C 2 + (β j /R 1 )(µ j /µ 1 )U ] = from the base flow shear stress balance, [C 2 + (β j /R 1 )(µ j /µ 1 )U ] = from the base flow x-momentum equation, [C 2 ] = G(β 1 β 2 ), [P ] = in the basic flow, and u x + v z = from incompressibility. The balance of normal stress yields T [T 22 ] [p] h xx + h[p ] + [ 2β jµ j v z ] =. (2.18) mr 1 µ 1 R 1 The kinematic free surface condition holds 2.2 Stability Analysis For the linear stability analysis, we discard terms that are quadratic or higher in the perturbations and seek normal mode solutions u, v, w, p, T ij and h which are proportional to exp(iαx+σt), where σ denotes complex-valued eigenvalues which are solved with the other parameters given. The details of the discretization, such as the Chebyshev-tau scheme [?], are given in [25] and are not repeated here Bifurcation to traveling wave solution At the onset of an instability of the base flow, the weakly nonlinear amplitude equation admits modes proportional to exp(iαx). These modes interact to form waves that travel in the x-direction. The methodology and notation are identical to that of [25] and [26]. The details are summarized as follows. Let Φ represent the set of unknowns (u, v, p, T 11, T 12, T 22, h). The equations and boundary conditions are represented in the schematic form LΦ = N 2 (Φ, Φ) + N 3 (Φ, Φ, Φ), (2.19) 13

29 where the real linear operator L has the form A + Bd/dt and L(σ) = A + σb. N 2 contains quadratic terms and N 3 contains cubic terms from the right hand sides of the equations, boundary conditions and interfacial conditions. The components f 1,..., f 15 of N 2 + N 3 are written in Appendix of [25] for the upperconvected Maxwell (UCM) liquid case and are modified here for the Oldroyd- B case. The modifications emanate from the terms involving β j, namely the base stress components, the Laplacian terms in the momentum equations, terms from the symmetric part of the velocity gradient in the constitutive equations, additional terms in the shear stress balance and normal stress balance. As in [25], we denote the nonlinearities of the momentum equations in fluid 1 by f 1, f 2 those of the constitutive equations in fluid 1 by f 3, f 4, f 5, the corresponding notation for fluid 2 is f 6,..., f 1, f 11 for the continuity of u, f 12 for the continuity of v, f 13 for the shear stress balance, f 14 for the normal stress balance, f 15 for the kinematic condition. We use λ to denote the bifurcation parameter, which can be any of the parameters, e.g. the Reynolds number or Weissenberg number. At λ =, there is one eigenvalue, the interfacial mode, at σ = ic, c >, for α = α c > and a corresponding eigenvalue σ = ic for α = α c, and the rest of the eigenvalues are stable (Re σ < ). The eigenfunction with wavenumber α is denoted by ζ(λ) and that with wavenumber α by ζ(λ), where the overbar denotes the complex conjugate. For λ >, ic becomes s(λ). The eigenfunction ζ satisfies Aζ(λ) = s(λ)bζ(λ). The adjoint eigenfunction with wavenumber α is denoted by b(λ) and is calculated from the discretized matrix representations of the operators A and B, by using the complex conjugate of the transpose of those matrices. The normalization condition is (b, Bζ) = 1. ζ and b are proportional to exp iαx. On the center manifold, the perturbation solution Φ can be decomposed as follows Φ = Zζ + Zζ + Z 2 η + ZZχ + ZZχ + Z 2 η + higher order terms. (2.2) 14

30 Here, the Z(t) is the complex-valued amplitude function, χ represents the distortion to the mean flow and η is the second harmonic. (A 2s(λ)B)η = N 2 (ζ, ζ), (2.21) where, in actual computations, η is proportional to exp 2iα c x and λ is set to zero. Similarly, the equation for the mean flow component χ simplifies to Aχ = N 2 (ζ, ζ). (2.22) The equations governing χ are detailed in Section (2.2.2). The final equation for the amplitude function is dz dt + s(λ)z = κ Z 2 Z, (2.23) κ = (b, 4N 2 (ζ, χ) + 2N 2 (ζ, η) + 3N 3 (ζ, ζ, ζ)). (2.24) This is the Suart-Landau equation and κ is the Stuart-Landau coefficient. If the real part of κ is negative, then the bifurcating solution is supercritical and the travelling wave solution would be stable for small amplitudes. If the real part of κ is positive, then the bifurcating solution would be unstable. To reconstruct the nonlinear waveform, we refer to the interface perturbation component in Eq. (2.2), and use the component h in the eigenfunction ζ and the second harmonic η. We may picture the total interface perturbation as Φ h = 2Re[Z(t)h ζ exp(iαx + iimσt) + Z 2 (t)h η exp(2iαx + 2iImσt)], (2.25) where we think of the Z(t) as an amplitude factor. The effect of the nonlinearity can be exaggerated by choosing Z(t) to see the trend of whether the waves steepen in the front or the back. The second harmonic term η contributes sin 2αx to the interface shape. 15

31 2.2.2 Mean Flow Component Since χ is not periodic in x, its component v satisfies dv/dz = by incompressibility. Since v = at z =, 1, v = in the entire domain. Denote the quadratic terms on the right hand sides of the momentum and constitutive equations as before by f 1,..., f 1 Denote the quadratic terms on the right hand sides of equations (??)-(2.21)by f 11 to f 15, respectively. We note that f 15 in N 2 (ζ, ζ) vanishes. We carry out the formulation for χ keeping the pressure gradient in the x-direction fixed in the entire nonlinear analysis. Putting d/dx =, d/dt =, and v = in the equations of motion, the components in χ satisfy: ρ 1 T 12 ρ j z β 2 u R j z = f 2 1 or f 6 of N 2 (ζ, ζ), (2.26) ρ 1 ( p ρ j z T 22 z ) = f 2 or f 7, (2.27) T 11 + W ( u 2C 2 z 2T 12U ) = f 3 or f 8, (2.28) T 12 W T 22 U µ u (1 β) µ 1 R 1 z = f 4 or f 9, (2.29) T 22 = f 5 or f 1. (2.3) u = at z =, 1, (2.31) h[u ] + [u] = f 11, (2.32) [T 12 + βµ µ 1 R 1 u z ] = f 13, (2.33) [T 22 p] h[p ] = f 14. (2.34) To preserve the given volumes of the fluids, we set h = for χ. To fully determine p requires an additional condition: we set p for fluid 1 equal to zero at z = l 1. The problem for the pressure decouples: p z = f 5 z + f 2 in fluid 1 (2.35) p z = f 1 z + 1 r f 7 in fluid 2 (2.36) [p] = [f 5 ] z=l1 [f 1 ] z=l1 f 14, p 2 (l 1 ) =. (2.37) 16

32 The problem for u and T 12 consists of Eqs. (2.26), (2.29), (2.31) - (2.33). T 11 is then calculated from Eq. (2.28). 17

33 Chapter 3 Implementation 3.1 Volume Of Fluid (VOF) Method We have developed a code for 2D simulation of two-layer shear flow of immiscible fluids, named fluid 1 and 2, where either one or both fluids may be viscoelastic. The Oldroyd-B model has been used for the viscoelastic phase. The code is based on the Volume of Fluid (VOF) method and is described in detail in succeeding sections. In the VOF method, the interface between the two fluids is not tracked explicitly as in the case of interface markers. Instead, we use a volume fraction field/function C: 1 in fluid 1, C(t, x, z) = (3.1) in fluid 2, (also commonly referred to as the concentration, color or component indicator function) to track the interface that is transported by the velocity field u = ui+vj: C t + u C =. (3.2) The interface, therefore, passes through the computational cells in which < C < 1 and equation (3.2) allows for the calculation of density ρ, solvent and polymeric viscosities (µ s and µ p ), relaxation time λ and Giesekus non-linear parameter κ. 18

34 In fact, the average values for these quantities are interpolated as follows: Ψ = Ψ 1 C + Ψ 2 (1 C), Ψ is ρ, µ s, µ p, λ or κ (3.3) The VOF method is one of the most popular methods for tracking deformable interfaces because it provides a simple way of treating the topological changes of the interface, including viscous fingering between the two fluids Piecewise Interface Calculation (PLIC) In the volume of fluid (VOF) method, the location of the interface is approximately represented by the volume fraction C of fluid 1 in the cell. We have < C < 1 in cells cut by the interface and away from the interface, C = in fluid 2 and C = 1 in fluid 1. Since we lose interface information when we represent the interface by a volume fraction field, the interface needs to be reconstructed approximately in each cell. In our code, the interface is reconstructed from the volume fraction field by the Piecewise interface calculation (PLIC) method, [11], [19]. The main idea behind the interface reconstruction is to calculate the approximate normal n s to the interface in each cell, since this determines one unique linear interface with the volume fraction of the cell. At each, a plane curve is then constructed within a cell where the volume fraction C satisfies, < C < 1. This plane curve divides the cell into two parts, the first corresponding to fluid 1 with volume equal to the product of the value of C in the cell and the cell volume. The second part corresponds to fluid 2 and has volume equal to the difference between the cell volume and the volume of the first part. The coordinate of the intersection points between the plane curve and the cell boundaries, x n, are calculated from the equation for the tangent line with normal n s. 19

35 The normal to the interface n s and hence by extension the total interface curvature κ are calculated from the mollified color function c(t, x, z) that varies smoothly over a thickness h across the interface (as compared to the discontinuous volume fraction function C used in [11]): n s = c c, κ = n s [n s (n s )] n s. In general, the mollified color function is obtained by taking the convolution of the volume fraction function C with a kernel K as: c(t, x) = S C(t, x )K(x x; h)ds, where x = (x, z), x = (x, z ), ds = (dx, dz ), S is the area (volume in 3D) of the interfacial region (of radius h), and the kernel K(x x; h) is the interpolation function (mollifier), which decreases monotonically with increasing x, is equal to zero outside the interfacial region and is differentiable. In our code, the interpolation function is selected in the form: A ( 1 1 x x 2) 4 S(x 4h, if x x < 2h x; h) = 2, if x x 2h. (3.4) Here A is a normalization constant to ensure S C(t, x)ds = 1 and h is mesh size. Once the interface has been reconstructed, its motion by the flow field is modeled via the topological equation (3.2). In our code, a Lagrangian form of this equation is used since the interface evolution is governed by a transport equation. In particular, we calculate the advected coordinates of the intersection points between the plane curve and cell boundaries, sequentially in x- and z-directions, i.e. the new position of the interface is calculated using the formula: x n+1 = x n + u( t), where ( ) u = u l 1 xn x n + u r. 2

36 Here is x or z and u l and u r are the velocities on the left and right edges of the cell respectively. Substituting the new coordinates into the equation of the tangent line gives the updated normal to the interface in the cell. We then check whether the interface has protruded at the neighboring cells, and if so, we calculate the volumes moved into those cells using the relationship between the volume fraction and the coordinates of intersection (if the normal is known, the volume of fluid 1 is a function of these coordinates, [5]). The advected volume of fluid 1 in the cell is found by using the same relationship and then by subtracting the volume moved to the neighboring cells and adding the volume that comes from them. The details of the PLIC method can be found in [5] and [19] Spartial Discretization and MAC Method v(i,j+1) T12(i+1,j+1) u(i,j) p, C, T11, T22 (i,j) u(i+1,j) z x v(i,j) T12(i+1,j) Figure 3.1: MAC cell The momentum equations are finite differenced on a staggered Marker-and-Cell 21

37 (MAC) grid Figure (3.1), i.e.pressure p, volume fraction function C, and extra stress components T 11 and T 22 are located at the center of a computational cell, the velocity components u and v (v corresponds to w in 3D) are defined on its edges and non-diagonal stress components T 12 = T 21 (or T 13 in 3D) are located on the corners (in 3D case u and w would be defined on the faces parallel to the x and z directions respectively and T 13 is defined on the top front-face edge parallel to the y-axis). The spartial derivatives for pressure and velocity in the momentum equations are calculated using second-order central finite differences over a single mesh spacing, where possible. For example, p/ x in (2.1) and u/ x in (2.3) are evaluated in cell (i,k) as p x p(i, k) p(i 1, k) =, x u x However, the advective term u u/ x is discretized as u u x u(i + 1, k) u(i, k) =. x + 1, k) u(i 1, k) = u(i, k)u(i. (3.5) 2 x The basic idea in equation (3.5) is to weigh the derivatives by cell size such that the correct order of approximation is maintained in a variable mesh. This type of approximation is used in our code for all convective terms appearing in equation (3.8). 3.2 Temporal Discretization, Projection Method The two-fluid flow is modelled with the Navier-Stokes equations (2.1) which we write here in the form: ( ) u ρ t + u u = p + (T + 2µ s S) + F, (3.6) where S is the viscous stress tensor: S ij = 1 2 ( uj + u ) i, x i x j 22

38 and F the source term for the momentum equation, which includes the gravity and interfacial tension forces. the velocity field is subject to the incompressibility condition (2.14) which we restate in vector form as u =. (3.7) We proceed to solve equations (3.6) and (3.7) by decoupling the solution of (3.6) from the solution of (3.7) by Chorin s projection method [3]. In this projection method, the Navier-Stokes equations are first solved for an approximate u without the pressure gradient: u u (n) t = (u n )u n + 1 ρ (T(n) + 2µ s S (n) ) + F (n), (3.8) where n is the time-step index and u (n) is assumed known. In general, the resulting flow filed u does not satisfy the continuity equation. However, we require that u (n+1) = and u n+1 u t = p ρ. (3.9) Taking the divergence of equation (3.9) then leads to the Poisson equation for pressure: ( ) p ρ = u, (3.1) t which is solved to find the pressure field. Next, u is corrected by this pressure field and the updated solution u n+1 is found from equation (3.9). This algorithm is easier to solve than the original fully coupled set of equations. We consider no-slip boundary conditions in the z-direction and periodic boundary conditions in the x-direction. It should be noted that in the marker-and-cell (MAC) grid, see Section (3.1.2) below, the Neumann condition for pressure is automatically involved in the numerical solution, i.e., no numerical boundary condition is needed. 23

39 3.2.1 Semi-Implicit Stokes Solver To avoid the problem of viscous diffusion instability, which imposes strict restrictions on the size in the case of small Reynolds number, we use the semi-implicit scheme, [11] to calculate the intermediate velocity, u, in the Navier-Stokes equations: u u n t = (u n )u n + 1 ρ T(n) + F (n) = + 1 ρ x (2µ u s x ) + 1 ( ) v n ρ z µ s x + u, (3.11) z v v n t = (u n )v n + 1 ρ T(n) + F (n) = + 1 ρ z (2µ v s z ) + 1 ( ) v ρ x µ s x + un. (3.12) z The semi-implicit scheme dictates that the terms with asterisks should be implicit. It is precisely these terms that are responsible for viscous diffusion instability, [31]. All other terms in the Navier-Stokes equations (with superscripts (n)) are left in the explicit part. This can be expressed as { I t ρ { I t ρ [ ( ) 2µ s + ( )]} µ s u = explicit terms, (3.13) x x z z [ ( ) 2µ s + ( )]} µ s v = explicit terms. (3.14) z z x x This procedure decouples the u component from the parabolic systems (3.11 & 3.12). The above semi-implicit scheme was demonstrated in [11] to be unconditionally stable. As the full explicit scheme, this semi-implicit scheme is first order in precision, [11]. Although it is easier to solve than the coupled system, it still requires the 24

40 inversion of a large sparse matrix. What makes the method very efficient is a factorization technique, [4], that is applied to the left-hand side of equations (3.13 & 3.14): { I t ( µ s ρ x x )} { I t ρ ( z µ s z )} u = explicit terms, (3.15) { I t ( )} { µ s I t ( )} µ s v = explicit terms. (3.16) ρ z z ρ x x This factorization technique is also applied to the Oldroyd-B constitutive equations and so we will give more details in the next section. In the VOF method, the boundary conditions at the interface cannot be applied directly and the only way to take into account the force due to interfacial tension is to include it into the Navier-Stokes equations as some body force. We use the Continuous Surface Force (CSF) method, [1] for modeling interfacial tension effects. In this method, the discontinuity present at the interface is smoothed artificially, i.e., interfacial tension is assumed to act everywhere within the transition region (of finite thickness) between fluids 1 and Multigrid Method The solution of the discreet counterpart of the Poisson equation (3.1) is the most time consuming part of our Navier-Stokes solver and, consequently, an efficient solution is crucial for the performance of the whole method. The performance of some classic iterative methods, such as Gauss-Seidel, Cholesky incomplete factorization (CIF) and preconditioned conjugate gradient (PCG) methods suffer from the degradation of the convergence rate when the mesh size increases. Moreover, the system can be very ill-conditioned when a large density ratio of the two fluids involved causes a sharp variation of the coefficients. Potentially, the multigrid method is the most efficient method: to reduce the error to a constant 25

41 factor, the multigrid method needs a fixed number of iterations, whatever the mesh size. The multigrid method achieves this convergence rate independent of mesh size by combining two complementary algorithms: one iterative method to reduce the high-frequency error and one course grid correction step to eliminate low-frequency error [11]. In our code, the Poisson equation for pressure is solved by the multigrid method as in [11], with the two-color (four-color for 3D case) Gauss-Seidel method for iteration and the Galerkin method for course grid correction. 3.3 Modeling interfacial tension force (CSF) In the VOF method, interfacial tension is posed as a body force over the interfacial cells. In the continuous surface force (CSF) formulation, which is used in our algorithm, the body forces in the Navier-Stokes equations include the interfacial tension force F s which in turn is approximated by F s = σκn s δ s, where as usual, σ is the interfacial tension, κ is the total interface curvature, n s is the normal to the interface and δ s = c where C is the volume fraction function. An alternative way to model the interfacial tension force would be to use is the continuous surface stress (CSS) formulation, in which F s = T = σδ s κn s, and T = [(1 n s n s )σδ s ]. At the continuum level both methods are equivalent. 26

42 Renardy et. al., [28] recently demonstrated elimination of spurious current by applying a parabolic reconstruction of the surface tension (PROST) formulation instead of the above two formulation. 3.4 Oldroyd-B Constitutive Equation The Oldroyd-B constitutive equation is implemented into the code to capture viscoelasticity of the fluid(s). This equation is for the extra stress tensor T, which represents the polymer contribution to the shear stress: the shear stress tensor τ = 2µ s S + T, where 2µ s S is the Newtonian part of the stress tensor due to the solvent. We also recast the Oldroyd-B model in terms of S = [( u) + ( u) T ]/2 and the relaxation time λ = W l /U i : [ ] T λ + (u )T ( u)t T( u)t + T = 2µ p S, (3.17) t where µ p is the polymer viscosity. The term (u )T is the advection part of the constitutive equation and ( u)t T( u) T can be considered as the contravariant part of the constitutive equation because it appears due to the use of the contravariant time derivative. If µ s =, this model reduces to the Upper-Convected Maxwell (UCM) model. The major problem with numerical integration of differential constitutive equations is the numerical instability caused by the advection term. This instability is generated if the advection term is treated explicitly. As a result, the explicit schemes can only be used for very small relaxation times. Indeed, if we consider the differential equation λ [ T t + (u )T ] + T = 2µ p S, on the MAC mesh and treat the advection term explicitly (the remaining terms in the left-hand side being implicit), the Von-Neumann stability analysis will give 27

43 the following stability condition: λ t (CF L). 2 Here CFL is the Courant-Freidrich-Levy number. the right-hand side of this condition tends to infinity only at negligibly small viscosities or if t is zero, which is impossible. The advection instability does not disappear if the contravariant terms are added. It should be noted that the contravariant terms should always be in the explicit part. The only way to avoid the advection instability is to treat the advection terms of the Oldroyd-B constitutive equation implicitly. In this case, the numerical scheme will be unconditionally stable, i.e., it can be used for any value of the relaxation time. We have also developed a semi-implicit scheme for the more general Giesekus constitutive equations. according to this scheme, the advection terms and the last term in the left-hand side of equation (3.17) are treated as implicit, but the contravariant and nonlinear terms are in the explicit part. The Giesekus constitutive equation in the semi-implicit scheme can then be expressed as ( λ 1 + t + u (n) t x + v(n) t ) T (n+1) = explicit terms. (3.18) z Stability analysis shows that on the MAC mesh the semi-implicit scheme is unconditionally stable. However, it is extremely time consuming to solve the resulting discretized equations directly because this would necessarily require the inversion of a large sparse matrix. An efficient way would be to use a factorization technique to the left hand side of (3.18) similar to the one described in the previous section. In this case, we rewrite equation (3.18) as ( (λ+ t) 1 + λ λ + t u(n) t ) ( 1 + λ x λ + t v(n) t ) T (n+1) = explicit terms. z (3.19) 28