Inertial effect on stability of cone-and-plate flow Part 2: Non-axisymmetric modes

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1 J. Non-Newtonian Fluid Mech., 78 (1998) Inertial effect on stability of cone-and-plate flow Part 2: Non-axisymmetric modes Yuriko Renardy *, David O. Olagunju 1 Department of Mathematics and ICAM, Virginia Polytechnic Institute and State Uni ersity, Blacksburg, VA , USA Received 21 June 1997 Abstract We consider torsional flow of a viscoelastic fluid in a cone-and-plate device. This flow is known to undergo a purely elastic instability when the Deborah number reaches a critical value. Beyond this critical value a Hopf bifurcation to spiral vortices occurs. In this paper we consider the stability of the flow to non-axisymmetric disturbances when the Reynolds number is non-zero. We examine the effect of inertia on the critical value of the Deborah number at the onset of instability, the winding number of the spiral waves, as well as the wave number of the vortices. The constitutive model of Oldroyd-B is used in the present analysis. Our results show that in general when the cone angle is small the stability characteristics of the flow do not change much with inertia, indicating that the creeping model is indeed a very good approximation in such cases. We show that the critical Deborah number tends to increase with inertia in the case of non-axisymmeric disturbances. One important implication of our results is that whereas the creeping flow approximation gives a good prediction of the onset of instability the post critical bifurcations will be influenced by the inertial terms. In particular, since inertia tends to stabilize non-axisymmetric modes while destabilizing axisymmetric modes, the interaction of the two modes could be more significant than is predicted by the creeping flow results. Indeed, experimental results reported in McKinley et al., 1995, J. Fluid. Mech. 285, 123, show that for parameter values for which the creeping flow equations predict bifurcations to spiral vortices, purely axisymmetric modes were also observed. An energy analysis of the non-axisymmetric modes shows the mechanism driving the instability to be the coupling between the perturbation polymeric stress and the base velocity Elsevier Science B.V. All rights reserved. Keywords: Torsional flow; Viscoelastic fluid; Cone-and-plate device; Spiral vortices; Axisymmetric modes * Corresponding author. 1 Present address: Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA /98/$ Elsevier Science B.V. All rights reserved. PII S (97)75-X

2 28 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Introduction Consider a viscoelastic fluid placed in the gap between an inverted cone and a flat plate (Fig. 1). Assume that the cone is rotated at a constant angular speed,. The resulting flow has important applications in rheometry. In experiments with cone-and-plate flow of Boger fluids, McKinley et al. [1] observed that the base torsional flow was stable as long as the Deborah number (a dimensionless measure of the fluid s relaxation time) was not too large. As the Deborah number increased the base flow became unstable and instead of the steady azimuthal flow they observed unsteady flow with spiral vortices. In the same paper, a linear stability analysis of the governing equations confirmed the experimental observations. In their analysis they assumed that the Reynolds number is zero. In this paper we consider the effect of inertia on these spiral instabilities. In ref [2] the effect of inertia on axisymmetric disturbances was analyzed. It was shown that inertia tended to destabilize the flow. Our results show that for the range of values considered in this paper inertia has a stabilizing effect on non-axisymmetric disturbances. Although counterintuitive, this result is not unusual. Similar stabilizing behavior have been found in Taylor Couette flow when the flow is driven by rotating the outer cylinder [3]. An energy analysis similar to that employed in Refs. [3 5] is used to explain the instability mechanism. 2. Governing equations The equations of motion are the equation of mass conservation for an incompressible fluid, ṽ=, (1) and the momentum equations: Dṽ = p + T. (2) Dt For the constitutive law we use the Oldroyd-B model. This model gives a reasonable prediction of the rheological behavior of a class of dilute polymer solutions called Boger fluids in shear flows [6]. The stress in this case can be written as T =2 s D+, (3) where, the Maxwell stress, satisfies the equation Fig. 1. Cone-and-plate geometry.

3 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) LT + D Dt L =2 p D (4) Here ṽ is the velocity, p is the pressure, is the density, is the relaxation time, and p is the polymer viscosity; L is the velocity gradient tensor and D its symmetric part. The above system of equations is to be solved subject to no slip conditions at the solid boundaries. Let (r,, ) be spherical coordinates. The problem is to be solved in the domain r a, ( /2 ) /2, 2. Following [7] we nondimensionalize as follows: r = ar, t =t, p = p, (5) ṽ= a (ru, r, rw), (6) =, (7) / where is the angular speed of the cone, = / is the shear rate, = s + p, and a is the plate radius. Fig. 1 is a sketch of the problem configuration It has been shown [8,1,7] that the most dangerous modes have short wavelengths with wave numbers of the order O ( 1 ). Consequently in [7] we adopted the scaling r=, and for convenience we let =( /2 )/ so that = on the plate and =1 on the cone. Note that is a rescaled radial variable and is not related to the density. The domain of the problem is now r 1/, 1, 2. The dimensionless quantities appearing above are the Deborah number De, the Weissenberg number We, the local Reynolds number Re ( r 2 )/, and the retardation parameter p /( s + p ). In the following analysis we shall assume that, the gap between the cone and the plate, is small. A perturbation analysis in small will then be used to obtain model equations which are more tractable than the full equations. Let q=(u,, w, p,,,,,, ). We now seek an expansion of the form q=q + q 1 +. (8) Note that in this limit the domain becomes to leading order r. The scaling adopted here has been shown to give the correct leading-order behavior of the cone-and-plate flow as the gap approaches zero. Our results agree very well with those of [1,9],

4 3 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) both of which were obtained from a numerical solution of the full equations. This is further confirmation that the scaling is indeed the correct one for the short-wave elastic instabilities seen experimentally. Moreover, by defining Re in our manner, we allow for the possibility that Re may be O (1) even though is small. 3. Linear stability analysis 3.1. Base solution A base solution to the problem is given by u= =, w=, (9) = = = =, =, =2 De. (1) This solution represents a purely circumferential flow Perturbation equations We seek solutions u 1, 1, w 1, p 1, 1, 1, 1, 1, 1, 1 which are proportional to (ik) exp( t+in ), where k is a rescaled radial wavenumber, and the temporal factor. The equations governing linearized stability are given in Appendix A. The Chebychev- scheme [1] is used for the discretization in the -direction. The velocity component, 1, is expressed as a superposition of Chebychev polynomials T n (z)=cos n(arccos z), z=2 1, of degree n= ton. The continuity Eq. (A1) suggests that u 1 and w 1 be expanded up to the same degree as 1 /, i.e. up to degree N 1. In the momentum Eq. (A2), we expand p 1 to the same degree as u 1. In Eq. (A3), we expand 1 / to the same degree as 1 so that 1 is expanded up to degree N 1; 1 and 1 are expanded up to degree N 2. The variables 1, 1 and 1 are expanded to degree N 3. This results in 1N 7 coefficients that make up the discretized eigenvector. This expression is checked to be consistent so that each equation is satisfied for Chebychev polynomials up to the appropriate degree. The continuity equation is expressed in terms of Chebychev polynomials of degree up to N 1, the momentum Eqs. (A2), (A3) and (A4) to degrees N 3, N 2 and N 3, respectively, and the constitutive Eqs. (A5), (A6), (A7), (A8), (A9) and (A1)) to the same degrees as the corresponding stress variables, i.e. N 3, N 2, N 3, N 1, N 2 and N 3, respectively. Together with the six boundary conditions, this yields a 1N 7 square matrix generalized eigenvalue problems Aq= Bq. The numerical results of this paper have been convergence tested. Tables 2 4 of [2] give critical eigenvalues for the axisymmetric case with a variety of non-zero Reynolds numbers, for the case without the terms 2De u 1 in Eq. (A9) and 4De 2 u 1 in Eq. (A1). His c listed in the tables correspond to our We, or the of [7]. Here, the onset conditions depend on 1 =DeWe and 2 =Re/We. We have verified our code against these tabulated results. The transformation n n, k k yields. The transformation n n keeping the same k yields the same

5 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 2. Re=, =.41, vertical axes 1/ c, horizontal axes k/. (a) De=2. The terms 2De u 1 in Eq. (A9) and 4De 2 u 1 in Eq. (A1) are not used, as in (b) De=1. (c) De=2, the two terms are included, as in (d) De=1. Re but different Im. This is because of up-down symmetry across the gap when the cone angle is small. When the problem is reflected vertically, then the wall speed needs to be transformed accordingly, and this accounts for the different in the Im. The perturbation equations of [1] are derived under the assumptions that the base flow is purely azimuthal. If the gap angle is small it has been shown that the base flow is indeed azimuthal to leading order in [11 13]. This assumption is not however used anywhere else in their linearized stability analysis so that they solve the full equations. However, due to the approximation in the base flow their solution will not be valid for all values of. Instead it will be valid to the extent that the purely azimuthal flow satisfies the equations of motion. For creeping flow the assumed base flow is correct to order O ( ) therefore we expect their linear stability results to be correct to the same order in. The equations used in our analyses are leading terms in an expansion in small. Hence our results are correct to order o (1) in. In comparing our results with those of [1] we expect some differences of order at most O ( ) between their results and ours. Fig. 2(a, b) show the neutral stability curves for parameters as in Fig. 1 of [1]. The neutral situations have been computed to satisfy Re 1 6. The vertical axes are the reciprocal of the critical cone angle, 1/ c. The horizontal axes are k/. The parameters are Re=, =.41, De=2 for (a), De=1 for (b). In light of the remark above, we expect better agreement with [1] for the case De=1 since the critical cone angle is smaller than for the case De=2. For De=2, the general features of Fig. 2(a) and Fig. 1(a) of [1] are similar, such as the cross-over of modes and 2 atk/ slightly below 2. However, the curves for azimuthal modes n=, 2, 1 are consistently lower in the 1/ achieved. For instance, in [1],

6 32 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 3. Re=, =.41, vertical axes are reciprocal of the critical cone angle, De=1. (a) Higher modes for the case of Fig. 2(b), (b) minimum value of 1/ attained by each azimuthal mode n=, 1,, 9. De=2, modes and 2 achieve minima around 1/ slightly above 6, while in our Fig. 2(a), the minima are 5.6. In order to achieve more accuracy, the theory of [2] is fine-tuned in this paper with the addition of the 2De u 1 terming in Eq. (A9) and 4De 2 u 1 in Eq. (A1) which had been dropped to permit analytical solution of the linear stability problem. Note that although the two terms are proportional to De and thus are O (1) they are still smaller compared to the terms proportional to We which is order O(1/ ) and so many neglected as. The corresponding results are shown in Fig. 2(c, d). It is evident that the inclusion of these terms leads to better agreement for the 1/ at the minima of the curves. However, it gets rid of the cross-over of modes and 2 around k/ =17 for De=2, and modes and 4 for De=1 around k/ =6. For De=1, the n= 5 curve remains above the n= curve in [1], but Fig. 2(d) shows a cross-over. For the De=2 case shown, azimuthal modes n=, 2 are close for k/ 2, with 2 being the onset mode. For the case De=1, modes n=, 3, 4 are close with 4 being the onset mode. The inclusion of the two terms improved the agreement with [1] for the minima attained by each curve. However, some details, such as cross-overs of the critical modes mentioned previously are different. Fig. 3(a) shows results for the higher modes corresponding to Fig. 2(b) and 3(b) shows the minima attained for each mode n=, 1,, 9. Fig. 4 shows Fig. 4. Re=, =.41, vertical axes 1/ c, minima of 1/ attained by each azimuthal mode n, corresponding to Fig. 2(c, d) (a) De=2 (b) De=1.

7 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 5. Re=, De=1, 2. Onset conditions are shown. (a) We=De/, mode (line) and non-axisymmetric modes (dash) for onset conditions, (b) azimuthal mode numbers n for onset conditions are shown. the onset values for 1/ for each azimuthal mode corresponding to Fig. 2(c, d). Fig. 5 shows the variation with of the critical Weissenberg numbers, We=De/, for the axisymmetric ( ) and the non-axisymmetric onset modes (- - -). This agrees well with Fig. 12(a, b) of ref [1]. As evident, the onset conditions at Re= reported in [1] are reproduced well with our system of equations. According to these results, mode interactions would be expected. 4. Effect of inertia 4.1. Continuous spectra In ref [7], the case of an axisymmetric zero Reynolds number is treated. A stream function is introduced, the stress components and pressure are eliminated and a coupled set of two equations for the eigenvalues in the two variables (q 1, q 2 ) is derived (see his Eq and 4.17). His or De denotes our We. We combine the equations into a single one, and find that the coefficient of the highest derivative vanishes when We+1= or ( 1) We 1=. This yields eigenvalues = 1/We and =1/(( 1)We), which are the counterparts to the continuous spectra found in parallel shear flows [14 16] and which correspond to eigenfunctions that are spatially spiked; for the singular structure of modes for the continuous spectra in plane Couette and Poiseuille flows for the upper convected Maxwell liquid, see [17]. These eigenvalues are stable. For the non-axisymmetric modes, they are also continuous spectra. When is close to, the continuous spectra come close to the real axis and may cause numerical difficulty. The set of numerically computed eigenvalues are shown in Fig. 6(a) for onset conditions for Re=1, n=, =.1, =.7, k=3.2 performed with N=15, and Fig. 6(b) for onset conditions at Re=1, n= 3, =.1, =.7, k=3.4 performed with N= 25. At these values, there is no difficulty in convergence for the onset mode. However, when is close to, the critical Deborah numbers are large and the continuous spectra are closer to the real axis.

8 34 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 6. Numerically computed eigenvalues for (a) Re=1, n=, cone angle =.1, =.7, k=3.2. (b) Re=1, n= 3, =.1, =.7, k= =.41, De=1, 2 Table 1 shows the effect of inertia on the modes shown in Fig. 2(c) for De=2, =.41. The critical radial wavenumber was searched in increments of.2 and found to be k=3.4 for azimuthal modes n=, 1, 2, k=3.8 for n= 3, k=4.4 for n= 4. Inertia decreases the critical 1/ for the axisymmetric mode, while it increases it for the non-axisymmetric modes. The frequencies are decreased by inertia for both the axisymmetric and non-axisymmetric modes. the critical value of the radial wavenumber k remains approximately 3.4 for the data shown. For De=1, Table 2 shows the trends are the same for the axisymmetric mode; 1/ is equal to the critical Weissenberg number. Both the critical We and Im decrease as Re increases. The critical radial wavenumbers for each azimuthal mode at Re= usually remain critical for the range of Re shown; these are 3.2 (n=, 2, 3), 3.4 (n= 1, 4, 5), 3.6 (n= 6), 4. (n= 7), 4.2 (n= 8). However, there is some dependence on Re; e.g. at n= 5, the critical k is 3.4 for Re= and is 3.6 at Re=1. The tabulated values there are at k=3.4 since the critical We is the same to three digits at k=3.4 and 3.6. The critical radial wavenumbers were searched in increments of.2. We find that it is larger for the higher azimuthal modes. There are some differences in the trends between Tables 1 and 2 for the non-axisymmetric modes. At De=1, for azimuthal mode n= 1, the critical We decreases with increasing inertia as for the axisymmetric mode, and the frequency increases as well. For De=1, n= 2 as well as higher modes n= 3, 4, 5 behave as do the non-axisymmetric modes at De=2; i.e. the Table 1 The effect of inertia on modes n=, 1, 2, at De=2, =.41, k=3.4 Re n= n= 1 n= 2 k/ 1/ Im k/ 1/ Im k/ 1/ Im

9 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Table 2 The effect of inertia on modes n=,, 5, at De=1, =.41, k=3.2 for n=,, 3, k=3.4 for n= 4, 5 Re n= n= 1 n= 2 k/ 1/ Im k/ 1/ Im k/ 1/ Im n= 3 n= 4 n= critical We increases with increasing inertia and the frequency decreases. According to this at De=1, mode becomes more unstable than mode 3 for Re around 1. Modes, 3 and 4 then are close, but the critical mode remains 4. Our formulation would not extend to the large Re case, but if these trends persist, then this indicates that the axisymmetric mode may become the most unstable mode for larger Reynolds numbers Variation with In the previous section, the Deborah number was fixed at a specific and the critical value of 1/ was obtained. In this section, we fix the cone angle ; this would typically be fixed in an experiment. Thereafter, we have We=De/ and we search for the critical Deborah number. Fig. 7(a d) show the critical conditions for zero Reynolds number. The cone angle is fixed at.1, De= We, and at each k, the neutral De is found where Re 1 6. We scan through k and graph the critical De. The onset mode is for less than about.37, then mode 2 for a small interval of between.37 and.47, then mode 3 upto =1. The onset wavenumber k is seen from Fig. 7(c) to be approximately 3.1 over the entire range. Azimuthal modes, 2 and 3 are close, suggesting that mode interactions are likely to take place. With the addition of inertia, mode is destabilized and the non-axisymmetric modes are stabilized. As a result, the window where mode 2 is critically shifts. This is illustrated in Fig. 8(a d) for the case Re=1, =.1. where (a) shows the critical Deborah numbers for azimuthal modes n=, 1, 2 (-.-), 3, 4 (--); (b) shows the frequencies Im ; (c) shows the radial wavenumbers k and (d) shows the azimuthal modes at onset. The zero Reynolds number case corresponding to these figures are given in Fig. 7. Table 3 shows specifically the effect of inertia at =.4 on the critical Deborah numbers on mode 2 which is critical at Re= as the Reynolds number is increased, the modes and 3 which are close by. Our system of equations is valid up to a moderate Reynolds numbers, but if these trends persist, then at values of where the non-axisymmetric mode onsets close to the axisymmetric mode at Re=, these modes will cross over as Re is increased.

10 36 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 7. Critical conditions for Re=, cone angle =.1, against. (a) Critical Deborah numbers for azimuthal modes n=, 1, 2 (-.-), 3, 4 (--), 5. (b) Frequencies Im at criticality for azimuthal modes n=, 1, 2, (-.-), 3, 4 (--), 5. (c) Radial wavenumbers k at criticality for modes n=, 1 ( ), 2 (-.-), 3, 4 (--), 5. (d) Onset azimuthal modes n. Fig. 9(a, b) shows the effect of inertia for mode n= 3 in more detail. This is the critical mode at Re= for larger than approx..47. When k is fixed, the frequencies increase with Re. The value of k for the data points vary between 3.3 and 3.4 and adds to variations in Im. The stabilizing effect of inertia on the non-axisymmetric modes appears to be uniform in. Figs. 7 9 show that the frequency for modes and 1 decrease slightly overall with the addition of inertia. The frequencies for mode 2, 3 increase slightly with the addition of inertia. The specific values at =.6, =.1 are given in Fig. 1. Here the critical mode at Re= is n= 3. The critical Deborah number and frequencies are shown for n=, 1, 2, 3. These changes are continuous for larger Re as we would expect. At Re=1, for instance, the critical frequencies are.72,.278,.4661,.596, respectively, for n=, 1, 2, and 3. The critical radial wavenumbers remain constant as indicated in the figures over the Reynolds numbers shown, but may differ slightly for larger Re when the variation of the critical De with k flattens; e.g. k=3.3 at Re=1, n= 2. As the Reynolds number increases from to 1, mode is destabilized by 2.8%, and modes 1, 2, 3 are stabilized by.7,.7 and 1.6%, respectively.

11 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 8. Critical conditions for Re=1, cone angle =.1, against. (a) Critical Deborah numbers for modes n=, 1, 2 (-.-), 3, 4 (--). (b) Frequencies Im at criticality. (c) Radial wavenumbers k at onset for modes n=, 1 ( ), 2 (-.-), 3, 4 (--). (d) Azimuthal wavenumber n at onset Variation with We next examine the variation with cone angle. Fig. 12 of [1] indicates that at Re=, the onset mode is axisymmetric for less than.4 for a range of cone angles, and for between.4 and 1, the critical modes are non-axisymmetric. For small cones angles, it is shown in [2] that the critical frequency for the axisymmetric mode is proportional to. Additionally, the product 1 =DeWe at criticality remains the same as with other parameters fixed. This is illustrated in Fig. 11 for Re= and 1. Fig. 11(a, d) show the critical DeWe for Re= and 1 for varying cone angles. In [2], it is indicated that the small regime is defined as 2 = Re/De small. Fig. 11(b, e) show that Im / remains the same as decreases with other parameters fixed. Fig. 11 (c, f) show the critical radial wavenumbers. Table 3 The effect of inertia on critical Deborah numbers for mode n=, 2, 3, at =.1, =.4 Re n=

12 38 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 9. Variation of azimuthal mode n= 3, cone angle =.1, against for increasing Re. (a) Critical frequencies; (b) Critical Deborah numbers for Re= (line), 1 (+), 1 ( ). At Re=, the onset modes for between.4 and 1 are non-axisymmetric, modes can cross, and results are sensitive to changes in the cone angle. For instance, Fig. 12 of [1] shows the critical We vs. our (1 ) at fixed De=.5, 1 and 2, and at =.5, the critical mode can be n= 7 for =.125, and n= 4 for =.5. The variation with of the non-axisymmetric mode n= 1 is shown in Fig. 12 at Re=1: (a) for the critical DeWe vs., and (b) for the frequency Im divided by vs.. The data represent =.1 (--), =.2 (--), and =.1. At =.1, the results at R= and R=1 overlap on the graph; the variation with Re is small on the scales of the plot. the plot of Im / for small cone angles shows an asymptotic behavior, together with a vertical shift. The shape of the DeWe vs. curves are similar for small angles. Fig. 13 shows the variation with cone angle for n= 2, 3. The non-monotone behavior in the curves is due to modes crossing. It is not unusual to have two modes that are close as the critical radial wavenumber is varied, with other parameters fixed. On these scales, the behavior at Re=1 is very close to that of Re=. 5. Mechanism of instability The instability reported in this paper for small to moderate values of the Reynolds number is related to the purely elastic instability that has been found in shear flows in a number of different geometries. The principal mechanism driving this type of instability is the coupling between the base polymeric stresses and the perturbation velocity gradient [1,9]. In [2] the effect of inertia on this instability was considered for axisymmetric flows. It was shown that inertia tends to destabilize the flow. In this section we examine the mechanism of instability by means of an energy analysis [18]. With respect to elastic effects, this type of analysis has been used in [3,4] to explain the instabilities observed in Taylor Couette and the Taylor Dean flow of fluids described by the Oldroyd-B model. It was also used by Byars et al. [9] to explain the instabilities which occur in purely elastic instabilities of viscoelastic flow between two parallel plates. For the Oldroyd-B constitutive model considered here the stress tensor can be written as =( s + p )D+ p. (11)

13 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Following [3] we substitute Eq. (11) into the linearized momentum equations, multiply by the velocity vector and integrate over a volume to obtain the disturbance energy equation. Integrate over the volume D: r, 1 and 2 /n where n is the azimuthal wave numbers, respectively. The disturbance energy equation is then given by Fig. 1. Efect of inertia on De and critical frequencies Im for azimuthal mode n= (a, b); 1 (c, d); 2 (e, f); 3 (g, h), at =.1, =.6

14 4 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 11. Effect of inertia on critical conditions for the axisymmetric mode n= against. (a) DeWe vs. for varying cone angles =.1 (line),.1 ( ), for Re=. (b) The ratio (Im / ) vs. for Re=. (c) Critical radial wavenumbers k vs. for Re=. (d) DeWe vs. for varying cone angles =.1 (line),.1 ( ),.2 (+), for Re=1. (e) The ratio (Im / ) vs. for Re=1. (f) Critical radial wavenumbers k vs. for Re=1. K E + Re = vis We p + p. (12) Here K E is the kinetic energy, Re the Reynolds stress energy term and vis is the rate of viscous energy dissipation. The term p is the disturbance power due to polymeric stresses and p is the rate of energy production caused by the coupling between the base and perturbed flows. As in [3,4] the last term can be decomposed as Fig. 12. Variation with cone angle for non-axisymmetric mode n= 1 against at Re=1. (a) DeWe vs. for =.1 (--), =.2 (--), and =.1. (b) Critical frequency Im scaled with vs. for varying cone angles.

15 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Fig. 13. Variation with cone angle for non-axisymmetric modes n= 2, 3 against at Re=1. (a) Critical radial wavenumber vs. for =.1, n= 2, and =.1. (b) Critical DeWe vs. for =.1,.1, n= 2. (c) Critical frequency Im scaled with vs. for varying cone angles. (d f) for n= 3. p = p1 + p2 + p3. The term p1 is the rate of energy production due to the coupling between the perturbation polymeric stress and the base velocity. The term p2 on the other hand is the rate of energy production caused by the coupling between the base polymeric stress and the disturbance velocity while p3 is the rate of energy production due to the coupling between the base polymeric stress and the disturbance velocity gradient. The terms have all been nondimensionalized and simplified using the small angle approximation. See Appendix B for details. Table 4 shows the energy distribution for the instabilities shown in Fig. 1, for =.6, =.1, n= 1, 2, 3 at Re= and 1. The table lists the results for Weissenberg numbers below, at and above criticality. The trends in the energy terms as the Weissenberg number is increased past criticality is shown: viscous dissipation decreases, p3 decreases, and p1 increases. The energy balance is mainly between the viscous dissipation term vis and the term p1 which originates from the perturbational stress and the base velocity. In the table, p3 is negative and stabilizing, while p2 can be either stabilizing or destabilizing, as the term We p. Viscous dissipation is always stabilizing, and the instability is driven by p1. This reiterates the familiar scenario for instabilities in the presence of curved streamlines and a first normal stress difference in the base flow.

16 42 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Table 4 The energy distribution for the instabilities shown in Fig. 1, at =.6, =.1, for n= 1, 2, 3 n k We Re vis p1 p2 p3 K E We p Re= i i i i i i i i i Re= i E i i E i E i i E i E i i E 4.5

17 Acknowledgements Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Y.R. is funded by ONR N14-92-J-1665 and NSF-CTS D.O. is funded by NSF-DMS and would like to thank Professors Robert Olin, Michael Renardy and Yuriko Renardy of Virginia Tech. for their kind hospitality during his sabbatical leave when this work was done. Appendix A. Governing equations The equations for the linearized stability analysis are as follows; continuity, the three momentum equations, and six constitutive equations. Primes denote d/d. Boundary conditions are u 1 = 1 =w 1 =at =, 1. iku 1 1 +i w 1 =, ((k )(1 )+Re +ire )u 1 (1 )u +ikp 1 1 ik 1 i =, ((k )(1 )+Re +ire ) 1 (1 ) 1 p 1 ik 1 i =, ((k )(1 )+Re +ire )w 1 (1 )w ik 1 Re 1 in 1 +i p 1 =, (1+We +inde ) 1 2i ku 1 =, (1+We +inde ) 1 i k 1 + u 1 +inde u 1 =, i kw 1 +We 1 +(1+We +inde ) 1 i( +2WenDe) u 1 We u 1 =, (1+We +inde ) inDe 1 =, i Denw 1 +We 1 +(1+We +inde ) 1 i( +2WenDe) 1 ( w 1 +We 1 ) +2De u 1 =, (A1) (A2) (A3) (A4) (A5) (A6) (A7) (A8) (A9) 2i (2nDe n)w 1 +2De 1 +(1+We +inde ) 1 2De w 1 4De 2 u 1 =. (A1) Appendix B. Energy equation The terms in the energy equation are given in detail. Here a superscript denotes the complex conjugate. These equations have been derived for non-axisymmetric solutions, and would need to be modified for the case n=. The functions u 1, 1 etc. are the same as in Appendix A. 1 K E =2ReR( ) ( u w 2 1 ) d, (B1)

18 44 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) Re = Re ( 1 w* 1 +w 1 * 1 )d, vis = 2 (k n 2 ) n 2 I u 1 w* 1 d, p = 2R w* 1 ( u w 1 2 )d + 1 d * 1 u1 d + d * 1 1 d +w 1 du 1 d 2+ d 1 d d * 1 d + n 1 u 1 * 1 d d n 1 1 2nI ( u 1 * * 1 +w 1 * 1 )d 2kI p1 =2De nkr R u 1 * 1 d ni +2We R 1 2+ dw 1 2 d n d (u 1 * * 1 +w 1 * 1 )d n, 1 (u 1 * * 1 +w 1 * 1 )d +R n 2 ( u 1 * * 1 +w 1 * 1 )d w 1 d * d I 1 d * 1 u 1 d u d * 1 1 * 1 1 d w d * 1 1 d kw 1 * 1 d n, +w1 * 1 d n 1 1 du* 1 p2 =4 DeR (w 1 d 2De u )d 4 DeI ( n 1 u* 1 2nDew 1 u* 1 )d, 1 1 p3 = 4 DeWe n ( u w 1 2 )d +2 DenR (k 1 u* 1 + n 1 w* 1 )d 2nkDew1 u* 1 dw 1 2 WeR 1 +2 DenI 1 +2 WeI 1 du* 1 2u1 d +3 1 du* 1 kw1 d d 2Denw 1 d * d 1 d d * 1 d +3w 1 dw* 1 d d (B2) (B3) (B4) (B5) (B6) d * d 1 d. (B7) References [1] G.H. McKinley, A. Oztekin, J.A. Byars, R.A. Brown, Self-similar spiral instabilities in elastic flows between a cone and a plate, J. Fluid Mech. 285 (1995) [2] D.O. Olagunju, Inertial effect on stability of cone-and-plate flow, J. Fluid Mech. 343 (1997) [3] Y.L. Joo, E.S.G. Shaqfeh, The effects of inertia on the viscoelastic Dean and Taylor Couette flow instabilities with application to coating flows, Phys. Fluids A4 (1992a) [4] Y.L. Joo, E.S.G. Shaqfeh, A purely elastic instability in Dean and Taylor Dean flow, Phys. Fluids A4(3) (1992b)

19 Y. Renardy, D.O. Olagunju / J. Non-Newtonian Fluid Mech. 78 (1998) [5] D.D. Joseph, Y. Renardy, Fundamentals of Two-Fluid Dynamics, Parts I and II, Springer-Verlag, New York, [6] R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, 2nd ed., Wiley, New York, [7] D.O. Olagunju, Elastic instabilities in cone-and-plate flow: small gap theory, Z. Angew. Math. Phys. 46 (1995) [8] G.H. McKinley, J.A. Byars, R.A. Brown, R.C. Armstrong, Observations on the inelastic instability in cone-and-plate flow of polyisobutylene Boger fluid, J. Non-Newtonian Fluid Mech. 4 (1991) [9] J.A. Byars, A. Oztekin, R.A. Brown, G.H. Mckinley, Spiral instabilities in the flow of highly elastic fluids between rotating parallel disks, J. Fluid Mech. 271 (1994) [1] S.A. Orszag, Accurate solutions of the Orr Sommerfeld stability equation, J. Fluid Mech. 5 (1971) 689. [11] G. Hueser, E. Krause, The flow field of newtonian fluid in cone and viscometers with small gap angles, Rheol. Acta 18 (1979) [12] D.O. Olagunju, Asymptotic analysis of the finite cone-and-plate flow of a non-newtonian fluid, J. Non-Newtonian Fluid Mech. 5 (1993) [13] K. Walters, N.D. Waters, On the use of rheogoniometers, Part I Steady Shear, Polymer Systems, Macmillan, New York, 1968, pp [14] M. Renardy, Y. Renardy, Linear stability of plane Couette flow of an upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech. 22 (1986) [15] Y. Renardy, Spurt and instability in a two-layer Johnson Segalman liquid, Theor. Comput. Fluid Dynamics (1995), pp [16] R. Sureshkumar, A.N. Beris, Linear stability analysis of viscoelastic Poiseuille flows using an Arnoldi-based orthogonalization algorithm, J. Non-Newtonian Fluid mech. 56 (1995) [17] M.D. Graham, Effect of axial flow on viscoelastic Taylor Couette instability, submitted for publication, [18] D.D. Joseph, Stability of Fluid Motion, I and II, Springer-Verlag, New York,