Memory Phenomena in Extrudate Swell Simulations for Annular Dies

Transcription

1 Memory Phenomena in Extrudate Swell Simulations for Annular Dies X.-L. LUO and E. MITSOULIS, Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada, KIN 9B4 Synopsis Streamline finite-element results are presented for the extrudate swell of a high density polyethylene (HDPE) melt flowing through straight, converging, and diverging annular dies. Viscometric data have been fitted using a spectrum of relaxation times and the K-BKZ integral constitutive equation. For the elongational viscosity a reasonable strain-thinning behavior has been assumed. Numerical calculations for three die designs show the dependence of diameter and thickness swell on the taper angle. It is found that the K-BKZ equation is capable of capturing memory phenomena exhibited by polymer melts in such geometries, which are in sharp contrast with Newtonian or Maxwell fluid simulations. The present results show that the diameter swell is highest for the converging, followed by the straight and then the diverging dies, in qualitative agreement with experimental findings available in the literature. While the simulations for converging and diverging annular dies failed to converge at high flow rates, converged results were obtained for straight annular dies at high flow rates (very high shear rates on the wall upstream), and the predictions for the diameter swell are in satisfactory agreement with experimental data. INTRODUCTION In the continuing effort to better understand the swelling phenomenon in polymer melt flows through extrusion dies, both experiments and numerical simulation have been used. Results for Newtonian fluids extruded from capillaries show a very good agreement between theory and experiments. For polymer melts, recent simulations by Luo and Tannerzs3 with integral constitutive equations of the K-BKZ type clearly demonstrated the power of such models in capturing long-range memory effects in extrusion through long and short capillaries in good agreement with experimental results for a fully characterized IUPAC-LDPE melt.4 In the case of contraction flows, numerical simulations by DuPont and Crochet, using the same integral constitutive equation, predicted vortex growth for low-density polyethylene (LDPE) melts in close agreement with experimental findings by White by The Society of Rheology, Inc. Published by John Wiley & Sons, Inc. Journal of Rheology, 33(8), (1989) CCC 0148~6055/89/ $04.00

2 1308 LUO AND MITSOULIS and Kondo. A detailed thorough account of the K-BKZ model can be found in a recent survey article by Tanner. Another challenging viscoelastic flow is the extrusion through tapered annular dies used in such important polymer processing operations as pipe extrusion, film blowing, blow molding, and wire coating. Experimental research has therefore shifted from the simplest die design of a straight capillary tube to more sophisticated annular dies.s* Converging and diverging annular dies are commonly used in the plastics industry for extrusion. The Newtonian fluid behavior in such geometries has been examined by Mitsoulis for straight and Mitsoulis and Heng for tapered annular dies, while Crochet and Keunings12 3 have considered the Maxwell fluid. These simulations have been performed in the same die designs employed by Orbey and Dealy, who carried out an experimental investigation on the parison swelling of high-density polyethylene (HDPE) melts. The major conclusions reached from the numerical simulations for Newtonian and Maxwell fluids were that the diameter swell was highest for the diverging, followed by the straight and then the converging annular dies, The reverse was observed for all HDPE resins studied by Orbey and Dealy. Obviously, other mechanisms take place in flow of viscoelastic fluids through tapered annular dies that cannot be explained by Newtonian or linear-viscoelasticity fluid mechanics. Apparently the flow geometry is responsible for molecular orientation phenomena, and polymer melts extruded from different die geometries will show different swelling behavior associated with their previous flow history. Since the K-BKZ integral constitutive equation has shown such promise in other viscoelastic simulations, it is our purpose here to use such a constitutive equation to describe the flow of HDPE polymer melts in extrusion through tapered annular dies. Special streamline finite elements are used to integrate the stress field and the results are compared with the experimental findings provided by Orbey and Dealy. MATHEMATICAL FORMULATION AND METHOD OF SOLUTION A schematic diagram of extrusion through a tapered annular die is given in Figure 1, along with the notation used. Two independent swell ratios can be defined in the case of an annular extrudate: the diameter swell B1, and the thickness swell B2, given by

3 EXTRUDATE SWELL FOR ANNULAR DIES 1309 Fig. 1. Schematic diagram of extrusion through a tapered annular die and notation for the numerical analysis. (1) A third swell ratio, the inner diameter swell B,, follows from the above definitions B3 = D, - 2h, D, - 2h, (3)

4 1310 LUO AND MITSOULIS The flow is governed by the general equations for conservation of mass and momentum. For an incompressible fluid under isothermal, creeping flow conditions (Re = 0) we have v-v=0 (4) o=-vp+v.7 (5) where v is the velocity vector, T is the extra-stress tensor, and p is the scalar pressure. The constitutive equation that relates T to the deformation history is a simplified form of the K-BKZ equation proposed by Papanastasiou et al.l* and is written as. C;l(t ) dt (6) where Xk and ak are the relaxation times and relaxation modulus coefficients at a reference temperature To, respectively, a and /I are material constants, and I,,I,-1 are the first invariants of the Cauchy-Green strain tensor C, and its inverse Cl, the Finger strain tensor. Luo and Tanner3 used a modification of Eq. (6) that includes the C, term to take into account a non-zero second normal stress difference, namely T(t) = j$,i.[c : exp(-y)]. (a - 3) + pip-1 + (1 - PY, * [c,- (f) + ec,(t )]dt where 0 is a constant controlling the constant ratio of the second normal stress to the first normal stress difference in simple shear flow, namely N A=- 8 Nl 1-O Equation (7) is used in the present work because a non-zero 8 value is considered more realistic for polymer melts such as HDPEa3 Integral-type constitutive equations need special numerical methods for their integration. These include a streamline finite-element method (SFEM) developed for this purpose by Luo and Tanner for flows with open streamlines (no recirculation present) and standard FEM with a special stress integration (7) (8)

5 EXTRUDATE SWELL FOR ANNULAR DIES 1311 technique developed by DuPont et a1.15 for flows with both open and closed streamlines (recirculation can be handled5). In the present work, SFEM is used, since in extrusion through annular dies only flows with open streamlines occur. The numerical method has been fully described elsewhere, and the only extension from previous work is the inclusion of the second free surface to account for the inner and outer parison walls. The flow domain is axisymmetric and cylindrical coordinates F,Z, 6 are used with two velocity components v,, u, and four extra-stress components, rz,,,7,, (normal stresses), re8 (circumferential or hoop stress) and T, (shear stress). The solution of the above conservation and constitutive equations requires the imposition of both essential and natural-type boundary conditions. A fully developed velocity profile corresponding to a given flow rate is imposed upstream. Such a profile has to be calculated numerically, since no analytical solution exists for fully developed pressure-driven flow of a K-BKZ fluid in an annulus. The momentum equation for the fully developed flow in the present case reduces to + $ [m(r)] - z = 0 which can be integrated to give where F, is an unknown integration constant and its physical meaning is the location of the maximum velocity in the annular gap. The following flow rate condition must also be satisfied (10) 0 2?? I ru(r)df = Q (11) 4n O where K,, is the entrance diameter ratio (inner/outer) and r. is the outer radius of the annulus. In the solution procedure, both the pressure gradient dp/d.z and the location of the maximum velocity F, are guessed, so that from Eq. (6) or Eq. (7) and Eq. (10) an expression for the shear rate as a function of F is obtained, g (r2 - r,2)

6 1312 LUO AND MITSOULIS where the viscosity is given by the integral in the denominator of the RHS and a change of variables has been introduced for the time s = t - t. Note that the integral on the RHS of Eq. (12) contains the unknown shear rate Jo and hence an iterative numerical process is employed to solve for +. With $J known as a function of r, the velocity profile across the annular gap is obtained by using a fourth-order Runge-Kutta integration. In general, the solution will not satisfy the flow rate and the no-slip boundary conditions on both inner and outer walls simultaneously, so a new guess for both the pressure gradient and maximum velocity position r, is provided for the next iteration, until the flow rate as well as the boundary conditions are all satisfied. The shear rate and the velocity as functions of position r provide a complete description for the strain history upstream. On the free surfaces and downstream plane, zero surface tractions are imposed (in the absence of drawdown as was the case in the experiments). The unknown free surfaces are found iteratively by constructing streamlines to satisfy the kinematic boundary condition of no crossflow normal to the surface, that is, n - v = 0. The SFEM formulation employs as primitive variables the velocities u, (or u), U, (or w), and pressure p (u-ur-p formulation). A Picard iterative scheme of direct substitution is used for the solution of the nonlinear set of equations (decoupled method). A zero-order continuation in the elasticity level is used for each flow rate, that is, for a given flow rate, we start with the Newtonian flow stress field and increase slowly the elasticity level to reach the full pseudo-body-force values. This procedure can be described by 7 = COP + (1 - C.+(O) (13) where r is the pseudo-body-force for the current iteration, r(o), 7(l) are the Newtonian and non-newtonian stress fields calculated from the latest velocity solution, and w is increased slowly from 0 to 1 through iterations. This process was found to be more stable than increasing the flow rate using a previous viscoelastic solution. Special care must be taken when selecting the domain length and finite-element grid. Figure 2 shows the discretization of the domain in its entire length for the converging and diverging dies. In all cases, the finite-element grid was extended about loh, upstream and exactly 17h, downstream of the exit. Such entrance and exit lengths were found to be sufficient to justify the imposition of a fully developed velocity profile upstream and v, = 0 downstream. From previous experience with Newtonian

7 EXTRUDATE SWELL FOR ANNULAR DIES P (a) L z/h, Fig. 2. Initial finite-element grid for tapered annular die swell simulations: (a) Converging die, K,, = 0.75, K, = 0.5, $ = -2O, L, = 5.5/z,, (b) Diverging die, K. = 0.5, K, = 0.75, c#j = 20, L, = 5.5ho. flows and after several trials, a grid was chosen having 300 quadrilateral elements, 1013 nodes (13 points across, 101 points along) with a total of 2383 non-zero degrees of freedom (unknowns) for the variables u - v - p. The grid was selectively dense where the solution is most sensitive, namely near the die exit and free surfaces. The size of the element near the walls at the die exit was about O.lh,. The adequacy of the grid to give accurate results was tested against previously established Newtonian solutions.lo* MATERIAL DATA AND THEIR FIT In the experiments by Orbey and Dealy, three blow moldinggrade HDPE resins were used for which viscometric data were obtained for the shear viscosity and first normal stress difference at 170 C in a limited range of shear rates. These are shown in Figures 3 and 4 along with their fit as obtained from the K-BKZ equation (7) with a spectrum of 8 relaxation times. The corresponding values of A,, ak, a, ~3, and 8 are summarized in Table I. These parameter values have been obtained through a simple trial and error numerical procedure, making use of the spectrum distribution of all the relaxation time contributions.3 Note that the value of 8 = corresponds to a constant ratio of the second to the first normal stress difference NJN, = -0.2, which is considered a reasonable approximation for the HDPE melts used in the experiments. It is found that the shear properties based on

8 1314 LUO AND MITSOULIS Fig. 3. Shear and elongational viscosities for HDPE melts at 170 C as predicted by Eq. (7) in comparison with experimental data: A: Resin 29; 0: Resin 27; 0: Resin 22A. Solid line: shear viscosity prediction; dashed line: elongational viscosity prediction. Eq. (7) compare well with the experimental data. The first normal stress difference is also in good agreement with the experimental data in the limited shear rate range of the experiments. Another point concerns the elongational viscosity. It has been shown2s16 that for the simulations to be successful it is necessary that the constitutive equation must at least give good predictions for steady shear and elongational measurements. Unfortunately there are no elongational viscosity data for the three HDPE resins concerned. Nevertheless, Laun17 has found that the elongational behavior of HDPE is in sharp contrast with that for LDPE melts, in the sense that HDPE melts have a strain-thinning elongational viscosity for moderate elongational rates, while LDPE melts show a strain-thickening behavior. Figure 3 also shows the prediction of Eq. (7) for the elongational viscosity. As expected, the model can adequately describe such a behavior and is, therefore, deemed

9 EXTRUDATE SWELL FOR ANNULAR DIES ' Fig. 4. First normal stress difference predicted by Eq. (7) in comparison with experimental data (same symbols as in Fig. 3). Solid line: prediction. qualitatively appropriate for the numerical simulation of HDPE melts. A relatively large value for the elongational parameter j3 is required in this case to prevent strain-thickening behavior in elongational flow. RESULTS AND DISCUSSION Straight Annular Die The first runs were carried out for the straight annular die employed by Orbey and Dealy. The annular gap IL, is mm and the diameter ratio (inner/outer) K is 0.75, with an outer diameter D, of 12.7 mm. To verify the adequacy of the finiteelement grid and the domain length, the Newtonian solution was first obtained. The diameter swell of 5.82% and thickness swell of 17.7% compare very well with previous calculations.10-12

10 1316 LUO AND MITSOULIS TABLE I Material Parameter Values in Eq. (7) for Fitting Data of HDPE Melts at 170 C k Ak (s) ak (Pa) x lo x x lo x x x lo x lo x 10-l a = 5.1, p = 0.1, B = -0.25,$ = The total pressure drop in the system AP was also used to evaluate the exit correction (or exit pressure loss) n, = AP - APO 27, (14) where APO is the pressure drop for fully developed flow in a straight annulus having a diameter ratio K,, at the entry and the same length L as the die, and 7, is the corresponding shear stress at the outer wall. The value of nex was found to be 0.161, also in good agreement with previous results. OS The experimental data for diameter and thickness swell were presented as functions of the volumetric flow rate,8 and owing to the very small annular gap used in the experiments, the shear rate on the wall reached high values (more than 300 s-l) even though the flow rate appears always small. Table II shows the shear rates on the outer and inner walls corresponding to each flow rate calculated from the fully developed solution for the TABLE II Shear Rate Values at Both Outer and Inner Annular Walls Calculated by Eq. (7) Along with Recoverable Strain Values Evaluated at the Outer Wall and Corresponding to Experimental Flow Rates for a Straight Annular Die with K = km3/s) Ll u/ vu,t O/d % )u Source: Ref. 18.

11 EXTRUDATE SWELL FOR ANNULAR DIES 1317 annular flow using the integral Eq. (7). Also shown are the corresponding values of the dimensionless recoverable strain (stress ratio) S, given by calculated for fully developed flow at the outer annular wall. As can be seen, SR never reaches high values (say of the order of 10 or more) but viscoelastic phenomena are still prominent even at such low S, values (~2). Although it is customary to present swelling results as a function of a dimensionless number, we have opted for the volumetric flow rate instead, both because of its use in the experimental results and because we believe that a single dimensionless number based on shear properties is not adequate to describe viscoelastic phenomena. Despite the high shear rates encountered, calculations for straight annular die extrudate swell were stable throughout the entire experimental flow rate range, and convergence was good as expected from the low SR values. For example, at the highest flow rate Q = cm3/s, the predicted outer free surface radius values in the last two iterations were and , respectively, and the values for the inner free surface were and , respectively. Figure 5 shows our numerical prediction for the diameter swell B, as a function of the volumetric flow rate in comparison with experimental data for all the three resins. The overall agreement between prediction and experiments is good. However, the thickness swell prediction on the whole is not as close to the experimental data as the diameter swell prediction. As shown in Figure 6, the calculations give satisfactory predictions at low flow rates, but overpredict the thickness swell at medium flow rates, and this is followed by a decrease in thickness swell at high flow rates. The discrepancies may be attributed to possible inadequacy of the model to accurately predict normal stresses at high shear rates and/or elongational behavior, which has been assumed in lieu of lacking experimental data. However, it is felt that the predictions capture well the major findings of the experimental work. The numerical simulations also provide the pressure drop in the system from which the exit correction n, can be calculated as defined by Eq. (14). Special care must be taken to consistently calculate ner from the finite-element solution, especially the AP, term for fully developed flow. Inaccuracies can be caused due to subtraction of two big numbers in the numerator (AP and APO)

12 1318 LUO AND MITSOULIS Volumetric Flow Rate, Q (cm3/s) Fig. 5. Prediction of the diameter swell B, for straight annular dies as a function of flow rate in comparison with experimental data for HDPE melts at 170 C (same symbols as in Fig. 3). as well as the existence of a second normal stress difference N2, which creates a radial pressure distribution across the die section. Thus, averaging had to be used for the pressure gradient dp/&. Figure 7 shows the exit correction versus volumetric flow rate. It is seen that n, rises substantially at low flow rates, but then it decreases slightly at higher flow rates. It is interesting to note that the reduction in exit correction occurs at about the same flow rate where the thickness swell also decreases. The overall pressure drop for the die length considered in the computations is plotted versus volumetric flow rate Q in Figure 8. It is seen that the pressure drop levels off at the highest flow rates after an initial rapid increase. Unfortunately, no pressure drop measurements were conducted in the experimental runs, so no comparison can be made at this point between simulation and experiments. Figure 9 shows the final streamline finite-element grids for several runs at different flow rates. An interesting feature to be

13 EXTRUDATE SWELL FOR ANNULAR DIES m - = 1.9 $ ? E I Volumetric Flow Rate, 0 (cm3/s) Fig. 6. Prediction of the thickness swell B2 for straight annular dies as a function of flow rate in comparison with experimental data for HDPE melts at 170 C (same symbols as in Fig. 3). noticed from these simulations is that, at high flow rates [Fig. S(c)1 the free surfaces near the die exit lips clearly exhibit a flat neck before swelling sideways. It is not clear at this point whether this reflects the real behavior of polymer melts, although there has been evidence of delayed swelling in the literature.20 Tapered Annular Dies The major features of the flow geometry for converging and diverging annular dies have been given in Figure 2. As described by Orbey and Dealy: the 20 diverging die had at exit the same annular gap and outside diameter (and therefore the same diameter ratio K) as the straight die. The converging die starts with the same dimensions as the straight die and both its inner and outer walls converge at an angle of 20 until the diameter ratio (inner/outer) K reaches 0.5. In our numerical work, the geometry is nondimensionalized so that the annular gap in all cases is unity.

14 1320 LUO AND MITSOULIS z= t s w Volumetric Flow Rate, Q (cm3/s) Fig. 7. Prediction of the exit correction n, for straight annular dies as a function of flow rate for HDPE melts at 170 C. For converging and diverging dies, our numerical results at low flow rates always predicted the right trend for the effect of geometry on annular extrudate swell. As found out experimentally and corroborated numerically, the converging die extrudates swell the most, followed by straight dies, and the diverging ones swell the least. This behavior is in sharp contrast with Newtonian and Maxwell models,13 in which case the diverging die extrudates swell the most, followed by the straight and then the converging die extrudates. Obviously, the viscoelastic behavior of polymer melts such as HDPE becomes dominant in tapered geometries and the memory phenomena associated with viscoelasticity manifest themselves beautifully in this case. Thus, in the diverging die a longer and wider channel is provided for the melt to relax and to decelerate its flow, so its swelling, which is a stress relief mechanism, becomes moderate: On the other extreme, the converging die geometry helps accelerate the flow and increase the stress levels of the melt, so that upon exit, a

15 EXTRUDATE SWELL FOR ANNULAR DIES CJ ' - 25 ' P B a Volumetric Flow Rate, Q (cm3/s) Fig. 8. Prediction of the pressure drop AP for straight annular dies as a function of flow rate for HDPE melts at 170 C. higher swelling occurs as relaxation takes place. These phenomena cannot be observed with inelastic or viscoelastic models of the Maxwell type. Figure 10 shows the final streamline finite-element grids for the converged solutions at flow rate Q = cm3/s for the converging, straight and diverging dies. The predicted diameter swell and thickness swell values for the three dies at this flow rate are compared with experimental data for the three resins in Table III. The agreement between our predictions and the experimental data is seen to be qualitatively satisfactory. From the numerical point of view, the simulations for both the converging and diverging dies failed to yield converged results at the higher experimental flow rate values. This is most likely attributed to the considerable swelling occurring in the case of the converging die, even for the most minute increase in flow rate. For the converging die, the experimental results showed that the diameter swell increased very sharply at very low flow

16 (a) & -;:; (b) *Cl LO A P z/h, z//h, Fig. 9. Final streamline finite-element grids for the straight annular die at different flow rates: (a) Q = cm3/s; (b) Q = cm3/s; (c) Q = cm3/s (cl

17 EXTRUDATE SWELL FOR ANNULAR DIES II & Go 0.0 b z/h, so 0.0 -L L z/h, Fig. 10. Final streamline finite-element grids for converging (a), straight (b), and diverging Cc) annular dies at flow rate & = cm /s.

18 1324 LUO AND MITSOULIS TABLE III Comparison between Predicted Swell Values and Experimental Data8 for Converging, Straight and Diverging Annular Dies (Q = cm3/s) Predicted Experimental data Resin 27 Resin 29 Resin 22A Geometry Bi (%) Bs (%) B, (%,) Bz (o/o) B, (%I Br (o/o) Bi ( lo) Bz (%) Converging Straight Diverging Source: Ref. 18. rates; >loo% swelling was found even at the lowest flow rate Q = cm3/s for Resins 27 and 29, and it increased to 150% at the next flow rate Q = cm3/s, and then more than 200% diameter swell was reported at higher flow rates. Such a steep initial increase in diameter swell was reflected in our simulations, and consequently resulted in large changes of the free surface profiles between iterations when the flow rate was increased. In other words, in these cases the problem nonlinearities, due to unknown location of the domain outside the dies, are so severe that the free surfaces can easily sway widely, unless one is very close to the correct configuration. An illustration of the large increase in the location of the free surfaces is given in Figure 11, which shows a typical streamline finite-element grid during iterations, that is, before divergence occurred, for the converging die at the next flow rate value Q = cm3/s. If we start from the previous solution at Q = cm3/s, divergence occurs readily since the jump in the solution process is quite large. Instead the new flow rate is set, and starting from the Newtonian solution, we increase slowly the viscoelastic contribution to reach the required level. Once this level has been reached, iterations have to be performed to obtain a converged solution that satisfies a preset tolerance for the velocities and the free surface location. However, as seen in Figure 11, the large increase and highly nonlinear shape of the extrudate makes it difficult to accurately calculate the velocity gradients near the free surfaces, owing to the need to extrapolate the field solutions to the newly updated locations. Thus, convergence becomes marginal, in a sense that the solution will diverge if more iterations are performed. Figure 11 corresponds to the iteration at which the solution gave the minimum error.

19 EXTRUDATE SWELL FOR ANNULAR DIES 1325 P L I i z/h, Fig. 11. A typical streamline finite-element grid during iterations for the converging die at flow rate Q = cm3/s. For the diverging die, the convergence difficulties at high flow rates were associated with the free surfaces showing a decrease in diameter before they became flat and parallel to the axis near the downstream end. Thus a sort of swaying inwards of the extrudate was observed close to the domain end. It is not clear whether this decrease is a reflection of the true physical behavior of HDPE melts in diverging annular extrusion flow or simply a numerical peculiarity. Further experimental work is needed, including extrudate photographs to clarify this point. CONCLUSIONS SFEM calculations using the K-BKZ integral model have given results in good agreement with experimental diameter swell data for HDPE melts extruded through straight annular dies. Also, the right trend for the geometrical effects on annular extrudate swell was predicted by our numerical simulations; in other words, the important memory phenomena exhibited by HDPE melts in tapered annular die extrusion were captured by the SFEM simulations using appropriately fitted shear viscosity and normal stress data and a realistic elongational viscosity behav-

20 1326 LUO AND MITSOULIS ior. These phenomena relate to obtaining the highest diameter swell from converging dies, followed by straight and then diverging dies. These findings are in sharp contrast with earlier predictions using the Newtonian and Maxwell models. S 3 Thus the present work has further demonstrated that the K-BKZ integral model is a practically useful constitutive equation for simulating some complex flows of commercially used polymer melts (e.g., LDPE and HDPE)?,3S5,7 Further work is needed to overcome the convergence difficulties at high flow rates in the numerical simulation for converging and diverging annular extrusion. Perhaps a different way of free surface updating is required (such as the one proposed by Kistler and &river?) when dealing with very steep diameter increase of the free surfaces. Furthermore, a full quantitative characterization of the three HDPE resins used in the experiments would be necessary before a closer quantitative agreement between predictions and experimental data for thickness swell as well as diameter swell can be obtained. In particular, normal stress data at high shear rates and elongational viscosity data which are lacking in this case would be indispensable to differentiate the swelling behavior among different HDPE melts. Financial assistance from the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. References 1. R. E. Nickell, R. I. Tanner, and B. Caswell, J. Fluid Me& 65, 189 (1974). 2. X.-L. Luo and R. I. Tanner, J. Non-Newt. Fluid Me&., 22, 61 ( X.-L. Luo and R.I. Tanner, Znt. J. Num. Meth. Eng., 25,9 ( J. Meissner, Pure Appl. Chem., 42, 551 (1975). 5. S. DuPont and M. J. Crochet, J. Non-Newt. Fluid Mech., (1988). 6. J. L. White and A. Kondo, J. Non-Newt. Fluid Me& 3,41 (1977). 7. R. I. Tanner, J. Rheol., 32, 673 (1988). 8. N. Orbey and J. M. Dealy, Polym. Eng. Sci., 24, 511 (1984). 9. P.R. &key, The Modeling of Shear and Axisymmetric Extensional Polymer Flows, Ph.D. thesis, Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA ( E. Mitsoulis, AZChE J., 32, 497 (1986). 11. E. Mitaoulis and F. L. Heng, Rheol. A&, 26, 414 (1987). 12. M. J. Crochet and R. Keunings, J. Non-Newt. Fluid Mech., 7, 199 (1980). 13. M. J. Crochet and R. Keunings, Proc. 2nd World Congr. Chem. Eng., Vol. 6, Montreal, 1981, p A. C. Papanastasiou, L. E. Striven, and C. W. Macosko, J. Rheol., 27, 367 (1983). 15. S. DuPont, J. M. Marchal, and M. J. Crochet, J. Non-Newt. Fluid Mech., 17, 157 (1985).

21 EXTRUDATE SWELL FOR ANNULAR DIES E. Mitsoulis, J. Vlachopoulos, and F.A. Mirza, Polym. Eng. Ski., 25, 677 (1985). 17. H. M. Laun, J. Rheol., 30,459 (1986). 18. N. Orbey, Sag and Swell of Extrudate from Annular Dies, Ph.D. thesis, Department of Chemical Engineering, McGill University, Montreal, Quebec (1983). 19. J. M. Marchal and M. J. Crochet, J. Non-Newt. Fluid Me&, 26, 77 (1987). 20. D. D. Joseph, J. E. Matta, and K. Chen, J. Non-Newt. Fluid Me&., 24, 31 ( S. F. Kistler and L. E. Striven, Int. J. Num. Meth. Fluids, 4, 207 (1984). Received January 27, 1989 Accepted May 19, 1989