Journal of Non-Newtonian FIuid Mechanics, 29 (1988) l-8 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

Lake Arrowhead Workshop Special Issue Papers-Introduction

The Fifth Workshop on Numerical Methods in Non-Newtonian Flow was held at the University of California Conference Center in Lake Arrowhead from June 7-11, 1987. The papers in this special issue of the Journal of Non-Newtonian Fluid Mechanics are based on papers presented at the Workshop. All manuscripts were received following the Workshop and were reviewed according to the normal practices of the Journal, so this is not a Workshop “proceedings” in the usual sense. Some papers were revised prior to acceptance, and some authors of these papers and others took the opportunity to update results for the published paper.

Four plenary lectures covering major workshop themes were presented on the first morning, as follows:

M. Renardy: Recent Advances in the Mathematical Theory of Steady Flows of Viscoelastic Fluids

W.R. Schowalter: The Behavior of Complex Fluids at Solid Boundaries E.J. Hinch: Do We Understand the Physics in the Constitutive Equa-

tions? M.J. Crochet: Improper Numerical Tools and the High Weissenberg

Number Problem. Most other papers were presented as ten-minute oral summaries of the major points, followed by poster sessions; discussion sessions followed the posters. Discussion was lively, and there was a general feeling that major advances had been made regarding both algorithm development and the understanding of underlying mathematical and physical issues since the Fourth Workshop in Spa, Belgium two years previously (this Journal, Vol. 20, 1986). In particular, there was a general sense that convergent calcula- tions can now be obtained in a range that overlaps meaningful experiments. Roger Tanner’s semi-logarithmic plot of the critical Weissenberg number for convergent calculation of isothermal extrusion from long dies as a function of workshop number (Fig. 1) is a light-hearted illustration of this progress.

Workshop participants broke up into three working groups on the final morning in order to prepare evaluations of progress and to identify out- standing problems. The groups were Numerical Techniques, Constitutive Equations and Modeling, and Interplay between Experiments and Numeri- cal Solutions. The reports were presented by the discussion leaders to the entire Workshop for further discussion, and the discussion leaders then drafted final reports after the close of the Workshop. Edited versions of the

0377-0257/88/$03.50 0 1988 Elsevier Science Publishers B.V.

2

(Wdmox

h%,

loL

I ISOTHERMAL EXTRUSION FROM LONG DIES

01 1 2 3 I 5 6 WORKSHOP NUWBER

1979 1981 1983 1965 1987 1989 OATE

Fig. 1. Approximate value of maximum Weissenberg number as a function of Workshop number. (R.I. Tanner).

reports follow. The general optimism regarding progress is reflected in all three working groups’ reports.

Organization of a workshop of this type requires a great deal of effort in communication, scheduling, etc. The major share of this effort before, during, and after the Workshop was ably borne by Kathy Lewis, of the Caltech Chemical Engineering Department, and we wish to express our thanks.

L.G. Lea1 MM. Denn

R. Keunings

Working group on numerical techniques (0. Hassager)

The discussion was divided into three issues. First, a classification of flow problems based on key features was proposed. The purpose of such a classification is to give a rough evaluation of the numerical difficulties that can be expected in a given flow problem. Second, a number of benchmark

3

problems were proposed. Such benchmark problems are helpful in the comparison of the reliability and accuracy of different numerical techniques. Finally, the numerical techniques were discussed to point out the state of the art, and designate areas where future work will be needed. The three topics are discussed below.

Flow Problems

The working group tentatively proposed the following classification: (1) Problems with smooth boundaries and without stress boundary layers.

Not many viscoelastic flow problems belong to this category. Examples include the leveling of disturbances in thin viscoelastic films and the breakup of viscoelastic jets. These flow problems are solvable with transient techniques that use the standard mixed formulation [l]. (2) Problems with smooth boundaries and with stress boundary layers.

Examples of such problems are the eccentric rotating cylinder geometry [2], the rounded entry flow, and the flow past a translating sphere. These problems are considered very difficult. (3) Problems with non-smooth boundaries that include stress singularities.

Examples of such problems are entry flow with a sharp corner and stick-slip flow. Such problems are considered very difficult.

It is recognized, of course, that some flow problems may contain features of both group (2) and (3).

Benchmark problems

The following benchmark problems were proposed for the comparison of different numerical techniques:

(1) The steady motion of a sphere along the axis of a cylinder filled with fluid. The ratio of sphere to cylinder radius should by 0.5. The quantity to compute is the nondimensional drag as a function of the Deborah number. This problem was designated as a benchmark after the Third Workshop

[3,41.

(2) The flow in a tube of sinusoidally varying cross section. The ratio of the amplitude of the sinusoid to the average radius of the tube should be 0.1 and 0.4. The ratio of the disturbance wavelength to the average radius of the tube should be 2.0. The quantity to compute is the non-dimensional pressure loss as a function of the Deborah number.

(3) The eccentric rotating cylinder, with parameters as found in [2]. The quantities to compute are the load and the phase angle.

4

(4) The stick-slip problem The quantity computed should be the velocity on the slip line [5].

(5) The 4 : 1 planar and axisymmetric contractions [5]. The quantity com- puted should be the extra pressure drop.

In all reports of calculations for these problems the size of the smallest element next to a possible singularity should be given. The fine structure of the solution should be shown to the extent possible in addition to the macroscopic “benchmark” quantities. Constitutive equations should include the upper-convected Maxwell model and the Oldroyd-B model with X2/X, = 0.8. It would be of interest to extend the problems to include studies of change of type or stability of the steady flow.

Numerical techniques

It was generally agreed that considerable progress has been made since the 1985 Workshop. In particular, it is now possible to obtain a family of steady two-dimensional solutions that are smooth and show little variance with mesh refinement [3,5,6]. The progress has come from a deeper mathe- matical understanding of the equations involved. Promising beginnings have also been made towards the construction of stable numerical methods for time-dependent viscoelastic flows [6].

The numerical difficulties were divided into three parts, as follows: (1) The integration of stresses. (2) The coupling between stresses and velocities. (3) The iterative scheme.

For differential models real progress has been made in the integration of the stress equations. This progress has come from an understanding of the first-order hyperbolic nature of the differential-type constitutive equations and the application of special techniques such as the Hughes-Brooks streamline upwind techniques. Progress for differential models has also been made in the understanding of the coupling between the stress and the velocities. The key result here is the satisfaction of the LBB condition through the use of sub-elements for the stresses [5]. Another example is the reformulation [7,2] of the equation of motion in a way that preserves ellipticity.

With respect to integral models, it was recognized that techniques have been available for some time that give almost perfect integration of the stresses. Where the integral models perform poorly, however, is in the coupling between the stresses and the velocity field and in the iterative scheme. No progress was reported on these problems, and it was concluded that more work is needed on these two latter points.

References

1 R. Keunings, Workshop paper; also, J. Comput. Phys., 62 (1986) 199. 2 R.C. Armstrong and R.A. Brown, Workshop paper. 3 0. Hassager and C. Bisgaard, J. Non-Newtonian Fluid Mech., 12 (1983) 153. 4 F. Sugeng and R.I. Tanner, J. Non-Newtonian, Fluid Mech., 20 (1986) 281. 5 J.M. Marchal and M.J. Crochet, Workshop paper; also J. Non-Newtonian Fluid Mech., 26

(1987) 77. 6 A. Beris and B. Liu, Workshop paper; also J. Non-Newtonian Fluid Mech., 26 (1988) 341,

363. 7 M. Renardy, Workshop paper; also Z. Angew. Math. Mech., 65 (1985) 449.

Working group on constitutive equations and modeling (R.G. Larson)

There was a strong consensus that techniques of numerical simulation of viscoelastic flow are now sufficiently advanced that some attention can be directed towards solving real flow problems for their own sake, and not merely as test problems in the quest for attaining solutions at higher Deborah number. There are two goals of “realistic” polymer flow simula- tion. The first is to increase our understanding of the physics of viscoelastic- ity and improve our constitutive theories. The second is to simulate polymer processing flows and thus aid in the practice of polymer engineering.

To accelerate progress towards the first goal of increasing understanding of constitutive theory and viscoelasticity, simulation results should be di- rected towards flow problems that distinguish between the predictions of competing constitutive equation and allow experimental measurement of those competing features. More sophisticated molecular-based constitutive equations will also be called for. The upper-convected Maxwell and Oldroyd-B equations have been the constitutive equations most often used in numerical simulations of viscoelastic flows. There are many choices of constitutive equations that go beyond these simple Oldroyd models, in the sense of being better founded on molecular theory, or being better able to fit experimental data for polymeric fluids of interest, or in avoiding infinite stresses. Although general agreement was lacking as to which “improve- ments” to the simple Oldroyd models are most needed, there was a con- sensus that finite extensibility is a phenomenon that surely plays a role in real polymeric fluids; it can be at least qualitatively modeled and has measurable consequences for the rheology of the fluid, in particular alleviat- ing the stress singularity in extensional flow.

Since all polymeric fluids posess a distribution of relaxation times, it was felt that some numerical studies should consider the effect of a relaxation time distribution on polymer flow. This observation is motivated in part by the report at the workshop (Marchal and Crochet) that the Deborah number

6

required to produce vortex growth in a numerical simuation of contraction flow is an order of magnitude larger than the corresponding Deborah number in experiments with “Boger” fluids. Other molecular phenomena that play a role in real polymeric liquids and can be modeled to include to some extent hydrodynamic interaction and internal viscosity in dilute solu- tions and reptation in melts. It was felt that we are near to the day when numerical simulation can becomes a tool, along with rheometry, to test constitutive equations and ultimately test our understanding of polymer physics. More interaction is needed between molecular theorists, rheology and fluid mechanics experimentalists, and experts in numerical simulation.

The second goal of polymer flow simulation is to aid in the engineering of polymer processing. For this purpose, empirical or semi-empirical constitu- tive equations that describe available rheological data are also legitimate candidates for numerical experimentation, since these equations can be useful in describing commerical materials used in polymer engineering. Numerical experience will help identify the most tractable of these equa- tions, perhaps suggest new equations, and point to the rheological experi- ments that are most crucial in determining the processing characteristics of the material.

In sum, while continued work is required to resolve fundamental mathe- matical and computational problems of viscoelastic flow, the state-of-the-art is sufficiently advanced that it can be applied with some confidence to the problems that motivated the development of the numerical techniques in the first place. Simulation of carefully chosen problems, together with velocity and stress measurements on these flows, will allow quantitative testing of both constitutive equations and the molecular theories that underlie these equations. In addition, both molecularly-based and empirical constitutive equations can be used to simulate polymer processing flows and thus assist in polymer engineering.

Working group on interplay between experiments and numerical solution (D.G. Baird)

The working group discussons focused on the following questions: (1) What should be the experimental goals in order to facilitate the develop- ment of numerical methods for flow simulation? (2) What flow experiments should be carried out, and why? (3) What fluids should be used? (4) What rheological methods should be used to characterize the test fluids? (5) What data from flow experiments should be provided for comparison with numerical simulations? (6) How important is it to use rheological models other than the upper- convected Maxwell and Oldroyd-B?

7

The issues are interrelated, as discussed below. Some group members plan on meeting in August, 1988 to review progress in the hope that experimental information can be obtained and distributed far enough in advance of the next Workshop to be of use in numerical comparisons.

It was readily agreed that we need to find a polymeric material for flow studies, preferably a melt, which is well characterized, such as the IUPAC low density polyethylene [l]. The complete data set obtained in a number of laboratories for this IUPAC material includes steady and transient shear, elastic recoil, extensional viscosity, entry pressure correction, and extrudate swell. The challenge with regard to the IUPAC polyethylene is to identify an appropriate constitutive equation which fits the rheological data but is numerically tractable. Flow visualization data in contraction flow are availa- ble on similar polymers, but not on precisely the same material. One group member (Editors’ note: specific group members agreed to take on the tasks outlined here, but we have elected not to identify individuals.) agreed to attempt to obtain a similar LDPE and carry out rheological experiments to ensure similarity of the polymer to the IUPAC material, and then to carry out studies in a 4 : 1 planar contraction, using both flow visualization and birefringence, to obtained information about streamlines and the stress field. (It was noted that other IUPAC reports are also available on polytethylene terephthalate [2] and linear polyethylenes [3].)

Studies on other basic polymeric stystems were also recommended. Poly- isobutylene solutions will be prepared in one laboratory and distributed to several others for rheological characterization and fitting of an appropriate constitutive equation. Flow visualization studies in the 4 : 1 contraction, as well as other geometries, will be carried out. The parameters in the constitu- tive equation will be made availabe to those interested in numerical compu- tations. Finally, polybutadiene melts will be synthesized and characterized in one laboratory for use in flow studies. These melts, which flow at room temperature, have narrow molecular weight distributions and may require only one relaxation time to fit rheological properties.

Other flow problems besides the 4 : 1 planar contraction were also recom- mended for further study. The first is to round the re-entrant corner on the 4 : 1 contraction to remove the singularity. (The degree of rounding was not specified.) Another problem of interest is the corrugated channel. (Editors’ note: cf. the second of the recommended numerical benchmark problems.) The value of this problem is that it is numerically tractable, experimentally accessible, and represents a flow with various degrees of Lagrangian un- steadiness but without sharp corners.

Finally, flow geometries involving slip-stick were considered. (Editors’ note: cf. the fourth benchmark problem.) It was observed that it might be possible to construct a planar contraction with lubricant on the re-entrant

8

comer, as well as a channel in which slip could be initiated half-way down the channel by the injection of a lubricant or by coating the walls with Teflon. A planar slit flow was proposed in which the stream is divided into two parts and then rejoined; in this way slip could be introduced at the interface between the two streams.

In summary, it was agreed that numerical algorithms have evolved to the point where it is time to test their quantitative predictions against experi- mental results. It will be necessary, therefore, to extend the range of choice of constitutive equations beyond the upper-convected Maxwell and Oldroyd-B to models which can fit a wider range of rheological properties. The constitutive equation should at least predict the extensional viscosity of the test fluid, for example. It is important that data from flow experiments include more than just velocity fields and streamlines; stress fields are particularly important, since calculated streamlines can remain smooth long after calculated stresses have become erratic. The stress-optical law can be questioned, but it is the only technique available at present to obtain stress field data, and such data may provide the critical test of numerical predict- ions.

References

1 J. Meissner, Pure Appl. Chem., 42 (1975) 551. 2 J.L. White and H. Yamane, Pure Appl. Chem., 57 (1985) 1441. 3 J.L. White and H. Yamane, Pure Appl. Chem., 59 (1987) 193.