COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING, VOl. 10, 135-148 (1994)

ACCURACY AND EFFICIENCY OF MULTIVARIANT

SIMULATION OF VISCOUS INCOMPRESSIBLE FLOWS FINITE ELEMENTS FOR THREE-DIMENSIONAL

MAHESH GUPTA'f AND TAI H. KWONi Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscata way,

NJ 08855, U.S.A.

SUMMARY The performance of the univariant finite element QlPo and the multivariant finite elements QtPo and RiPo are compared for simulation of three-dimensional incompressible viscous flows. Incompressible flow vortices for the R;Po element are presented. For the two flows simulated, the RZPo element is found to be not only the most accurate but also computationally the most efficient of the three types of finite element used. For one of the two flows simulated, the pressure distribution obtained by using the QlPo element is found to be amicted with the checker-board spurious pressure mode. For the uniform finite- element mesh used, the checker-board pressure mode is eliminated by taking a weighted average of the pressure over the neighbouring QlPo elements.

1. INTRODUCTION

The primitive-variable formulation, ' using the velocity and the pressure as the unknown variables, is the most popular approach for simulating incompressible ff ows by the finite- element method. However, some understanding of the convergence properties of the elements used is essential in order t o avoid erroneous results from the primitive-variable formulation. Another popular approach is to satisfy the incompressibility constraint by expressing velocity as the curl of a streamfunction. The disadvantage of the stream-function formulation is that the velocity is obtained by differentiating the streamfunction, which necessitates the use of C' continuous elements. This, in turn, requires the use of higher-order piece-wise polynomials. Another method, known as the penalty-function method, l S 3 uses a penalty function to incorporate the incompressibility constraint. In this case, pressure does not appear in the calculation. However, it was shown by Malkus and Hugues4 that the penalty-function method works if and only if the associated hybrid method works. This brings back the question of the existence and uniqueness of the solution obtained by the primitive-variable formulation of the incompressible flow problem. The question was first addressed by BabuSka' and Brezzi. The compatibility condition introduced by them is known by their names as the 'BabuSka-Brezzi condition'. To ensure the existence of a unique solution to the variational form of the Stokes-flow problem, the finite-element type used must satisfy the Babus'ka-Brezzi condition. For simulation of two-dimensional incompressible flows, a variety of finite elements

t Present address: Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, U.S.A. $ Present address: Department of Mechanical Engineering, Pohang Institute of Science & Technology, Pohang, South Korea.

0748-8025/94/020135- 14 0 1994 by John Wiley & Sons, Ltd.

Received July 1991 Revised July 1993

136 M. GUPTA AND T. H. KWON

(a) (b) (c) Figure 1 . Three-dimensional finite elements: (a) QIPo; (b) Q[Po; (c) RZPO

have been used so far. Many of these elements satisfy the BabuSka-Brezzi condition and, therefore, give a reliable simulation. For instance, PIPI- 1 triangular elements? and QIPI- quadrilateral elements ( I > 2) satisfy the BabuSka-Brezzi condition.

With the development of supercomputers, there has been some work done recently to simulate the three-dimensional flow of incompressible, viscous flows. 7 - lo However, a straightforward generalization of the two-dimensional interpolations to three dimensions makes the computations very expensive. For instance, the QtZ7)P1 element, which is the three- dimensional equivalent of the QZPI element, gives 24 velocity degrees of freedom$ and four pressure degrees of freedom. This particular element satisfies the Babudka-Brezzi condition, but it is very expensive computationally. To ensure cost-effectiveness, lower-order elements have to be enriched by adding enough velocity nodes such that the spurious pressure modes are eliminated. The idea is to add the minimum number of degrees of freedom to the 'standard' elements such that the BabuSka-Brezzi condition is satisfied.

Fortin" introduced a series of new elements, obtained by enriching the lower-order standard elements, which satisfy the Babuika-Brezzi condition and are also coputationally very efficient. The simplest member of the series, Q{Po, is derived from the QlPo (trilinear velocity and constant pressure) brick element by adding one velocity node on each of the six faces of the QIPO element (Figure 1). Each of the nodes on the faces has only one degree of freedom, which is the velocity component normal to the face. Owing to the restricted degrees of freedom at the mid-face nodes, the Q?Po element is multivariant in nature; that is, different velocity components vary differently within the element. Velocity components parallel to one of the six faces of the Q?Po element vary bilinearly over the face, whereas the variation of the velocity component normal to a face is biquadratic over the face. The pressure is constant within a Q?Po element. Another multivariant finite element, called R ~ P o , is shown in Figure l(c). It is obtained from the Q?Po element by adding one velocity node on each of the 12 edges. Each of these new nodes has only two degrees of freedom, which are the velocity components normal to the edge. Therefore, on an RZPO element the velocity components normal to an edge vary quadratically over the edge, whereas the velocity component along an edge varies linearly over the edge. The pressure is constant within an RJPo element.

t In the terminology used to name the various element types, the first letter denotes the interpolation for velocity and the second letter denotes the interpolation for pressure. The subscripts specify the order of interpolation used. p k means kth order polynomial interpolation, whereas Qk means kth order polynomial interpolation in each direction. 18 corners x of a corner node + 12 edges x f of a mid-edge node + 6 faces x 4 of a mid-face node + 1 centre node = 8 nodes; 8 nodes x 3 velocity components = 24.

ACCURACY AND EFFICIENCY OF MULTIVARIANT FINITE ELEMENTS 137

2. INCOMPRESSIBLE-FLOW VORTICES FOR THREE-DIMENSIONAL FINITE ELEMENTS

The basic structure of a divergence-free vector field is a vortex, which is circulating flow around a point. I1*l2 Qualitatively, a vortex can be represented by a circle, with the direction of circulation shown by an arrow. Vortices can be combined to form larger vortices or more complex structures like the twin vortex shown in Figure2.l' The basic structure of a divergence-free field in three dimensions is still a vortex, but it can take any orientation in space, leading to very complex structures. The simplest structure of a divergence-free vector field in three dimensions is a 'roll', which is obtained by putting the two-dimensional vortices parallel to each other (Figure 3(a)). A roll can be closed on itself to obtain a torus-like structure (Figure 3(b)), called a 'doughnut' by Fortin. Rolls and doughnuts can be combined to obtain very intricate three-dimensional structures. One of the factors which determine the ability of an element type to simulate complex incompressible flows is the number of elements required to form these basic structures of incompressible flows. l1

Fortin" presented the shape of the incompressible-flow vortices on a regular mesh of the three-dimensional QlPo and Q?Po elements. An n x n x n mesh of QlPo elements has 3(n - 1 ) 3 degrees of freedom with n 3 constraints corresponding to the continuity equation over each of

(4 Figure 2. Two-dimensional vortices: (a) a vortex; (b) two co-rotating vortices combining into a larger vortex; (c) two

counter-rotating vortices combining into a twin vortex (Fortin")

Figure 3. Three-dimensional vortices: (a) a roil; (b) a doughnut (Fortin")

138 M. GUPTA AND T. H. KWON

the n elements, but owing to 3n - 2 spurious pressure modes and 1 hydrostatic pressure mode, only n 3 - 3n + 1 of these n 3 constraints are linearly independent. This leaves 3(n - 2)'(n - 1) - (n - 2)3 linearly independent vortices. Thus, a 3 x 3 x 3 mesh of QIPo elements should have five independent vortices. As shown in Figure 4, a two-dimensional vortex can be defined in the mid-plane of each of the six 2 x 3 x 3 submeshes of the original 3 x 3 x 3 mesh. Since any one of these six vortices can be obtained as a linear combination of the remaining five, only five of these are independent. In general, on an n x n x n mesh, there are 3(n - 2)'(n - 1) such vortices, out of which only 3(n - 2)'(n - 1) - (n - 2)3 are linearly independent. These elementary vortices can be piled parallel to each other to obtain a 3 x 3 x k roll (k > 2). A doughnut can be obtained on a 4 x 3 x 3 mesh of the QIPO elements, but the torus obtained is somewhat crude.

+ 3n2(n - 1) degrees of freedom linked by n 3 - 1 constraints. This leaves 5(n - 1)3 + 3(n - 1)' independent vortices. Accordingly, a 2 x 2 x 2 mesh of Q :PO elements has eight independent vortices. Figure 5 shows the two types of vortices present. In particular, the vortex in Figure 5(a) is called a fountain. It consists of four two-dimensional vortices moving vertically up at the middle edge and recirculating through the four adjacent faces. Two more fountains similar to the one shown in Figure 5(a) can be formed in the remaining two co-ordinate directions. The vortex in Figure 5(b) is a recirculating flow around the middle edge. Six such vortices can be obtained around the six non-boundary edges of the 2 x 2 x 2 mesh, out of which only five are linearly independent.

An n x n x n mesh of Q:Po elements has 3(n -

(a) (b)

Figure 4. Vortex on a 3 x 3 x 3 mesh of QlPo elements: (a) a 2 x 3 x 3 submesh of the 3 x 3 x 3 mesh; (b) a vortex in the plane M (Fortin")

(a) (b) Figure 5 . Vortices for QlPo element: (a) a fountain; (b) a vortex around an edge (Fortin")

ACCURACY AND EFFICIENCY OF MULTIVARIANT FINITE ELEMENTS I39

These two types of vortices together give the expected eight independent vortices on a 2 x 2 x 2 mesh of QTPo elements. Even though the vortex in Figure 5(b) looks like a two-dimensional vortex, similar to the vortices for the QIPO element, since there are no nodes on the edges, the vortex cannot cross the element boundaries. Therefore, the vortices in Figure 5(b) cannot be piled parallel to each other to obtain a roll. The smallest roll on the QTPo elements is the 3 x 3 x k roll of the QIPo elements. Owing to the absence of the mid-edge nodes, the fountain in Figure 5(a) is not a doughnut. Similar to the QlPo element, a 4 x 3 x 3 mesh of Q:Po elements is required to obtain a doughnut. l 1

An n x n x n mesh of RZPo elements has 3(n - 1 ) 3 + 6n(n - 1)2 + 3n2(n - 1) degrees of freedom, which are linked by n 3 - 1 constraints. This leaves 1 l(n - 1 ) 3 + 9(n - 1)2 independent vortices. Accordingly, a 2 x 2 x 2 mesh of RgPo elements has 20 independent vortices. Eight of these vortices are the same as the eight independent vortices on a 2 x 2 x 2 mesh of the Q:Po elements (Figure 5 ) . Figure 6 shows three new types of vortices on a 2 x 2 x 2 mesh of RZPo elements. A copy of the fountain in Figure 6(a) and the vortex in Figure 6(b) can be constructed in each of the three co-ordinate directions. Two vortices similar to the one shown in Figure 6(c) can be formed around each of the six non-boundary edges of the 2 x 2 x 2 mesh. Even though the 12 vortices thus obtained are independent of each other, together with the other four types of vortices on a 2 x 2 x 2 mesh of RZPo elements (Figures 5, 6(a) and 6(b)), only six out of these 12 vortices are linearly independent. These five types of vortices together give the expected number of 20 independent vortices, which can be easily combined to obtain a roll and a doughnut on 2 x 2 x k and 2 x 2 x 2 meshes, respectively, of the R2Po element.

(b) (4 Figure 6. Vortices for R ~ P o element: (a) a fountain; (b) a vortex; (c) a twin vortex

140 M. GUPTA AND T. H. KWON

From this analysis, which element is expected to perform the best? The QlPo element does not satisfy the BabuSka-Brezzi condition. The pressure obtained by using this element is afflicted with spurious pressure modes, which may lead to instability. The QTPo and RZPo elements satisfy the Babugka-Brezzi condition and, hence, will always give a stable solution. However, the smallest independent vortices for the Q:Po element do not assemble easily into complex flows. Since we require the same number of QTPo elements as QlPo elements to form a roll and doughnut, if identical finite-element meshes are used, we expect the same order of accuracy from the QlPo and QTPo elements provided the QlPo element gives a stable solution. However, the use of the QtPo element almost doubles the number of degrees of freedom. Therefore, the QTPo element may be computationally less efficient than the QlPo element. The R $PO element satisfies the Babugka-Brezzi condition and will, therefore, always give a stable solution. The smallest vortices defined on a 2 x 2 x 2 mesh of R$Po elements assemble very well to give the rolls and doughnuts. Therefore we expect the R$Po element to be much more accurate compared to the QTPo and QIPo elements. However, from a user’s point of view, the efficiency of a finite-element type, which can be defined as the inverse of the computational time required to obtain the desired accuracy in the simulated results, can be more important than the accuracy of the element. Whether the RZPO element is also more efficient can only be determined from numerical experiments, which are presented in the rest of this paper.

Even though the numerical results presented in the following Section are focused on studying the accuracy and efficiency of the three types of finite element by analysing the convergence of the numerical results for two simple flows for which analytical results are available, the present computer program is capable of simulating complex three-dimensional flows and has been successfully used to simulate flow in a screw extruder and different types of extrusion dies. 13-15 Since the velocity component normal to a face (edge) does not remain normal to the face (edge) after transformation to the global reference frame, standard shape functions cannot be used for the multivariant elements over which different velocity components have different variations. Shape functions for the QTPo and RZPO elements have been introduced in Reference 14.

3. CONVERGENCE WITH MESH REFINEMENT

In the finite-element method, if analytical results are available we expect the numerical results to converge to their analytical values as the finite-element mesh is refined. In this Section, the numerical results obtained by using the QIPo, QTPo and R$Po elements are compared with their analytical values for two different flow problems. The problems considered are the pressure-driven flow in a square channel and the drag flow along a rectangular channel. For a Newtonian fluid such as employed in the present numerical study, analytical solutions to both isothermal flow problems are available. The results presented in the following two Sections have been normalised with respect to the shear viscosity, bulk velocity and characteristic length of the channel cross-section.

3. I . Pressure-driven j lo w in a square channel

For developed flow in a rectangular channel, the pressure varies linearly along the axis and is uniform over any cross-section. For a given axial pressure gradient, the velocity profile can be determined analytically.16 The finite-element mesh used to simulate the flow through the square channel is shown in Figure 7 . A purely axial fully developed velocity profile is imposed

ACCURACY AND EFFICIENCY OF MULTIVARIANT FINITE ELEMENTS 141

at the inlet, whereas the two transverse components of velocity and axial traction are set to zero at the exit.

The velocity profiles in the vertical plane through the centre of the channel, obtained by using the Q P o , QTPo and RgPo elements, are shown in Figures 8(a), (b) and (c), respectively. Figures 9(a), (b) and (c) present the corresponding normalized pressure distributions along the channel obtained by using the QIPo, QTPo and R ~ P o elements, respectively. As expected, the velocity and pressure distributions obtained by using the R ~ P o element are the most accurate. Even though, upon mesh refinement, the velocity profiles obtained by using the QTPo element converge to the analytical results, somewhat surprisingly these profiles are less accurate than the velocity profiles obtained by using the QlPo element on the corresponding finite-element meshes. However, the poor accuracy of the velocity distributions obtained by using the QTPo element is not contradictory to the error estimates for the BabuSka-Brezzi-compatible elements, l 7 since these error estimates bound the error in velocity and pressure together and not each of the two separately. It is noted that the pressure distributions obtained by using the Q [PO element (Figure 9) are more accurate than the pressure distributions obtained by using the corresponding finite-element meshes of the QlPo element. For flow through the square channel, the pressure distribution obtained with the QlPo element is free from spurious pressure modes. Figure 10 gives the error in the velocity distributions obtained from the three types of finite element.$ It is clear from Figure 10 that for simulating the developed flow in a square channel using a uniform mesh, the QlPo element is more efficient than the QTPo element, but the RZPo element is the most efficient among these three types of elements.

3.2. Drag jlow along a rectangular channel

For the drag flow due to the upper plate moving along a rectangular channel the pressure is zero everywhere in the channel. The velocity distribution along the channel for the Newtonian fluids can be determined analytically. l 6 To analyse the axial drag flow, the finite- element mesh used is shown in Figure 11. The normalized cross-sectional dimensions of the

Figure 7. Finite-element discretization of a square channel

§The error is defined as A lUth - Vnum I dA

flow rate Error =

where Uth is the theoretical value of the down-channel velocity at a given point, vnum is the down-channel velocity determined numerically and A is the area of cross-section.

142 M. GUPTA AND T. H. KWON

LEGEND 4~4x4 element mesh 6x6~6 element mesh

............................................. 8 x 8 ~ 8 element mesh 0 Theoretical velocity profile

Axial velocity

LEGEND 0 2~2~2elementmesh

_ 3x3~3 element mesh ............................................. A 4~4x4 element mesh

0 5 x 5 6 element mesh Theoretical velocity profile

_.-.-._._I__.___

------

0.0 0.6 1.0 15 2 0 2.5

Figure 8. Velocity profile at the mid-plane of the square channel using (a) QIPo, (b ) Axial velocity (c)

and ( c ) R;Po elements (theoretical velocity profile from Reference 16)

ACCURACY AND EFFICIENCY OF MULTIVARIANT FINITE ELEMENTS

8 W

LEGEND 0 4~4x4 element mesh

-.. 6x6~6 . element mesh A 8x8~8 element mesh 0 Theoretidjressure variation

..................................................

8 0.0 0.5 LO l.5 2.0 26 3.0

Axial distance (4

I I D 0.5 1.0 15 2 0 2.5

Axial distance

LEGEND

_ 4x4~4 element mesh -. 3x3~3 element mesh

A 5 x 5 ~ 5 element mesh 6 6~6x6 element mesh

.................................................. -._._._._.-.-._____. -- y " & o r e t i pressure variation

LEGEND 0 =elementmesh

3x3~3 element mesh A 4~4x4 element mesh * 5~5x5 element mesh y Eeoretid pressure variation

-. ...... - . .................................................. -._._.-___-.-._._._. --

Axial distance

143

Figure 9. Pressure variation along the square channel using (a) QIPO, (b) Q{Po and (c) RZPo elements (theoretical pressure variation from Reference 16)

144 M. GUPTA AND T. H. KWON

rectangular channel are 2.41 x 1.0. For fully developed drag flow along a rectangular channel, the component of traction force in the down-channel direction is zero everywhere. The axial traction and the two transverse components of velocity have been set to zero at the inlet and the exit.

The velocity profile along the vertical plane at the centre of the channel, obtained by using the QlPo, Q[Po and RJPo elements, are shown in Figures 12(a), (b) and (c), respectively. Even for axial drag flow, the R ~ P o element gives the most accurate velocity profile. As expected, owing to the discontinuous boundary conditions the largest error for the axial drag flow occurs at the top two corners. Figure 13 shows the error in the velocity distributions obtained by using the three types of elements. Again, we find the R;Po element to be the most efficient, followed by the QlPo element with the Q:Po element being the least efficient of the three types of element. As shown in Figure 14, the pressure distribution obtained from the QlPo element suffers from the checkerboard pressure mode. l 8 However, after averaging the pressure over the

I b.0 10.0 eO.0 30.0 40.0 80.0

computation time (s) 1.0

Figure 10. Error in the axial velocity distribution for flow in a square channel (computations were done on the Cray-2 computer at NCSA)

Figure 1 1 . Finite-element discretization of a rectangular channel

ACCURACY AND EFFICIENCY OF MULTIVARIANT FINITE ELEMENTS 145

Axii velocity

Axial velocity

9

s-

LEGEND 0 4x2~2 elements mesh

- 6x3~3 . elements mesh A 8~4x4 elements mesh * 10x56 elements mesh --- m meoreti@ velocity prof&

0.0 0.5 1.0 15 20 25 3.0 Axial velocity (4

Figure 12. Velocity profile along the mid-plane of the rectangular channel for axial drag flow using (a) QIPo, (b) Q;Po and (c) R2fPo elements (theoretical velocity profile from Reference 16)

146 M. GUPTA AND T. H. KWON

LEGEND 0 Using Q,Pn elements

Using .... Q 1 'P ...... 0 - elements A Us iqR'P elements ................ L..P.......- ......

0

"I

,< .............. .. -.... - ......._..... .- R

*%. Y---- ....................

0 d

I I r I 0.0 5.0 10.0 15.0 20.0 25.0 30.0 :

Computation time (s) i.0

Figure 13. Error in the down-channel velocity distribution for axial drag flow (computations were done on the Cray-2 computer at NCSA)

LEGEND 0 Using Q,Po elements at x = 1.09. y = 0.5

... t ...... V S ~ " Q ~ P * . ~e ,men~-a t . . .~= . .~04 !~ .=~:5

* ~ S ~ . C ? d ! ~ % t s

.................................. After ave- on Q 1 P .. 0. elements * ......................

Using R,'P, elements -_

.o Axial distance

Figure 14. Pressure variation for axial drag flow. Co-ordinate-frame origin is located at the lower left corner of the channel cross-section with positive x and y directions towards the right and vertically up, respectively

ACCURACY AND EFFICIENCY OF MULTIVARIANT FINITE ELEMENTS 147

neighbouring QlPo elements, the checker-board pressure mode is eliminated and the resultant pressure obtained is zero everywhere. As expected, zero pressure is obtained by using the QTPo and RZPO elements (Figure 14).

4. CONCLUSIONS

The accuracy of a given finite-element type to simulate complex incompressible flows depends upon the minimum number of elements required to construct the elementary vortices. For two different incompressible flows analysed here, for which analytical solutions are available, the numerical solutions obtained by using the QlPo, QTPo and RZPo elements converged to the analytical results. For both flows, the RZPo element is not only the most accurate but also the most efficient. Between the Q:Po and QIPO elements, which have comparable accuracy, the QIPo element is computationally more efficient. For one of the two flows simulated, the pressure distribution obtained by using the QIPO element exhibits the checker-board pressure mode. However, for the uniform finite-element mesh used here, the checker-board pressure mode could be easily eliminated by averaging the pressure over the neighbouring elements.

ACKNOWLEDGEMENTS

The authors are grateful to Professor R. S. Falk for his helpful discussions. We wish to thank the National Center for Supercomputing Applications (NCSA) at the University of Illinois at Urbana-Champaign for making their computing resources available to us.

REFERENCES 1 . 0. C. Zienkiewicz, The Finite Element Method, McGraw Hill Book Company, New York, 1971. 2. M. D. Olsen, ‘Variational finite element methods for two-dimensional and axis-symmetric Navier-

Stokes equations’, in Finite Elements in Fluids, R. H. Gallagher, G. F. Carey, J. T. Oden and 0. C. Zienkeiwicz (Eds.) Vol. 1 and 2, Wiley, 1975.

3 . R. S. Falk, ‘An analysis of the finite element method using Lagrange multiplier for stationary Stokes equations’, Math. Comput., 30, 241-249 (1976).

4. D. Malkus and T. Hugues, ‘Mixed finite element method - reduced and selective integration techniques; a unification of concepts’, Comput. Methods Appl. Mech. Eng., 15, 63-81 (1978).

5 . I . BabuSka, ‘Error bounds for finite element method’, Numer. Math., 16, 322-333 (1971). 6. F. Brezzi, ‘On the existence, uniqueness and approximation of saddle point problems arising from

Lagrangian multipliers’, RAZRO, Anal. Num., 8, R2, 129-151 (1974). 7, P. A. Tanguy, A. Fortin and F. Bertrand, ‘A 3-D finite element model for polymer melt flow’, Adv.

Polymer Technol., 8(2), 99-1 13 (1988). 8 . A. Kiani, R. Rakos and D. H. Sebastian, ‘Three dimensional computational analysis of fluted mixing

devices’, SPE ANTEC Tech. Papers, 35, 62-65 (1989). 9. R. Rakos, A. Kiani and D. H. Sebastian, ‘Computer design aids for non-axis-symmetric profile dies’,

SPE ANTEC Tech. Papers, 35, 66-68, (1989). 10. H. H. Dannelongue and P. A. Tanguy, ‘Application of adaptive meshing to polymer melt flow

modeling’, in Polymer Processing: Modeling & Validation, H. P. Wang (Ed.), Vol. MD-15, ASME, 1989.

1 1 . M. Fortin, ‘Old and new finite elements for incompressible flows’, Znt. j . numer. methodsjuids, 1,

12. D. F. Griffiths, ‘The construction of approximately divergence-free finite elements’, in Mathematics 347-364 (1981).

of Finite Elements and Applications, J. R. Whiteman (Ed.), Academic Press, 1979.

I48 M. GUPTA AND T. H. KWON

13. M. Gupta and T. H. Kwon, ‘3-D flow analysis of non-Newtonian viscous fluids using ‘enriched’ finite element’, Polym. Eng. Sci., 30, 1420 (1990).

14. M. Gupta, T. H. Kwon and Y. Jaluria, ‘Multivariant finite elements for three-dimensional simulation of viscous incompressible flows’, Int. j . numer. methods fluids, 14, 557-585 (1992).

15. M. Gupta, Y. Jaluria, V. Sernas, M. Esseghir and T. H. Kwon, ‘Numerical and experimental investigation of 3-D flow in extrusion dies’, Polym. Eng. Sci., 33, 393-399, 1993.

16. Z. Tadmor and I. Klien, Engineering Principles of Plasticating Extrusion, Krieger Publishing Co. , Malabar, Florida, 1978.

17. V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer- Verlag, Berlin, 1986.

18. R. Sani, P. M. Gresho, R. L. Lee, D. F. Griffiths and M. Engleman, ‘The cause and cure(?) of the spurious pressures generated by certain FEM solutions of incompressible Navier-Strokes equations’, Int. j . numer. methodsfluids, 1, 17-43, 171-204 (1981).