3-D flow analysis of non-newtonian viscous fluids using “enriched” finite elements

3-D Flow Analysis of Non-Newtonian Viscous Fluids Using “Enriched” Finite Elements

MAHESH GUPTA and TAI H. KWON*

Department of Mechanical and Aerospace Engineering Rutgers, the State University of New Jersey

Piscataway, New Jersey 08855

‘Enriched’ element, Q:Po, and ‘standard’ element, QIPo, are compared for the simulation of 3-dimensional flow of non-Newtonian fluids. Several 3-dimensional polymer flow problems are analyzed. The pressure field obtained by using QIPo elements suffers from spurious pressure modes. For complex flows, depending upon the flow geometry and the boundary conditions used, QIPo elements may fail to simulate even the velocity field. Q:Po elements, which satisfy the Babugka- Brezzi condition, give accurate velocity and pressure distributions for all the problems analyzed here.

INTRODUCTION

o improve the design and control of polymer pro- T cessing equipment a good understanding of the transport phenomena in the processes such as extrusion, injection molding, compressing molding, cast- ing, forging etc., is very important. To thoroughly understand the polymer melt flow during these manufacturing processes, which involves flow through complex geometries, a 3-dimensional study of the flow with provisions for free surfaces and for the non-Newtonian rheological behavior of the polymer melt is required. Some CAD packages have been introduced recently into the market for analyzing the flow in the manufacturing processes. Unfortunately, the requirements mentioned above are beyond the capability of the software available commercially for analyzing the flow during polymer processing. The present-state-of-the-art includes only 2-dimensional simulation of the flow of the polymer melt. Research- ers are beginning to realize the importance of the 3- dimensional flow simulation, especially for the fluids with complex rheological behavior. There has been some development recently in this direction (1, 2).

A direct use of the finite element method (FEM) to simulate the flow of non-Newtonian fluids leads to a set of nonlinear equations due to the nonlinearity introduced by the rheological models. The problem must be solved by a n iterative technique, solving a sequence of linear problems.

The choice of finite element type to be used for simulating the 3-dimensional flow of non-Newtonian fluids can be different depending upon the process being simulated. If the primary interest lies in the velocity field only and the flow is relatively simple

*Currently at Department of Mechanical Engineering. Pohang Institute of Science & Technologv. Korea.

then one of the ‘standard’ elements might be a good choice. But flow domains in most of the polymer processing equipment are quite complicated and for simulating these processes knowing the velocity field is not enough. For instance, during polymer extrusion a back pressure is developed due to flow obstruc- tion caused by the die a t the end of the extruder. The back pressure is one of the major factors controlling the flow of the polymer melts in extruders. A n accurate calculation of the pressure distribution is therefore very important for a good simulation of extrusion process. For flow simulation by FEM, using velocity and pressure both as unknown variables, it is well known that a random combination of the velocity and pressure interpolations over an element cannot be used for a reliable velocity and pressure profiles. Velocity and pressure interpolations must satisfy the Babugka-Brezzi condition at the element level to avoid the spurious pressure modes (3, 4). I t is also very well known that a series of so called ‘enriched’ finite elements satisfy the Babugka-Brezzi condition (5). The simplest member of the series-Q:Po, is derived from the QIPo (trilinear velocity and constant pressure) ‘standard’ brick element. The Q:Po element is obtained by adding one node at the center of each of the six faces of Q,Po element (Fig. 1 ) . The nodes

Fig. I . (a) Q I P o element (b) @Po element.

1420 POLYMER ENGlNEERlNG A N D SCIENCE, NOVEMBER 1990, Vol. 30, No. 22

3-0 Flow Analysis of Non-Newtonian Viscous Fluids

at the center of the faces have only one degree of freedom-the velocity component normal to the face. Pressure is constant over a Q:Po element, whereas the velocity component parallel to one of the six faces varies bilinearly in the face and the variation in the velocity component normal to the face is biquadratic in the face. With the use of ‘enriched’ elements a very stable pressure profile, free from spurious pressure modes, is obtained.

Nonetheless, to the best of our knowledge, no ‘enriched’ element has been used so far in the literature to simulate the 3-dimensional flow of non-Newtonian fluids. It seems that the complexity of treating the normal velocity degree of freedom at the center of six faces has kept the researchers away from using the ‘enriched’ elements.

In the present work, a comparative study has been made on the performance of QTPo type and QIPo type of finite elements for the simulation of 3-dimensional flows of non-Newtonian fluids. Several 3-dimensional flow problems have been analyzed. For relatively simple flows like axial flow through a circular pipe or flow through a rectangular channel, the velocity profile obtained by using either of the two elements are close to the theoretical values. But even for these simple flows, the pressure variation obtained by using QlPo elements suffers from check- erboard pressure mode. In some cases, the check- erboard pressure mode may be eliminated by averaging the pressure over the neighboring elements. But for more complicated flows like circulating drag flow across a rectangular channel, QIPo element fails completely to simulate the flow. For such a flow, the velocity as well as the pressure (even after averaging) obtained by using QIPo elements is completely different from its correct value. But by using QTPo elements we got very accurate velocities and pressures for all the flows we have analyzed so far.

All the simulations reported in this paper have been run on ETA10 supercomputer located at John von Neumann National Supercomputer Center. A typical simulation with 300 elements, 462 corner nodes, and 1040 mid-face nodes requires about 4 minutes of CPU time.

MATHEMATICAL FORMULATION

For flows inside various polymer processing equipments, due to the highly viscous nature of the polymer melts and the relatively low velocities, inertial forces are negligible compared to the viscous forces. For such creeping flows, neglecting the inertial term in the momentum equation we get

d i v G + Z = O i n Q (1) where

C? = stress tensor X = body force per unit volume Q is a bounded region in R 3

For generalized Newtonian fluids (1 0)

a = 2pF - p l (2)

where

p = viscosity of the fluid F = strain rate tensor p = pressure i = identity tensor

The strain rate tensor is given by

= $(grad ri + (grad LI)~) (3)

where LI = velocity a t a point in Q. For the flow of polymer melts, body forces like

gravitational force are negligible compared to the viscous forces. Neglecting the body force term in E q I and using Eq 2

div(2pZ) - grad p = 0 (4)

Polymer melts during processing can be regarded as incompressible. Incompressibility can be represented as

div li = 0 in R (5)

For a well posed problem, boundary conditions to be satisfied are

- 0 . f i = T on rT

and

G = LI on r, where

r i = outward unit normal vector at the boundary rT = part of the boundary where the traction vector

r, = part of the boundary where the velocity is given + = prescribed value of the traction vector on rT li = prescribed value of velocity on r, For Newtonian fluids viscosity is constant for the isothermal flows. Whereas for the generalized New- tonian fluids, in addition to temperature, viscosity also depends upon the shear strain rate.

is given -

p = 4 0 . 4 (8) where 0 = temperature ee = effective shear strain rate

ce = @7) (9)

p = Kc:-’ (101

In particular for power law fluids

The variational form of the incompressible flow problem described in Eqs 4-7 is

L2pZ(LI):i(C) dx - p div d dx (1 1)

L = i ? . . d d s V d E V

l q div ii dx = 0 Vq E Q (12)

1421 POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1990, Vol. 30, No. 22

Mahesh Gupta and Tai H . Kwon

where V and Q are the solution spaces of velocity and pressure, respectively. An element in V must not only be square integrable but its gradient must also be square integrable, whereas the only restriction on Q is that it is a function space of square integrable functions. Defining

L”(0) = {ul lu(x) I2 d x ]

as the space of square integrable functions and H’(Q) being the usual Sobolov space,

s H’(0 ) = ( u I u E L2(0), grad u E (L’(Q))”)

we have V = {a16 E (H’(0))”. 6 = fi on I’,) and Q = L’(n). If v,, and Q h are the finite dimensional sub- spaces of V and Q, respectively then the discrete formulation of the problem is:

Find ( i ih. ph) E vh X Q,, such that

P (13)

where

COMPATIBILITY CONDITION FOR DISCRETIZATION OF VELOCITY

AND PRESSURE

A s mentioned before, for reliable velocity and pressure profiles, the finite dimensional solution spaces vh and Qh of the velocity and pressure, respectively, cannot be selected randomly. A s reported by Sani (6), a random combination of velocity and pressure interpolations may lead to spurious pressure modes. For instance, the QIPo element frequently exhibits the checker-board pressure mode. With the checker- board pressure mode, for a constant pressure flow, different values of pressure are obtained on adjacent elements. These values are repeated alternately over the complete flow domain.

Another problem reported by Fortin (7) is that even Eq 14, which is an approximation of Eq 5, might be too strong to satisfy. This is called the ‘locking phenomena. ’

These problems in the simulation of incompressible flows can be avoided if the combination of the velocity and pressure interpolations used over a n element satisfy the Babugka-Brezzi condition, which says that there must exist a positive constant k such that

where

IIqhIIQ/R = inf IIqh -k c l lQ c E R

must be used because pressure is defined only up to a n additive constant.

However, the Babugka-Brezzi condition is rather abstract and is difficult to check directly. It was shown by Fortin (8) that the Babugka-Brezzi condition will be satisfied by a combination of discrete velocity and pressure solution spaces if for any given velocity field ii E V , one can construct a discrete vector field i i h in the velocity solution space such that

l d i v ( f i - rih) qh du = 0 vqh E Qh (16)

and such that i i h continuously depends on G.

duces to For constant pressure finite elements Eq 16 re-

l d i v i i h d x = l div fi d x (17)

for any element e, that is t ih has the same mass balance as G on each element.

The presence of the nodes at the center of the faces of the QTPo element, with the velocity component normal to the face as a n explicit degree of freedom, gives control of the flow through each of the six faces of the element. An interpolation on Q:P, element satisfying Eq 17 is

&(Cr) = G(Ct) i = 1. . .8 (18)

Where ci are the eight corner nodes and ft are the six faces of a QTP, element. Therefore QTP, element satisfies the Babugka-Brezzi condition.

Whereas due to the absence of the mid-face nodes, any change in the velocity at one of the nodes of a QIPo element affects the flow rate through three faces of the element. There is no obvious interpolation on the QIPo element for which Eq 17 is satisfied. Indeed, a s reported by Sani (6), the pressure field obtained from the QIPo element suffers from the checker-board pressure mode, which implies that the QIPo element does not satisfy the Babugka-Brezzi condition.

FINITE ELEMENT FORMULATION

Using the finite element method, velocity and pressure inside a n element can be represented as

lule = [ r V l { ~ l e (20)

Pe = INpllple (21)

where

{ U I ~ , = column matrix of three velocity components at a point inside the element

1422 POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1990, Wol. 30, No. 22


(& = column matrix of velocity components at the

p,, = pressure at a point inside the element = column matrix of pressure at the nodes of the

Differentiating Eq 20, the six independent strain rate components can be arranged in a column matrix as

node of a n element

element

( € 1 = "'l(ule (22)

Using these notations, discretized linear momentum and continuity equations can be expressed in the matrix form as follows:

IKI(4 - [GllPI = If1 (231

IGI'lul = (01 (24)

where [K] is the global stiffness matrix, [GI is the global matrix for the incompressibility constraint ( f) is the column matrix for the work equivalent force due to the traction and body forces, (0) is a zero column matrix, (u ) and ( p ) are the column matrices of unknown velocities and pressure at nodes. For each element [K), [GI, and {f] are:

[Kle = Le "'1 ID"'] dx

PIe = ~[N,ITlhJT~N'1 $2, dx (26)

(25)

Neglecting the body forces

( f l = JIWT{TI 1'e ds (271

where (TI = column matrix of the traction force at a point

on the surface of the element

O l r % 2 r r 0 0 0 0 0 0 0 0

0 0 0 4 p o 0 O 1 0 0 2 p o o

and

{h)= 11 1 1 0 0 O J '

In E q 20. for a QIPo element [W is the matrix of standard shape functions such that if Ni is shape function for the it" node then j'" velocity component is given by

u, = N,uj j = 1 . 2, 3 (28)

where uj is the j t h velocity component at the if" node. But the standard shape functions cannot be used for 'enriched' finite elements like QTP,. By separating the normal and the tangential components of velocity at the mid-face nodes of a QTPo element, we have developed a new type of shape functions to handle the normal velocity degree of freedom at the mid-face nodes of QTP, elements. Details of the shape func-

tions are presented in (9). With these shape functions the velocity component parallel to one of the six faces of a QTP, element varies bilinearly in the face, whereas the velocity component normal to the face varies biquadratically in the face.

EXAMPLES

In this section performance of QIPo and Q:Po elements is compared for three different flow problems. For all three problems, the power law model has been used to model the nonlinear behavior of the polymer melt.

Flow of Polymers through a Circular Pipe

In this example, flow of polymer melts through a circular pipe for a given pressure gradient is analyzed. The finite element mesh used for this example is shown in Fig. 2. For flow through a circular pipe, a n analytical solution for velocity is known for New- tonian as well as for power law fluids (10). Therefore exact velocities for the power law fluids can be specified at the inlet. Traction is specified to be zero at the exit. But a s we move to more complex geometries and more involved boundary conditions in the follow- ing examples, analytical solutions are not available for the velocity distribution. To determine the effect of inaccuracies in the boundary conditions on the velocity field determined from the F E M analysis, the velocity profile for the Newtonian fluids has been specified at the inlet, even though the flow of power law fluids is being simulated in this example. The velocity variation along a diameter of the circular pipe is shown in Fig. 3. I t is clear from Fig. 3 that even though a very inaccurate velocity distribution has been specified at the inlet, velocities obtained at the exit are very close to the exact values obtained analytically for the power law fluids. Both QTPo and QIPo elements give accurate velocities for this problem. Of course, if the exact velocity profile is specified at the entrance, the developed flow is simulated ac- curately by both types of elements.

For axial flow through a circular pipe, the pressure gradient is constant along the axis and the pressure is constant across the pipe. Fig. 4 shows the pressure variation along the axial direction at four different locations obtained by using QIPo elements.

Fig. 2. Finite e l emen t discretization of a circular pipe.

POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1990, Vol. 30, No. 22 1423

M a h e s h G u p t a and T a i H . Kwon

LEGEND Velocity pmfile for Newtonian fluids specified at=

. . .. Velocity,,prgfile .. .. . . .using Q,P, elements . . . .,

AnaAtical velocity profile for power law fluids ..~... ve!~~~~~.P~~~o”e.us!ng.Q~p*e~.men~ ...................... ---.

0 ______--_-

Fig. 3. Velocity profile along a diameter ofa circular pipe.

LEGEND

At x = -0.001m, y = 4.001m 0

x = 0.0, y = 0.0 at the cenkr of a cirmlar crosssection ’b- 2 z-

Fig. 4. Pressure Variation along the axis of a circular pipe using g , P , elements.

It is evident from Fig. 4 that the pressure distribution obtained by using QIPo element is suffering from the checker-board pressure mode. But the pressure variation obtained at the center of the pipe by averaging these four pressure variations is very close to its actual values (Fig. 5(a)). A perfectly linear pressure variation, very close to the actual pressure variation is obtained using Q:Po elements (Fig. 5(b]).

Flow of Polymers through a Converging Rectangular Channel

Flow through a converging rectangular channel is very common during polymer processing. For instance in extruders, just before entering the die, the polymer goes through a converging section. Also in injection molding, before entering the mold, the polymer passes through a converging channel. Since the analytical solution for the velocity field through a rectangular channel for the power law fluids is not

known, the velocity profile for the flow of Newtonian fluids through a rectangular channel, which can be calculated analytically (1 1). has been specified at the inlet. The finite element mesh used to study the converging flow is shown in Fig. 6. Velocity distributions in the vertical plane through the center of the channel obtained by using Q,Po and Q:Po elements are shown in Fig. 7(a) and Fig. 7(b) , respectively. Pressure variation along the axis of the channel is shown in Fig. 8. Both QIPo and @:Po elements give almost the same pressure variation along the converging rectangular channel. In this particular case, the pressure obtained even by using Q,Po elements is almost free from spurious pressure modes.

2 4 - a 0.00 0.01 O.W 0.03 0.04 OM

Adal distance (m)

Fig. 5. Pressure variation along the axis of a circular pipe la) using @Po element (bl using &Po element.

Fig. 6. Finite element discretization of a converging rectangular channel.



. a l

Fig. 7. Velocity field along a converging rectangular channel (a) using QIPo elements (b] using @Po elements.

for the comparison is a uniform mesh shown in Fig. 10. Before analyzing the combined flow in axial and cross-channel directions, we analyzed the two flows separately.

For axial drag flow, very similar velocity profiles are obtained by using QTPo and QIPo elements (Fig. 11 ). A s expected Q:Po elements give zero pressure gradient for the axial drag flow. Pressure distribution obtained by using QIPo elements suffers from checker-board pressure mode. After averaging over the QIPo elements, almost zero pressure gradient is obtained on the uniform mesh. Even though the weighted average of the pressure over the nonuniform mesh shows some fluctuations, the pressure

f 0.00 0.01 0.- 0.03 a04 0.00 0.00 0.m

-*-(m)

Fig. 8. Pressure Variation along a converging rectangular channel.

Flow of Polymers through a Screw Extruder Channel

In a screw extruder, polymer melt flows through the helical channel in the screw, a s the screw is rotated and the barrel is kept stationary. But it is easier to analyze the flow inside the extruder if the screw is assumed to be stationary while the barrel is treated as rotating about the stationary screw. The analysis is further simplified by unwinding the helical channel into a straight channel and considering the barrel as a n infinite plate sliding on top of the rectangular channel with a constant velocity a t a n angle (equal to the helix angle) with the down channel direction. The two flows are not exactly the same. There is a difference in radial pressure distribution due to centrifugal forces. However, if the channel is not very deep, the centrifugal effects can be ne- glected. McKelvey [ 12) obtained a curvature correc- tionfactor to estimate the permissibility of the sim- plifying assumption.

Extruder dimensions used in this example are: outer diameter = 3.07 cm, channel depth = 0.4775 cm, channel width = 1.152 cm, helix angle = 16.1888", and screw frequency = 1 Hz. LDPE is the polymer being extruded. For LDPE the power law coefficient n = 0.48 and K = 735.76 Pa.$.

Since the maximum velocity and pressure variation is at the two top corners, a good finite element mesh to analyze the flow, with finer elements at the two top corners, is shown in Fig. 9. Another mesh used

Fig. 9. Nonuniform finite element discretization of an extruder channel.

Fig. 10. Unqormfinite element discretization of an extruder channel.

0

Fig. 1 1 . Axial velocity distribution at the center of the extruder channel for axial dragflow.


Mahesh Gupta and T a i H. K w o n

values are negligible compared to the viscous forces, which are of the order of lo4 Pa (Fig. 12).

The characteristic which makes the cross-channel flow different from the flows studied so far is the large pressure gradients. There are two singularities a t the two top corners and the pressure changes from -w to +w from one end to the other. Velocity distribution across the channel obtained by using the nonuniform mesh (shown in Fig. 9) of Q:Po elements is given in Fig. 13. Pressure contours for the cross- channel drag flow obtained by using the nonuniform mesh of Q:Po elements are shown in Fig. 14. Velocity

LEGEND 0 Alter averaging on uniform mesh ol Q,P.

After averaging on nonuniform mesh of P P ....... ..... ........ -.L .I

..:.. . on.non~orm.m~~."'Q~~*e!e.men~ .........

I / I b 0.w 0.01 0:m 0:m 0:oc 0'06 0:oe AXM distance (m)

Fig. 12. Pressure variation along the extruder channel for axial dragflow (a) using QIPo elements (b) using QTPo elements and after averaging on Q , P , elements.

Fig. 13. Velocity field across the extruder channel for cross-channel drag flow using the nonunlJorm mesh of Q:Po elements.

vectors in Fig. 13 and pressure contours in Fig. 14 are as expected for the cross-channel drag flow.

On the other hand, the QIPo element fails completely to simulate the cross-channel flow. For the nonuniform mesh of QIPo elements velocities obtained do not even satisfy the constant velocity boundary condition at the top surface and zero velocity boundary condition at the three remaining sides, for the normal blasting factor of 10" used for forcing the boundary condition by the blasting technique. A s the blasting factor is increased, the boundary conditions are satisfied to a greater extent but then the velocity obtained at some of the inside nodes is as large as 1 O4 cm/s for the top surface drag velocity of 2.68897 cm/s. A better velocity distribution, satisfying the boundary conditions, is obtained on the uniform mesh of Q,Po elements (shown in Fig. 10). The velocity vectors across the channel by using QIPo elements on the uniform mesh are shown in Fig. 15(a). Even this velocity distribution obtained by using the uniform mesh of QIPo elements is very different from the expected velocity distribution for the cross-channel flow. Q,Po elements were next tried with a uniform mesh of 12 elements along the

Fig. 14. Pressure contours across the extruder channel for cross-channel drag flow using the nonunijorm mesh ofQ;P, elements.

I . . . . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . I

Fig. 15. Velocity field across the extruder channel for cross-channel dragflow using a unworm mesh of QIPo elements (a] using 7 elements in each direction across the channel (b) using 12 elements along the width and 8 elements along the depth of the channel.

1426 POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1990, Yo/. 30, No. 22

3-0 Flow A n a l y s i s of Non-Newtonian V i s c o u s F l u i d s

width and 8 elements along the depth of the channel. The velocity distribution obtained on this mesh by using QIPo elements is as absurd as shown in Fig. 15(b). As expected @:Po elements give a very good velocity even on a uniform mesh (Fig. 16).

In all the cross-channel drag flow simulations so far, the velocity at the top corners has been specified to be zero. If leakage flow is allowed at the top corners by specifying the top plate velocity at the corner nodes, both QIPo as well as Q:Po elements success- fully simulate the cross-channel flow. Figures 17(a) and (b) show the velocity distribution obtained by using QIPo and Q:Po elements, respectively, for the cross-channel drag flow with leakage flow at the top corners. Even with leakage flow, the pressure distribution for cross-channel drag flow obtained by using QIPo elements is afflicted with the checker-board pressure mode. Figure 18[a) shows the pressure contours for the pressure distribution for the cross-channel drag flow obtained from QIPo elements. Small cells in the pressure contours in Fig. 18(a) are due to the large fluctuations in the pressure values on the neighboring elements. The pressure distribution obtained after taking a weighted average of the pressure over the neighboring QIPo elements is shown in Fig. 18(b), which is similar to the pressure distribu-

/ - - - - - L

i

- - 1 - t h

\ , - - 4

\ L / ...

/ . - - / . - - - .. - L - - L L

Fig. 16. Velocity field across the extruder channel for cross-channel dragflow using a uniform mesh of Q:Po elements.

Fig. 17. Velocity field across the extruder channel fo r cross-channel dragflow with leakage at the top corners (aJ using QIPo element (bj using Q:Po element.

tion obtained by using Q:Po elements (Fig. 19). In Fig. 18(b), pressure values at the four corner nodes, which are obtained by averaging the pressure on the two neighboring QIPo elements only and are still afflicted with the checker-board pressure mode, have been discarded. That is why in Fig. 18(b) the extreme values of pressure are not at the top corners.

The axial and cross-channel drag flows are combined next to simulate the flow in the screw extruder. One difficulty which arises in simulating the general drag flow is the unavailability of the boundary conditions at the inlet and the exit. For the axial drag flow, velocity is zero across the channel and traction force is zero in the down channel direction on the inlet and outlet boundary surfaces. For the cross- channel flow, on the inlet and outlet boundaries, traction force is zero across the channel and the velocity is zero in the down channel direction. Noth- ing like this can be specified for the general drag flow. We have used velocities across the channel obtained from cross-channel drag flow analysis and velocities in the down channel direction obtained from the axial drag flow analysis to specify the boundary condition at the inlet and exit for the general drag flow problem. The velocity distribution

a

Fig. 18. Pressure distribution across the extruder channel for cross-channel drag flow with leakage at the top corners using Q I P n element (a) with checker-board pressure mode (bJ after averaging over the neighboring elements.

Fig. 19. Pressure distribution across the extruder channel for cross-channel dragflow with leakage at the top corners using Q:Po element.

POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1990, Yo/. 30, No. 22 1427

Mahesh Gupta and Tai H. Kwon

across the channel for the general drag flow obtained by using nonuniform mesh of Q:P, elements is shown in Fig. 20. Pressure contours on the same mesh of QTP, elements for the general drag flow are shown in Fig. 21. The velocity and pressure distributions across the channel for the general drag flow with leakage flow obtained by using QIPo elements are shown in Figs. 22 and 23, respectively. The pressure distribution in Fig. 23 has been obtained by averaging the pressure over the neighboring QIPo elements to eliminate the checker-board pressure mode. The velocity and pressure distributions across the channel for the general drag flow with leakage flow obtained by using Q:Po element are shown in Figs. 24 and 25, respectively. A s expected, the pressure contours are very similar to the contours obtained for the cross-channel drag flow. Because of the non-linear nature of the power law fluids, velocity distributions for the axial and cross-channel flows are expected to be different from the general drag

flow. Fig. 26(a) gives the variation in axial velocity a t the center of the extruder channel for axial and general drag flows. Variation in cross-channel velocity for cross-channel and general drag flows are given in Fig. 26(b).

CONCLUSIONS

Finite element simulation of certain simple flows of non-Newtonian fluids using QIPo elements gives quite accurate velocity field. But even for these simple flows pressure field obtained by using QIPo elements suffers from spurious pressure modes. For simple flows a reasonably accurate pressure distribution is obtained by averaging the pressure over the adjacent QIPo elements. The flow inside most of the polymer processing equipments is rather complex. The complex nature of the flow makes it essential to have a 3-dimensional flow analysis. QIPo elements may fail to simulate such complex flows depending

i:.:,;. .. - - - - - ' ,q - L

Fig. 20. Velocity field across the extruder channel for general drag flow using a nonuniJorm mesh of &Po elements.

Fig. 23. Pressure distribution across the extruder channel for general dragflow with leakage at the top corners using QIP, elements.

Fig. 21. Pressure contours across the extruder channel for general dragflow using a nonuniform mesh of QTP, elements.

Fig. 24. Velocity field across the extruder channel for general dragflow with leakage at the top corners using Q:Po elements.

Fig. 22. Velocity field across the extruder channel for general dragflow with leakage at the top corners using QIPo elements.

Fig. 25. Pressure distribution across the extruder channel for general dragflow with leakage at the top corners using Q:Po elements.


NOMENCLATURE

(f) = column matrix for the work equivalent force due to the traction and body forces

ity constraint [GI = global stiffness matrix for the incompressibil-

i = identity tensor k , K = constants [ K ] = global stiffness matrix ~~

r i = outward unit normal vector at the boundary [w = matrix of velocity shape functions [Np] = matrix of pressure shape functions [ N ’ ] = matrix relating strain rate components to the

velocity a t the nodes pe ( p J

= pressure at a point inside a n element = column matrix of unknown pressure at nodes

{pie = column matrix of pressure at the nodes of an element

q = a n element of Q qh = a n element of Qh Q Q,,

= solution space of pressure = finite dimensional subspace of Q

R = one dimensional Euclidean space R” -f

= three dimensional Euclidean space = prescribed value of the traction vector on rT -

3-D Flow Analysis of Non-Newtonian Viscous Fluids


kid velocity ( 4 s )

{Tj = column matrix of traction force a t a point on the surface of the element

d d

= velocity at a point in f? = prescribed value of velocity on r,

{u] = column matrix of unknown velocities at nodes (u], = column matrix of three velocity components

{ u ) ~ = column matrix of velocity components a t the at a point inside a n element

4 0

b -0.01 O h O h 1 O h 0.03 (Son channel velodty (I&)

Fig. 26. Velocity distribution at the center of the extruder channelfor dragflow using nonuniform mesh of fa] axial uelocity (b) cross-channel velocity.

upon the flow geometry and the boundary conditions used. Therefore, even though QIPo elements give rea- sonable results for some simple flows, one cannot rely on QIPo elements for general 3-dimensional flows. The flow simulations using the ‘enriched’ finite elements Q:Po, which satisfy the Babu6ka- Brezzi condition, have very accurate velocity and pressure distributions in all the flows analyzed here.

ACKNOWLEDGMENT

We wish to thank John von Neumann National Supercomputer Center for making their supercom- puting resources available to us. This work has been partially supported by a grant from N. J. Center for Advanced Food Technology (CAFT) at Rutgers University.

nodes of a n element i, i,,,

= a n element of V = a n element of Vh

V Vh X

= solution space of velocity = finite dimensional subspace of V = body force per unit volume

{o) = zero column matrix = part of the boundary where the traction vector

= part of the boundary where the velocity is is given

I‘, given

i = strain rate tensor ( t ) = column matrix of the six independent com-

ponents of the strain rate tensor tr

0 = temperature p 5 = stress tensor 0

= effective shear strain rate

= viscosity of the fluid

= a bounded region in R”

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3-D flow analysis of non-newtonian viscous fluids using “enriched” finite elements

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