Journal of Non-Newtonian Fluid Mechanics, 14 (1984) 279-299 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

279

NUMERICAL SIMULATION OF THE FLOW OF A FLUID THROUGH AN ABRUPT CONTRACTION *

R. KEUNINGS and M.J. CROCHET **

VISCOELASTIC

UnitP de MPcanique AppliquPe, Universitk Catholique de Louvain -la - Neuve (Belgium)

(Received May 31, 1983)

Summary

The flow of a Phan Thien-Tanner fluid through an abrupt 4/l contrac- tion is analyzed by means of finite elements. It is found that the choice of such a fluid gives rise to an important corner vortex growth and to an increasing Couette correction when the flow-rate increases.

1. Introduction

A deceptive aspect of the numerical simulation of viscoelastic flow has often been the lack of deviation from second-order behavior, which is characterized by a velocity field which is essentially Newtonian, while the stress components do indeed show strong deviations from their Newtonian counterpart; the exception has of course been the simulation of the extrusion from a die, where the stresses play a dominant role in the search for the free surface. The use of an upper-convected Maxwell fluid for solving the flow through an abrupt contraction is a typical situation where the flow field is hardly influenced by the elasticity of the flow. It had been hoped that the absence of deviation was due to the limited domain of elasticity where the calculations could be completed; however, the use of an Oldroyd-B fluid [l] rather than an upper-convected Maxwell fluid in [2] has allowed us to reach a relatively high value of elasticity with still little effect upon the kinematics.

It seems appropriate at this stage to wonder whether the use of the upper-convected Maxwell fluid or the Oldroyd-B fluid, characterized by a

* Dedicated to the memory of Professor J.G. Oldroyd. ** All correspondence should be addressed to: Prof. M.J. Crochet, Unite de Mtcanique Appliqute, 2 place du Levant, Bl348 Louvain-la-Neuve, Belgium.

0377-0257/84/$03.00 0 1984 Elsevier Science Publishers B.V.

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constant shear viscosity and a quadratic first normal stress difference, is not precisely the cause of this lack of deviation from Newtonian behavior. Several possibilities are of course available for enlarging the spectrum of constitutive equations of the differential type; Oldroyd [3-41 made the important step of proposing equations “which are linear in the stresses alone, and include terms of the second degree in the stresses and velocity gradients taken together”. Such equations provide the opportunity of consid- ering important deviations from Maxwellian behavior by means of a proper selection of the parameters.

A good example of this sort is the Johnson-Segalman fluid [5] which is based upon the molecular theory of polymeric solutions, and which can be generated by means of a special choice of parameters in the Oldroyd equation. Moreover, the addition of a retardation time, contained in the Oldroyd equation, allows for a monotonic increasing shear stress when the shear rate increases. The Johnson-Segalman fluid is a special case of the Phan Thien-Tanner fluid [6,7] which contains an additional scalar non-di- mensional parameter. In the present paper, we wish to investigate the implications of the use of such fluids upon the flow field in a problem where the kinematics is not trivial. Before embarking on the numerical simulation, we will however reconsider briefly, in Section 2, the steady shear and elongational properties of the Johnson-Segalman and the Phan Thien-Tanner fluids, and exhibit some curves which are unavailable in the literature.

After a very brief description of the numerical methods in Section 3, we study in Section 4 the flow of a Phan Thien-Tanner fluid through an abrupt 4/l axisymmetric contraction. The entry flow of a viscoelastic fluid is an important topic where many questions remain unanswered; important aspects have been recently reviewed by Boger [8]. The observations which can be made concern mainly the intensity of the recirculating vortex, its size, and the extra pressure loss, or Couette correction, due to the sudden contraction. In the present paper, we show important deviations from the Maxwell fluid results; the corner vortex grows appreciably and, for once, the Couette correction increases when the flow rate increases in the contraction. In view of these large deviations, it has been found essential to recheck the calcula- tions with the use of several meshes. Interesting numerical convergence phenomena are observed.

In Section 5, we then discuss the influence of the material coefficients upon the results, and investigate in particular the behavior of a White-Metzner fluid [9] which, in steady shear, would behave like a John- son- Segalman fluid.

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2. Constitutive equations

In his early work on the formulation of rheological constitutive equations in differential form, Oldroyd [3] observed that the simplest forms are linear in the stresses, and include quadratic terms in the stresses and velocity gradients taken together; the following equation was proposed by Oldroyd,

T+ A,?+ pOTrTD - p,( TD + DT) + Y,Tr( TD)I

= 2q0[D + x,d - 2pzD2 + v2Tr(D2)I], (2.1)

where T is the extra-stress tensor, D is the rate of deformation tensor, and f denotes the corotational derivative defined by

F=DT/D~++T(L-L~)++(L~-L)T, (2.2)

where L is the velocity gradient tensor. Although such a constitutive equation already presents a considerable

simplification with respect to general models of simple fluids, the presence of 8 material constants complicates the task of selecting an appropriate model. With the special choice

ru1 = X,(1 - 0, I”0 = V, =x2 = /12 = V2 = 0, (2.3)

eqn. (2.1) becomes

T+A, (1 [ V

where T

- 5/2) “T + (t/2) ; = 2n0D, I

(2.4)

A

and T denote, respectively,’ the contravariant and the covariant

derivatives of the stress tensor,

V

T= DT/Dt- LT- TLT, (2.5)

A

T= DT/Dt+ TL+LTT. (2.6)

Equation (2.4) is the single-relaxation-time form of the constitutive equa- tions proposed by Johnson and Segalman [5] allowing for non-affine defor- mations of the molecular network, and a special case of the equations simultaneously developed by Phan Thien and Tanner [6,7], who suggest the following form,

Y(T)T+A, (1 -<,2);+(1,2)+ =2n0D; [ 1 (2.7)

the definitions

Y(T) = 1 + (<A,/q,,)Tr T, (2.8)

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where c is a small parameter, and

Y(T) = exp((eJWq,)Tr T)

were suggested respectively in [6] and [7].

(2.9)

The scalar parameter 5 in (2.4) and (2.7) governs the ratio of the first to the second normal stress difference in viscometric flows; a value of 0.2 implies that A$ = -O.lN,. The scalar parameter E in (2.8) and (2.9) is usually small and of the order of 0.01. The viscometric functions for the model (2.4) are given by Petrie [lo]; Phan Thien and Tanner [6] calculated the behavior of (2.7) in the start-up of shear and elongational flows. In order to obtain the viscometric behavior of (2.7), it is necessary to solve a non-linear algebraic system of equations. Fig. 1 shows a graph of the shear-viscosity as a function of the shear-rate for various values of 5, and for c = 0 and 6 = 0.015; Figure 2 shows the graph of the first normal stress difference for the same values of the parameters 5 and C. It is found that a value of 6 like 0.015 has little influence upon the viscometric properties so long as 5 is not vanishing; moreover, the use of either (2.8) or (2.9) does not generate any graphically observable deviation.

Oldroyd [4] realized that restrictions are needed on the choice of parame- ters appearing in (2.1) to ensure that the shear stress be a monotonic

1.

! 0

0.8

0.6

0. 3. 4. h,P 5.

Fig. 1. The steady shear viscosity as a function of the shear rate for various values of the parameter 5; - : c = 0; a: z = 0.015.

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30..

Fig. 2. The first normal stress difference as a function of the shear rate for various values of the parameter 5; -: E=o; 0: <=0.015.

increasing function of the rate of shear; more specifically,

9[W, + I%(!% - 3V2) -P,(P* - %>I

2x: +cLob*- 3%/2) - PI (IT - v, > 2 0. (2.10)

A simple inspection shows that the choice of coefficients (2.3) does not satisfy the constraints (2. IO), except for 5 = 0 or 2. Since our objective is to solve arbitrary flows in which the rate of shear is not limited to small values, the models (2.4) and (2.7) are not acceptable as such for numerical flow simulation. Instead of (2.3), we will consider the following choice of parame- ters,

Pl = X,(1 - 0, P2 = A,(1 - 5), /Jo = V, = V2 = 0; (2.11)

Equation (2.1) may then be written as follows,

(2.12)

with the following constraint,

9h,5(2 - 6) > X,5(2 - 6) 2 0. (2.13)

A useful interpretation of (2.12) is obtained by a decomposition of the extra-stress tensor T as the sum of two partial stresses T, and T2, given by

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the following constitutive equations,

T=T,+ T2,

T , + X 1 l - f / 2 ) T 1 + ( ~ / 2 ) T, =2n~D, (2.14)

T 2 --- 21120 ,

with

112 : 1 1 0 ~ 2 / ~ 1 ' 110 "~- 171 + 112; (2.15)

it is easy to show that (2.12) and (2.14) are equivalent. Thus, the use of (2.12) amounts to the addition of a purely viscous component to the stress tensor given by the constitutive equation (2.4). In view of (2.13), the choice of partial viscosities is restricted by the inequality

811 2 ~ 711, (2.16)

whenever f differs from 0 and 2. The constitutive equation (2.7) may be modified in a similar way, and in

the subsequent sections we will consider the flow of fluids defined by the following equations,

T = L + T 2 ,

Y(Llr, + x, (1 - ~/21 T, + ( f /21 T l = 211,D, (2.17)

T2 = 2n2D,

with the restriction (2.16). Since we will not consider a spectrum of relaxation times, the inequality

(2.16) implies that, even for high shear rates, the shear viscosity cannot decrease below 1/9 of the zero shear rate viscosity. Moreover, the overall behavior of the fluid in steady flow is Newtonian for very low and very large values of the rate of shear.

Before closing this section, we wish to examine briefly the steady elonga- tional behavior of the fluids defined by (2.4) and (2.7). The Johnson-Segal- man fluid (2.4) is characterized by an infinite value of the steady elonga- tional viscosity for a finite value of the rate of elongation 0, given by

) ~ = 1/2(1 - () . (2.18)

Considering the Phan Thien-Tanner fluid (2.7) with the definition (2.8) for the factor Y(T), one finds analytically that there exist three distinct steady elongational viscosities for a given value of the rate of elongation 0. Figure 3 shows the three real branches which have been obtained for the pair

= 0.2, ~ = 0.015. Only one of these three branches is admissible in view of

200:

285

rl o

1 0 0 :

Branch

Branch 1 /

/ / 3. (Newtonian) o. I-- L_~.z'__i.

2, 4,

Branch 2

i I - I

O. 8. h i e lO.

-100 '

Fig. 3. The three branches of the elongational viscosity as a funct ion of the rate of elongation found for the Phan T h i e n - T a n n e r fluid, eqn. (2.8), with ~ = 0.2 and ~ = 0.015.

100.

q e m

qo

O.

-100

i

Branch /

( Newton ian )

Branch 1 £.

ol I I

4. 6. a . . . _ h 1 6 10.

Fig. 4. The two branches of the elongational viscosity as a funct ion of the rate of elongation found for the Phan T h i e n - T a n n e r fluid, eqn. (2.9), with ~ = 0.2 and ~ = 0.015.

286

its behavior for low values of the rate of elongation. When the definition (2.9) is used for the factor Y(T), it is impossible to find the elongational viscosity by analytical means; for the same values t = 0.2 and e = 0.015, the Newton-Raphson method allows us to find two branches for the elonga- tional viscosity, shown on Fig. 4. It is clear again that only one branch is admissible. Further details on the method used for obtaining Figs 3 and 4 may be found in [ 111.

The presence of multiple branches for the steady elongational viscosity is an anomaly which might sometimes be troublesome in the numerical simula- tion of the flow of the Phan Thien-Tanner fluid.

3. Numerical method and boundary conditions

We wish to solve the non-linear system of equations consisting of the constitutive equations (2.17) the momentum equations,

-~p+v*T+F=pa, (3.1)

wherep is the pressure, F the body force per unit volume, p the specific mass and a the acceleration, and the conservation of mass which, for an incom- pressible fluid, takes the form

v*u= 0. (3.2)

The numerical method is a mixed finite-element technique which has been exposed in [ 121, and is a slight generalization of the method developed in [ 131 for the case of an Oldroyd-B fluid. Briefly, a finite-element representation is assumed for the extra-stresses T,, the velocity components and the pressure in the form,

(3.3) i=l i= 1 i=l

where T:“, d’), p(j) are nodal values, and qi, nj are shape functions. The system of partial differential equations is discretized by means of the Galerkin method applied to the set (2.17) (3.1) and (3.2). One obtains

(4;; Y(TI”)Tr”+A, (I-[,2);;+((,2)& 1 1 -217,0*>=0,

< ( V#i)T; -p*I + 2q2D* + T: > + < 4,; pa* > = F(‘), (3 -4)

c 7rj; v w* > = 0,

withl<i<M, l<j<N,andwhereF (U denotes the nodal force vector at node i; the brackets indicate the L*-scalar product. The derivation of (3.4) in component form is straightforward in rectangular Cartesian coordinates;

287

volume integrals must be used for obtaining the relevant equations in cylindrical coordinates.

The elements used in the present paper are Lagrangian isoparametric quadrilaterals and six-node triangular elements; the shape functions for the velocity and stress components include complete second-degree polynomials and are Co-continuous. For the pressure, the shape functions include first degree polynomials and are also Co-continuous.

Equations (3.4) form a system of non-linear algebraic equations in terms of the unknowns T{“, u(j) and p(j). The system is solved by means of the Newton-Raphson iterative technique which has been found very efficient as long as the method converges; when the convergence is lost, the use of a Picard type of iteration has not allowed us to proceed further in elasticity.

The numerical method will be applied in Section 4 to the solution of the flow of viscoelastic fluids through an abrupt axisymmetric contraction in the absence of body forces and inertia effects. In order to impose boundary conditions without perturbing the flow in the contraction, we will consider fairly long entry and exit tubular regions allowing for fully developed entry and exit flows. In the entry section, the fully developed Poiseuille velocity profile is imposed together with the associated extra-stresses T,. Obtaining the fully developed values of the velocity and stress components is not a straightforward task. For the Johnson-Segalman fluid with a retardation time given by (2.14), an analytical method for calculating the fully developed Poiseuille flow has recently been given in [14]. For the Phan-Thien Tanner fluid (2.17), it is necessary to resort to a one-dimensional finite-element method for calculating the entry conditions. In the exit section, we impose the velocity components only, together with a vanishing nodal pressure on the axis of symmetry. The velocity components vanish on the walls of the contraction. Finally, on the axis of symmetry, we impose vanishing nodal forces in the axial direction, vanishing radial velocities, and vanishing shear-stress components.

4. Numerical results for the entry flow of a Phan Thien-Tanner fluid

In the present section, we wish to exhibit the numerical results which have been obtained for the flow of a Phan Thien-Tanner fluid through an abrupt contraction; the following parameters have been selected for the simulation,

6 = 0.2; c=0.015; 771 = 8772. (4.1)

We will discuss in Section 5 the influence of the parameter 5 and 6 upon the calculations. We will however, in the present section, present a comparison with results obtained in [2] for the flow of an Oldroyd-B fluid for which

<=o; r=O; ~1 = 8172. (4.2)

In view of the important modifications in the kinematics of the flow brought about by the elasticity of the fluid, it was found necessary to recalculate the results with three different meshes in order to allow for a careful assessment of the numerical results. Figure 5 shows a partial view of the meshes in the neighborhood of the contraction, together with the extent of the entry and the exit tubes. MESH1 and MESH2 have been designed for numerical tests, while MESH3 is a copy of the finite element mesh drawn by Viriyayuthakorn and Caswell [15] for solving the flow of a fluid of the integral type, where, however, the elements have been elongated in the downstream region. Table 1 shows the characteristic dimensions of the meshes. The last column is of particular interest: it shows the radial size of the small element located near the reentrant corner, which will be found to play a dominant role as far as the limit of convergence is concerned.

MESH1 z-r-0.

z=-20.

r--20.

MESH2

tr

z-20.

MESH3

-2-20.

r=-12. -

\ I \/---\

Fig. 5. Partial view of the three finite element meshes used for the calculations.

TABLE 1

Characteristic dimensions of the finite element meshes.

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Elements Nodes Degrees of freedom Size of the corner element

MESH1 130 594 3733 0.25 MESH2 218 959 6018 0.20 MESH3 129 577 3623 0.05

In order to progressively increase the elastic character of the flow, we will assume that the mean velocity in the downstream section is 1, and that the zero-shear-viscosity 710 is also 1 ; we will then slowly increment the zero-shear relaxation time )~l, while keeping the parameters ~ and c fixed. We need to evaluate the amount of elasticity of the flow by means of a non-dimensional number. For a Maxwell or an Oldroyd-B fluid, it has been customary to use the recoverable shear defined by the ratio

S R = N J 2 z w (4.3)

of the first normal stress difference to twice the shear stress on the wall in the fully developed downstream flow. When (4.2) applies, it is easy to see that

S R = ?t~J, w8/9 (4.4)

where ~'w is the shear-rate on the wall. However, the viscometric properties of the Phan Thien-Tanner fluid render such a measure inadequate. Figure 6 shows the curve of SR, defined by (4.3), as a function of the product Xf?w, which will increase together with X1. The fact that SR is not a monotonic function of )~fi'w prevents its use as an appropriate measure in the present problem. Instead, we will rather use a Weissenberg number defined by

W e = ) k l V / O , (4.5)

where V is the average velocity in the downstream tube of diameter D. We wish to stress that W e is a convenient measure, where, however, one uses the zero shear value of the relaxation time which is in no way related to the ratio V / D .

As in many other numerical solutions of viscoelastic flow, the iterative algorithm ceases to converge beyond some value of the Weissenberg number We. Table 2 gives the limit value of W e obtained for the meshes of fig. 5. It is interesting to note that, in the present simulation, the upper value of W e is related to the size of the element near the reentrant corner.

yw 0. 5. 10. 15. 20. 25.

Fig. 6. Behavior of the recoverable shear S,, eqn. (4.3), as a function of the product X,p,.

TABLE 2

Limit of the Weissenberg number beyond which the numerical method ceases to converge.

Mesh Limit value of We

MESH1 - 0.5 MESH2 - 0.7 MESH3 - 1.75

Since the linearized system obtained from the application of the Newton-Raphson method is solved by means of a Gaussian elimination, it is easy to detect changes of sign of the Jacobian after each increment of A,, by calculating the product of the signs of the pivots used during the elimination. It was found repeatedly that the Jacobian changes sign just beyond the value of A, for which the last converged solution has been found. This shows that the Jacobian matrix becomes singular in the neighborhood of the last converged solution; moreover, the value of A, at which the Jacobian matrix becomes singular is strongly mesh dependent. This tendency has been confirmed in further tests where a slight modification in MESH3 has produced dramatic changes in the limit value in We. It will be reassuring to find later in this section that the three meshes produce essentially the same results up to their lack of convergence. The odd dependence of the convergence properties upon the mesh is not understood at the present time.

291

Let us now consider the effect of elasticity upon the flow kinematics. Previous work on the flow of an Oldroyd-B fluid [2] has shown very little influence of the Weissenberg number upon the streamlines and the recir- culating corner vortex. Figure 7 shows the streamlines obtained for the Newtonian fluid, and the Oldroyd-B fluid at the relatively high value of S, = 32/9, or We = 0.5 with the definition (4.5). The stream function is normalized between 0 and 1; the intensity of the vortex indicates the ratio between the flow rates in the secondary flow and the main flow in the contraction. For an Oldroyd-B fluid, the size and the intensity of the vortex are only slightly enhanced by elasticity.

Fig. 7. Streamlines obtained for the Newtonian fluid and for the Oldroyd-B fluid at We = 0.5

(S, = 32/9).

The situation is entirely modified for the Phan Thien-Tanner fluid. Figure 8 shows the streamlines obtained respectively for We = 1 and We = 1.75. The growth of the vortex is evident, and its intensity reaches 0.125 of the main flow rate when We = 1.75. The streamlines of Fig. 8 should be compared to the photograph shown on Fig. 4(b) in [ 161. A good idea of the vortex growth may be obtained by considering on Fig. 9 the graph of the

292

vortex intensity as a function of We; on the same figure, we have indicated the values obtained with the Oldroyd-B fluid (on MESH3), and with the Phan Thien-Tanner fluid with MESH1 and MESHZ. The curve for c = 0 will be discussed in Section 5.

.Wed

We ~1.75

Fig. 8. Streamlines obtained for the Phan Thien-Tanner fluid at We=1 and We=1.15.

The size of the vortex may be evaluated by the ratio X = J&/D, where D, is the diameter of the upstream tube, and L, is the reattachment length of the cell (see Fig. 8(a)). Table 3 gives the values of X as a function of We; it is clear that, in the present problem, the results lie well beyond the second order approximation.

293

O_l5

t

Vortex intensity

0.10.. JOHNSON-SEGALMAN

x MESH1 o MESH2 - MESH3

0.05-m

0. 0.5 1. 1.5 We 2.

Fig. 9. The intensity of the corner vortex as a function of We for the Phan Thien-Tanner fluid (E = O.OlS), the Johnson-Segalman fluid (6 = 0). and the Oldroyd-B fluid.

TABLE 3

Dimensionless vortex size as a function of the Weissenberg number.

We 0 0.3 0.6 1 1.25 1.5 1.75 X 0.18 0.22 0.26 0.35 0.39 0.42 0.46

The change of curvature of the curve shown on Fig. 9 at We = 1 may find an explanation in a closer look at the velocity profiles in the section where the contraction occurs. Figure 10(a) shows the velocity profiles obtained in that section and in the exit section where one finds a fully developed Poiseuille flow at We = 1. The axial velocity overshoot in the contraction section is obvious; however, the natural boundary condition av,,Gr = 0 on the axis of symmetry is still fairly well satisfied. However, at We = 1.5, Fig. 10(b) shows that the velocity overshoot in the contraction section is so high as to prevent the satisfaction of the natural boundary condition. The finite element mesh which was primarily refined around the reentrant corner is clearly too coarse for an accurate representation of the phenomena taking place along the axis.

294

Fig. 10. Velocity profiles obtained in the section of the abrupt contraction and in the exit section, for We = 1 and We = 1.5.

The development of the axial velocity along the axis of symmetry is shown on Fig. 11. It is found that the maximum value of the axial velocity occurs just beyond the contraction. It is interesting to observe that the fully developed downstream value of the axial velocity is an increasing function of We beyond some value of the Weissenberg number. Such a behavior is due to the presence of a retardation time in (2.17); when X,? is large, T, becomes small with respect to T2, and the velocity profile tends to the Newtonian behavior.

A measurable quantity in experimental rheology is the Couette correction due to the entry flow. The Couette correction is defined as follows. Let L,, L, denote the lengths of the upstream and downstream tubes, respectively, and let Ape, Ap,, be the associated pressure gradients in the fully developed upstream and downstream flows. Let also 6p be the total pressure loss between the entry and the exit sections, and T,,, be the wall shear stress in the downstream tube. The Couette correction Sp,, is given by

apen = (6~ - L&P, - L,A~d/(2d. (4-b)

295

Fig. 11. Development of the axial velocity along the axis of symmetry for various values of

We.

While many experiments have revealed an increase of 6p,, when the flow rate (or the elasticity) increases [8], available numerical results dealing mainly with the Maxwell fluid have shown the opposite trend. The situation is reversed for the entry flow of the Phan Thien-Tanner fluid, and consistent results have been obtained with MESHl, MESH2 and MESH3 within their relative domains of convergence. Figure 12 shows the graph of 6p,, as a function of We; the Couette correction decreases up to We - 0.2 and from there on is a monotonic increasing function of We. On the same graph, we have indicated the curve obtained in [2] for an Oldroyd-B fluid, where the Couette correction becomes negative beyond We = 0.2. Table 4 indicates the Couette correction as a function of We for the fluid under consideration.

TABLE 4

Couette correction as a function of the Weissenberg number We for a Phan Thien-Tanner fluid.

We 0 0.2 0.50 0.75 1. 1.25 1.50 1.75

6p,,(r = 0.015) 0.56 0.30 1.07 1.40 1.75 2.11 2.44 2.79 Sf%“(C = 0.) 0.56 0.36 1.10 1.44 1.78 2.21 2.54 -

296

:r JOHNSON y

/ 1. 1.5 We 2.

I ‘t- OLDROYD - B

-1. \ x MESH1 o MESH2 - MESH3

-2.

Fig. 12. The Couette correction as a function of the Weissenberg number for the Phan Thien-Tanner fluid (e = 0.015), the Johnson-Segalman fluid (E = 0) and the Oldroyd-B fluid.

5. Influence of the material parameters upon the numerical simulation

The reasons why a numerical simulation may in some cases be pursued

over a wide range of parameters while in other the convergence fails for a low value of the Weissenberg number are not understood at the present time. The results obtained in Section 4 depend upon the selection of two parame- ters, 5 and e; [ determines in a sense the shear-thinning behavior of the fluid, while e removes the troublesome phenomenon of an infinite elongational viscosity at a finite rate of elongation. One may thus wonder about the effect of non-vanishing values for 5 and c upon the success of the calculation.

The set of calculations presented in Section 4 has been run again on MESH3 for a fluid characterized by the following parameters,

5=0.2; e = 0.; VI= 8772; (5.1)

these parameters determine in fact a Johnson-Segalman fluid with a retarda- tion time given by either (2.12) or (2.14). Converged solutions have been obtained up to We = 1.5; we have started the iterative procedure from the results obtained at E = 0.015, and the solution at c = 0 is reached within a few iterations. The results for the Johnson-Segalman fluid do not differ appreciably from those obtained for the Phan Thien-Tanner fluid. Figure 9 shows that the vortex intensity at E = 0 is even higher than what we found with 6 = 0.015. The couette correction is little affected by the value of e; this is confirmed by the values shown on Figure 12 and given in Table 4.

297

On the basis of these tests, a natural question is to wonder whether, starting from the solution obtained for the set of parameters (5.1), it is possible to decrease ,$ and to eventually obtain a solution for the Oldroyd-B fluid for high values of We. We have tried to decrease 5 from 0.2 to 0.15 for a value of We = 1.5; unfortunately, the Jacobian changes sign in the New- ton-Raphson procedure, and no solution can be obtained by means of the available algorithms.

A last question we have addressed is the influence of the constitutive equation upon the results. Given the viscometric functions of a

Johnson-Segalman fluid, the latter is not the only constitutive equation which is able to generate these functions. In order to verify whether the viscometric behavior for the choice of parameters (5.1) is mainly responsible for the results shown in Section 4, we have also considered a White-Metzner fluid [9] defined by the following equations,

T= T, -I- T2, (5 *2)

T, + X, ;, = 2pi(II,)D,

T, = 2112D,

where

II,= :D;~D~~,

and

(5.3)

(5.4)

It is easy to verify that (2.14) and (5.2) exhibit the same viscometric behavior, except for the second normal stress difference which vanishes for the White-Metzner fluid.

Two iterative techniques have been developed for solving the new numeri- cal problem. The first method is a combination of Picard and Newton-Raphson iterations where p,(II,) is calculated in terms of the velocity components of the previous iteration. The second method involves a full Newton-Raphson scheme.

Unfortunately, it has been impossible to obtain converged results beyond We = 0.325. At a somewhat higher value, the Jacobian changes sign, and no converged solution can be found. For such a low value, we know from Fig. 9 that we cannot possibly verify whether the corner vortex increases. However, on the basis of Figure 12, we should be able to detect an increasing Couette correction. Figure 13 shows the curve of the Couette correction for the Johnson-Segalman fluid defined by (5.1) and for the White-Metzner fluid

298

(5.2). The qualitative agreement between both curves is evident. The quanti- tative discrepancy may be due to the differing second normal stress dif- ferences of the models. An inspection of the behavior of the axial velocity component along the axis of symmetry shows an overshoot which is again very similar to the curve obtained with a Johnson-Segalman fluid.

1. ..6P,, JOHNSON - SEGALMAN

0.5

I 1

0 0.2 0.4 We

Fig. 13. The Couette correction as a function of the Weissenberg number for the Johnson-Segalman fluid (e = 0) and the White-Metzner fluid defined by (5.2); .$ = 0.2 and

1)2 =1),/g.

6. Conclusions

A special selection of the material parameters in the general constitutive equation proposed by Oldroyd [3] allows us to generate the constitutive equations proposed by Johnson and Segalman [5], which is a special case of the molecular model by Phan Thien and Tanner [6]. The use of such a constitutive equation instead of the classical Maxwell or Oldroyd-B models for calculating the flow of a viscoelastic fluid through an abrupt contraction permits the prediction of a large corner vortex and of an increasing Couette correction as a function of We; these experimentally known effects had not been simulated previously. However, the paper reopens the acute question of why the iterative procedure fails under some sets of circumstances related to the mesh, the fluid and the problem.

299

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