Entrance Effects for Power-law Fluid

MAHESH GUFTA, C. A. HIEBER, and K. K. WANG

Sibley School of Mechanical and Aerospace Engineering Cornell University

Ithaca New York 14853

Entrance pressure losses for the creeping flow of a power-law fluid are calcu- lated for an abrupt contraction of ratio 2 .4 .8 and m for both the axisymmetric and planar cases using PzPl and P l P , finite elements. Contrary to some earlier findings in the literature, the entrance pressure loss obtained by using the two different types of finite elements, both of which satisfy the BabuSka-Brezzi condi- tion, are found to converge to the same results. The present results also confirm that the variational method of Duda and Vrentas gives excellent upper bounds for both the axisymmetric and planar cases with infinite contraction ratio.

INTRODUCTION

he entry-flow problem has received considerable T attention in the literature, as evidenced by the reviews in (1-3). Despite the limitations of a pure- ly viscous modeling in handling such complex shear/elongational flow regimes, results based upon such modeling can still serve as a benchmark from which more realistic predictions based upon visccl elastic formulations can be compared. Somewhat sur- prisingly, however, for the simplest shear-thinning model, namely the power-law model, the results in the literature, such as in (4-101, are not consistent even in the creeping-flow limit.

In particular, the numerical results for the extra pressure loss obtained by Boger, et d (5) indicated that the creeping-flow upper-bound result of Duda and Vrentas (4) for the axisymmetric infinite contrac- tion becomes poor as the power-law index becomes small, being 55% higher than the numerical predic- tions for a power-law index of n= 1/6. I t was later shown in (91, however, that the numerical results in (5) were inaccurate at small n due to a correspond- ingly slow decay of the pressure field in the far field of the infinite reservoir which was not captured by the finite domain employed in (5). Numerical results for a finite abrupt contraction ratio ( p ) of 2, 4, and 8 were also obtained for the axisymmetric case in (9), with the results for p = 4 agreeing well with those from (7). Corresponding results for the planar case, including extension of the Duda-Vrentas approach to this case, were also presented in (91, where the resulting upper bound was found to describe the numerical results well in this case also.

More recently, Robichaud and Tanguy (10). un- aware of the results in (9), have obtained numerical predictions for the axisymmetric case with 0 = 4 and

found their results for the extra pressure loss to deviate markedly from those in (7) with decreasing n, lying about 28% below that in (7) at n = 1 / 5 . As an explanation for this discrepancy, the authors of (10) propose that their results are more reliable due to the use of a finite element which ensures a divergence- free velocity field at the element level, as well as the employment of a systematic mesh refinement.

Shown in Q. I are schematics of the three differ- ent elements employed in Refs. 7, 9, and 10. All three types of finite element satisfy the BabuSka-Brezzi compatibility condition (1 1- 131, which ensures the existence of a unique solution to the discrete viscous-flow problem. However, because of the dis- continuous pressure approximation, the Pz PI ele- ment ensures mass conservation at the element level (which cannot be ensured by the PzP, and Q2Q1 elements).

In the present investigation, an independent code has been developed by the first author for creeping flow of a power-law fluid which can handle either the P2 P, or P i P , element. By employing these two types of finite elements on the same mesh, it is found that results obtained from both types of element converge to the same results upon mesh refinement. Indeed, this is as would be expected given that the estimate of the combined error in velocity and pressure on any BabuSka-Brezzi-compatible finite element goes to zero as the mesh is refined (for instance, see page 114 in Ref. 14). In turn, these converged results for the extra pressure loss are found to be in good agreement with the results in Refs. 7 and 9.

RESULTS AND DISCUSSION

The extra pressure loss in entrance flow is gener- ally expressed in terms of the equivalent length of the

POLYMER ENGINEERWG AND SCIENCE, MID-FEBRUARY 1994, Vol. 34, NO. 3 209

Mahesh Gupta, C. A. Hieber. and K. K. Wang

A

(d Fig. 1. Schematic of Q2Q1(a). P2P,(b) and P;P,(c) elements as employed. respectively, in Refs. 7, 9, and 10; solid (oped circles denote velocity (pressure) nodes.

down-stream (smaller) channel:

AP - AP, - AP2 I dP/&l2

L, = (1)

where Ap is the total pressure drop, Apl and Ap2 are the pressure drops for fully-developed flow in the up-stream and down-stream channels, respectively, and I dp/&l2 is the magnitude of the fully-developed pressure gradient in the down-stream channel.

Three successively refined finite-element meshes used in the present study for axisymmetric and pla- nar flows with contraction ratio /3 = 4 are shown in Fig. 2. A purely axial, fully-developed velocity profile is imposed at the inlet whereas the transverse velocity and axial traction are set to zero at the exit.

Resulting predictions for the extra pressure loss for the axisymmetric case with /3 = 4 are shown in Fig. 3 based upon the three mesh configurations shown in Fig. 2 with either PzPl or PZP, element. It is seen that both formulations exhibit convergence to the same asymptote upon refinement. Further, the con- verged result agrees well with the corresponding pre- dictions from Refs. 7 and 9, whereas the results from Ref. 1 0 are seen to be anomalous. As a further illustration, Fig. 4 presents corre-

sponding predictions for the axial velocity at the con- traction plane for n = 0.2 and 0.4. In both instances, the present predictions using the P2P, and PZP, elements agree well with those from Ref. 7 whereas

the corresponding results from Ref. 10 are seen to be somewhat anomalous, particularly at n = 0.4.

Since the results in Figs. 3 and 4 indicate that both the P2 P, and PZP, elements give essentially the same results for a sufficiently refined mesh, the cal- culations reported below have all been based solely upon the P2 P, element with a finite-element configu- ration comparable to mesh C in refinement. In partic- ular, Fig. 5 gives the predicted center-line pressure variation at various values of n for the axisymmetric case with /3 = 4. It is seen from these results that the effect of the abrupt contraction is felt progressively further upstream as n decreases. In addition, an anomalous leveling off of the pressure gradient is seen to occur near the inlet for smaller values of n in Fig. 5. This latter anomaly is actually due to limita-

(C)

Q. 2. Finite-element-mesh conturutions used for contrac- tion ratio p = 4 for both axisymmetric and planar cases. Total number of elements = 114 (A), 456 (B). 1026 (C).

2.1 - Pz P, , Mesh A

- P2 P, , Mesh B

P,P, , Mesh C

P;? ,Mesh A

P:? , Mesh B - - - - - - - - _ P;? ,MeshC J'

1.7

i 0 Robichaud and Tanguy 4 tiieber 0 Kimet al.

nr. I I I I "... 1 2 3 4 5 6

l/n

Fig. 3. Predicted extra pressure loss us. power-law index for axisymmetric case with = 4.

210 POLYMER ENGINEERING AND SCIENCE, MID-FEBRUARY 1994, VOl. 34, NO. 3

Entrance Effects for Power-Law Fluid 1.4 I I 1 1 I I

0.0 ' I I I I

0.0 0.2 0.4 0.6 0.8 1 .o Radial distance, (r/Rz)

(a) 1.6 I I I I 1

0.0 ' I I I I

0.0 0.2 0.4 0.6 0.8 1 .o Radial distance, (r/Rz)

(b) Rg. 4. Predicted axial velocity profle at contraction plane for axisymmetrlc/P = 4 with n = 0.2 fa), 0.4 fb).

tions of the quadratic shape function to capture (with fixed mesh) the approach of the fully-developed pro- file towards a plug-like behavior at small n. In fact, this anomalous behavior can be shifted from the inlet to the outlet by switching the imposed fully-devel- oped velocity profile from the inlet to the outlet. For present purposes, the effect of the anomaly has been circumvented by evaluating Ap in Eq 1 between z/R, = - 12 (where a fully developed pressure gradi- ent has been established, as can be seen in Fig. 5) and z/R, = 4. As in Ref. 9, calculations have also been done for

p = 2, 8, and m. In every case, the stream-wise extent of the computational domain in the small tube has been taken as 4 X R, whereas the extent in the u p stream member has been taken as 4 X R, in the cases where p is fmite. On the other hand, for j3 = m,

the finite-element configuration is shown in Fig. 6 where the outer radius ( p ) of the computational do- main in the reservoir corresponds to 8 X R,. In order to capture the overall pressure drop in the reservoir,

f W

e n

.- n = 0.1667 -.- n=0.2

-16 -12 -8 -4 0 4

Axial distance, (zlR,)

0. 5. Predicted pressure variation along the center line for axisymmetricflow with P = 4 and various values of power- law index. 7he abrupt contraction corresponds to z / R , = 0, where R , is the radius of downstream tube.

Rg. 6. Finite-element mesh used for injkite-contraction ratio.

the mesh in Fig. 6 has been augmented by succes- sively adding rings of additional elements resulting in domains corresponding to values for p/R, of 12, 18, and 27. In this case, the fully-developed velocity pro- file is imposed at the exit whereas the traction is set to zero at the outer radius of the computational do- main in the reservoir. Similar to the behavior of the center-line pressure near the inlet for the fmite-con- traction cases, the center-line pressure for j3 = M was found to deviate from the linear behavior near the exit, where the fully-developed velocity is imposed. To avoid the effect of this anomaly at the smaller values of n. Eq I has been based on limiting the down- stream domain to z/R, = 3. In addition, it is noted that Apl is zero in the p = x. case. On the other hand, it has been confirmed, as in (9). that the pressure in the reservoir decays as p-3" in the axisymmetric case (and as p-'" in the planar case), thus allowing extrapolation to p -+

Resulting predictions for the extra pressure loss us. power-law index for j3 = 2, 4, 8, and a are shown

in calculating Ap.

POLYMER ENGINEERING AND SCIENCE, MID-FEBRUARY 1994, Vol. 34, No. 3 21 1

Mahesh Gupta, C. A. Hieber, and K . K . Wang

plotted in Fig. 7 and tabulated in Table 1. Results based upon the variational method of Duda and Vrentas (41, which serve as an excellent upper bound for p = m , have also been plotted in Fig. 7. Corre- sponding predictions for the planar case are shown in Fig. 8 and tabulated in Table 2. Interestingly, the variational upper bound actually improves with in- creased shear thinning, being high by = 22% at n =

1.0 compared with = 5% at n = 1/6 in Fig. 7 with corresponding respective values for the planar case of = 19% and 5% in Fig. 8.

ACKNOWLEDGMENTS

This work has been camed out under the Cornell Injection Molding Program (CIMP) which is supported by the CIMP Industrial Consortium. We wish to thank the National Center for Supercomputing Applications (NCSA) for making their computing resources avail- able to us.

0.0 L I I I I I 1 2 3 4 5 6

tin

Q. 7. Present predictions (solid curves) for extra pressure loss us. 1 / n for axisymmetric case and various p ; symbols indicate corresponding predictions &om (9): dashed line is upper bound for p = = from Duda and Vrentas (4).

Table 1. Entrance Pressure Loss (L , /R2) for Axisymmetric Flows.

$ n = l n=0.8 n=0.6 n=0.4 n=0.3 n=0.2 n=1/6

2 0.45 0.52 0.59 0.68 0.75 0.85 0.90 4 0.56 0.69 0.86 1.12 1.31 1.58 1.70 8 0.58 0.72 0.94 1.31 1.61 2.04 2.23 s 0.58 0.73 0.97 1.45 1.94 2.94 3.55

s 2"

6 -

5 -

4 -

3 -

I I I I

1 2 3 4 5 6 1 In

Fig. 8. Same as Fg. 7 but for planar case with dashed curve based on extension of Duda and Vrentas (4) to planar case as determined in (9): b, denotes the half-gap thickness of the downstream channel.

Table 2. Entrance Pressure Loss (L , /b2 ) for Planar Flows.

p n = l n=0.8 n=0.6 n=0.4 n=0.3 n=0.2 n=1/6

2 0.51 0.58 0.66 0.78 0.89 1.08 1.18 4 0.74 0.90 1.12 1.44 1.68 2.04 2.21 8 0.81 1.02 1.33 1.81 2.19 2.73 2.98 x 0.83 1.08 1.49 2.32 3.17 4.89 5.92

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