Asymptotic structure of the stress field in flow past a cylinder at high Weissenberg number

J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

Asymptotic structure of the stress field in flow past a cylinder at high Weissenberg number Michael Renardy* Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA Received 30 September 1998; received in revised form 1 June 1999

Abstract We study the flow of an upper convected Maxwell fluid past a cylinder in the limit of high Weissenberg number. We make the simplifying assumption that the velocity field is given and similar to the Newtonian case (as discussed below, this assumption is likely to be wrong) and consider the integration of the stress equations. The asymptotic behavior at high Weissenberg number leads to boundary layers along the edge of the cylinder as well as stress concentration in the wake. The stresses in the boundary layer are of order W while the width of the boundary layer is of order Wÿ1. The stresses in the wake are of order W3, and the width of the wake is of order W ÿ2. Just outside the boundary layer near the cylinder and wake, there is a region where the stresses are much larger (order W 3 near the cylinder and W 5 near the wake). The origin of these stresses is in the stretching flow near the upstream stagnation point. # 2000 Elsevier Science B.V. All rights reserved. Keywords: Asymptotics; Boundary layers; Streamlines

1. Introduction The flow past a cylinder or a sphere has been designated as a benchmark problem for the numerical simulation of viscoelastic flows, and it has been the subject of numerous studies [1±11]. The solutions obtained show steep stress gradients in a boundary layer near the cylinder (or the sphere) as well as in the wake. These stress gradients are most pronounced when the upper convected Maxwell (UCM) model is used as a constitutive hypothesis. The numerical simulations continue to encounter difficulties in resolving these stress gradients and lack of convergence. This has so far prevented successful numerical simulations at high Weissenberg numbers. To gain some insight into the nature of these stress layers, we shall examine a simpler problem in this paper. We shall assume a prescribed velocity field, which we assume to have similar characteristics as the Newtonian velocity field. We then consider the integration of the constitutive equations for the ÐÐÐÐ * Fax: +1-540-231-5960. 0377-0257/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 9 9 ) 0 0 0 5 0 - 6

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M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

stresses in this velocity field, assuming the UCM constitutive law, and we study the asymptotic behavior of the resulting stress field in the limit of high Weissenberg number. We shall see that we can identify an outer region consisting of streamlines where the time the fluid spends near the cylinder is small compared to the relaxation time. Close to the cylinder, where the velocity becomes small, we shall find a boundary layer with thickness of order 1/W and stresses of order W. This boundary layer is essentially like the boundary layers considered in [12]; one new feature is that the streamlines in the boundary layer come from a region of low Weissenberg number flow near the front stagnation point. In the wake, stresses of order W 3 are found in a region with thickness of order W ÿ2. The influence of the stretching flow near the upstream stagnation point produces stresses just outside the boundary layer and wake which are of a higher order of magnitude than those in the boundary layer and wake. We shall find that the stresses close to the cylinder reach order of magnitude W 3, and those near the wake are of order W 5. We can expect that the simplified problem considered in this paper produces some of the essential features of the stress field for the actual flow field that would arise for a UCM fluid, as in the reentrant corner problem [13,14]. On the other hand, the assumption of a Newtonian-like velocity field is certainly open to question. We note that the numerical simulations report a strong downstream shift of streamlines in the wake, suggesting that the high Weissenberg number velocity field is substantially different from the Newtonian. Also, the maximum stresses reported, for instance, in [6], are clearly of order W, not of order W 5 as the analysis of this paper would predict. It would thus appear that a significant change of the velocity field is necessary to satisfy the momentum balance equation. This remains an important issue for future research. A more detailed analysis of the numerical results may shed light on the crucial differences between the velocity field for the UCM flow and the Newtonian velocity field assumed here; the behavior near the two stagnation points on the cylinder is of particular interest. We shall focus our analysis on the two-dimensional flow past a cylinder; the axisymmetric flow past a sphere is analogous. The flow domain is the region

ˆ f…x; y† 2 R2 jx2 ‡ y2 > 1;

jxj < Rg

(1)

i.e. we have a cylinder of radius 1 inside a channel of half-width R. The fluid is flowing past the cylinder in the y-direction; we denote the streamfunction by (x,y), with the associated velocity given by u ˆ ÿ y, v ˆ x. Polar coordinates are introduced such that x ˆ r sin , y ˆ ÿr cos , so  ˆ 0 is upstream and  ˆ  is downstream. We assume that the streamfunction (x,y) is given with the following properties: 1. is an odd function of x, i.e. the flow past the cylinder is symmetric. 2. For y ! 1, asymptotes to the function x ÿ x3/(3R2). This makes the velocity equal to 1 ÿ x2/R2. This is a Poiseuille profile appropriate for a cylinder in a channel. As it turns out, the Poiseuille flow actually has little effect on the stresses near the cylinder and in the wake. 3. x is nonzero except on the cylinder and on the channel wall. This means that the velocity in the y-direction is always positive and there are no regions of recirculation. 4. Near x ˆ R, we have ˆ 2R/3 ÿ f(y)(x ÿ R)2 ‡ O(|x ÿ R|3), where f(y) > 0 and f(y) ! 1/R as y ! 1. This means the flow at the wall has the local shear rate 2f(y). 5. At r ˆ 1, the no slip condition applies, to that is proportional to (r ÿ 1)2. Moreover, we assume symmetry about the centerline, so that also vanishes at  ˆ 0 and  ˆ . It follows that is of the

M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

15

form ˆ K()(r ÿ 1)2 sin  ‡ O((r ÿ 1)3sin ), and we shall assume that K() is a nonzero positive function. For a given velocity field satisfying these qualitative properties (e.g. the Newtonian velocity field), we shall integrate the constitutive equation of the UCM fluid h i (2) W …v
r†T ÿ …rv†T ÿ T…rv†T ‡ T ˆ rv ‡ …rv†T : The stress integration is subject to the upstream condition of viscometric stresses in Poiseuille flow, i.e. 0 2x 1 0 ÿ 2 B R C C: (3) lim T…x; y† ˆ B @ A y!ÿ1 2x 8Wx2 ÿ 2 R R4

2. Preliminaries To simplify the stress integration, we use the same transformation of the equations as in [12]. This transformation expresses the stress tensor in a basis which is aligned with the velocity field. Specifically, if v ˆ (u, v) is the velocity vector and w ˆ (ÿv, u)/(u2 ‡ v2) is a vector perpendicular to v, we express the stress in terms of the dyadic products formed by these vectors: T ˆÿ

1 I ‡ vvT ‡ …vwT ‡ wvT † ‡ wwT : W

(4)

We now rewrite the constitutive law in terms of the new stress components ,  and . The transformed constitutive equations are W…v
r† ‡  ‡ 2W div w ˆ

jvj2 ; W…v
r† ‡  ‡ W div w ˆ 0; W…v
r† ‡  ˆ : W Wjvj2 (5) 1

The advantage of the transformation is that it yields a problem in which the stress components are decoupled: we can first solve the third equation for , then use the result in the second equation and solve for  and finally solve the first equation for . In each step, the equation to be solved is of the form W…v
r† ‡  ˆ f ; and the solution is obtained by integration along streamlines: Z 1 1 ÿs=W e f …y…x; s†† ds; …x† ˆ W 0

(6)

(7)

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M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

where y(x, s) is a point on the same streamline as x, such that the fluid takes the time s to move from y to x. Eq. (7) will play a crucial role in analyzing the asymptotic behavior of the stresses below. The goal of this paper is to analyze the asymptotic behavior of Eq. (5) as W ! 1. For this, it is important to see how |v| and div w behave in various parts of the flow domain. Clearly, |v| is of order 1 except close to the cylinder and close to the wall, where |v| goes to zero. Near the wall, |v| is proportional to R ÿ x. Near the cylinder, except near the two stagnation points, |v| is proportional to (r ÿ 1). The divergence of w is of order 1 expect near the wall, the cylinder and the centerline. Near the wall, div w becomes infinite; it is proportional to (R ÿ x)ÿ2. At the centerline, div w vanishes because of symmetry; we find that div w is of order x except when r is close to 1. Near the cylinder, div w is of order (r ÿ 1)ÿ2 except in a neighborhood of the two stagnation points which occur at r ˆ 1,  ˆ 0, . In the neighborhood of these stagnation points, is proportional to K(r ÿ 1)2 sin . It follows that the leading contribution to the velocity, in polar coordinates, is ! …r ÿ 1†2 cos ; 2…r ÿ 1† sin  : …vr ; v † ˆ K ÿ r We can approximate r in the denominator of vr by 1, and cos  by 1 and we find that |v| is of order q …r ÿ 1† …r ÿ 1†2 ‡ 4 sin2 : We can then calculate w and div w. We omit the details of this calculation, which shows that div w is of the order sin …7…r ÿ 1†2 ‡ 4 sin2 † ……r ÿ 1†2 ‡ 4 sin2 †2 …r ÿ 1†2

:

(8)

That is, if sin   r ÿ 1, then div w is of order 1/[(r ÿ 1)2 sin ], if r ÿ 1  sin , then div w is of order sin /(r ÿ 1)4, and if r ÿ 1 and sin  are of the same order, then div w is of order (r ÿ 1)ÿ3. 3. The outer flow region We begin by considering the behavior of the stresses along streamlines in the region away from the boundaries where W ÿ2  and 2R/3 ÿ  1/W. Along such streamlines, the time fluid particles need to go past the cylinder is small compared to the relaxation time W, and we can therefore neglect stress relaxation until the fluid is well downstream from the cylinder. We note that, in the region far upstream from the cylinder, we have jvj ˆ 1 ÿ

x2 ; R2

div w ˆ ÿ

2x R2 …1 ÿ x2 =R2 †2

;

and hence the stresses are (to leading order in W) given by  2 1 x2 2x 8Wx2 1ÿ 2 ;  ˆ 2;  ˆ : vˆ W R R R4 …1 ÿ x2 =R2 †2

(9)

(10)

M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

17

The same stresses apply far downstream from the cylinder. Except for streamlines close to the cylinder, where is small, the upstream values of the stresses given by Eq. (10) are indeed the dominant contributions to the stress. To see this, we merely have to note that, with the exception of streamlines which come close to the cylinder, |v| and div w do not change their order of magnitude along a streamline. As a consequences, the convected derivative of any stress component will be smaller in order of magnitude (by a factor 1/W) than its upstream value. (For instance, the upstream value of  is of the same order as |v|2/W, but (v
r) is only of the same order as |v|2/W2). As long as the time which the particle spends on the streamline is small relative to W, any change in the stress along the streamline is small relative to the initial value upstream. Consequently, if is of order 1 and 2R/ 3 ÿ  1/W, then the dominant contribution to the stress is !2 1 X… †2 2X… † 8WX… †2 1ÿ ; ˆ ; ˆ ; (11) vˆ W R2 R2 R4 …1 ÿ X… †2 =R2 †2 where ˆ X… † ÿ

X… †3 : …3R2 †

(12)

Let us now consider what happens on a streamline near ˆ 0, but still  1/W 2, so that the time a particle takes to go past the cylinder remains small relative to W and stress relaxation can be neglected until particles are well downstream from the cylinder. Then the dominant contribution to  is still given by Eq. (11), since the term |v|2/W in Eq. (5) only becomes smaller along the streamline. Hence we have   1/W. Recall, however, that the contribution of  to the stress is wwT , and w increases as the fluid approaches the cylinder. Hence if  remains approximately constant, then the stress is actually growing. This stress growth is caused by the stretching of the fluid transverse to the flow as the fluid decelerates when it approaches the cylinder. Now consider the equation for , i.e. the second equation of Eq. (5). There are two contributions to , one is the initial condition upstream, which is of order 2 /R2. The other contribution arises from integrating the term W div w. Let us parametrize the streamline by the angle . Along the streamline, we have …v
r† ˆ

1@  2K…†…r ÿ 1† sin  r @r

(13)

close to the cylinder. Moreover, at leading order in W, we have d …v
r†  div w ˆ ˆÿ : d …v
r† …v
r†

(14)

We have   1/W, and the behavior of div w near the cylinder is given by Eq. (8). We note that also, we have (r ÿ 1)2  /(K() sin ) near the cylinder. It follows that the order of magnitude of d/d is given by sin …7…r ÿ 1†2 ‡ 4 sin2 †=W……r ÿ 1†2 ‡ 4 sin2 †2 …r ÿ 1†2 : …2K…†…r ÿ 1† sin †

(15)

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M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

We note that K() and 7…r ÿ 1†2 ‡ 4 sin2 

(16)

…r ÿ 1†2 ‡ 4 sin2  are of order one, so that the order of magnitude of d/d is 1

: W…r ÿ 1† ……r ÿ 1†2 ‡ 4 sin2 † p Since r ÿ 1 is of order =sin , the order of magnitude of this term is 3

sin3=2  W…4 sin2  ‡ =sin †

3=2

:

(17)

(18)

This expression applies when r ÿ 1 is small, i.e. as soon as sin   . Clearly, the expression Eq. (18) behaves like sin5=2  W 5=2

(19) 1/3

if sin  

and like

1 1=2

W sin if sin  



1/3

3=2

. By integration, we find that the integral of d/d is of order

7=2 W 5=2 if  

1/3

(20)

(21) and of order

1=2 W 3=2 if   1/3. In contrast, the initial value of  upstream is of order . By putting these results together, we find the order of magnitude of  is   1=2   7=2 : max ; min ; W 3=2 W 5=2

(22)

(23)

When  takes on finite values, then  is of order 1/(W 3/2) if W 5/2  1 and of order if W 5/2  1. Hence there is a transition between a region where  is dominated by its upstream initial data and a zone where it is dominated by the integration of the term  div w along the streamline. This transition occurs for ˆ O(Wÿ2/5), i.e. at a distance from the cylinder of order Wÿ1/5. We note that this transition is not the boundary layer described in [12] and below; this boundary layer is characterized as the region where stress relaxation becomes important and has a thickness of order W ÿ1.

M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

19

In a similar fashion, we find that d/d is of the same order of magnitude as  div w ; …v
r†

(24)

and div w (v
rf) is as before. We shall skip over the discussion of the various cases which arise in dependence of the relative magnitude of , and W. Let us focus on the case where  is large relative to 1/3. In that case, the integral of  div w (v
r) is of order /(W 3) if   W 5/2; in contrast the initial value of  is of order W 2. Hence  is dominated by its initial value if W 2 5  , and it is dominated by the terms accumulated in the integration along the streamline if W 2 5  . For  of order 1, we conclude that the boundary between the region where  is dominated by its initial value and the region where it is dominated by the integral along the streamline is again where is of order W ÿ2/5, i.e. at a distance of order W ÿ1/5 from the cylinder. In summary, we therefore find a transition when is of order W ÿ2/5. Stress relaxation is negligible on both sides of this transition. However, for  W ÿ2/5, the dominant influence on the stresses is in the upstream initial values, and the change of the stresses along a streamline is associated entirely with the kinematics of the velocity field (i.e. in the representation Eq. (4) v and w vary along the streamline, but ,  and  are essentially constant). If  W ÿ2/5, on the other hand, the upstream shear rate becomes small, and the dominant influence on the  and  components is from the coupling terms in Eq. (5). The smallest value of the streamfunction for which stress relaxation can be neglected is of order W ÿ2. At this point, the magnitude reached by  is of order 1/(W 3) ˆ W 5. Since the stress associated with  is vvT , this leads to stresses near the cylinder which are of order W 3 (since |v| is of order Wÿ1), but stresses of order W 5 downstream from the cylinder where |v| is of order 1. 4. The boundary layer at the channel wall The analysis of the boundary layer at the channel wall is as in [12]. The velocity near the wall is, at leading order, given by v ˆ …f 0 …y†…R ÿ x†2 ; 2f …y†…R ÿ x††

(25)

and we conclude that the leading contribution to div w is ÿ

1 2f …y†…R ÿ x†2

:

(26)

We can now introduce the same scalings as in [12]. Let x ˆ R ‡ ~x=W; y ˆ ~y;  ˆ W ÿ3 ~; ~ Then the new equations in the rescaled variables are  ˆ ~;  ˆ W 3 . ~ @ ~ @ ~ ~ ÿ 2f …~y†~x ‡ÿ ˆ 0; @~x @~y f …~y†~x2 @ ~ @ ~ ÿ 2f …~y†~x ‡ ~ ˆ 4f …~y†2~x2 : f 0 …~y†~x2 @~x @~y

f 0 …~y†~x2

f 0 …~y†~x2

~ @ ~ @ ~ ÿ 2f …~y†~x ‡ ~ ÿ ˆ 0; @~x @~y 2f …~y†~x2 (27)

The Eq. (27) have a solution for which ~ is of order ~x2 ; ~ is of order 1, and ~ is of order ~xÿ2. From this, we find that  is of order W 3 =~x2, leading to a stress contribution vvT , which is of order W.

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M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

5. Boundary layer near the cylinder The difference between the boundary layer on the cylinder and the boundary layer on the channel wall is the presence of a separating streamline upstream of the cylinder: Particles close to the cylinder originate in an upstream region which is not close to a wall. This manifests itself in two competing influences in the solution of Eq. (5). The third equation has a forcing term |v|2/W. This term is much smaller near the boundary than it is in the outer flow. Hence the initial value of  upstream is much larger than the value which would be produced under the influence of the forcing near the boundary. We hence need to determine which contribution to  dominates: the effect of the upstream initial condition or that of the local forcing. This question is a matter of whether stresses have had time to relax. Indeed, we characterized the outer flow region above as the region where the initial condition remains dominant along the entire streamline. To analyze the flow near the cylinder, we rescale variables in a similar fashion as in Section 4. That is, we set r ˆ1‡

~r ; W

~  ˆ W 3 ;

 ˆ W ÿ3 ~:

(28)

In the rescaled variables, we have, at leading order q ÿ2 2 ÿ1 ˆ W K…† sin ~r ; jvj ˆ W K…†~r 4 sin2  ‡ W ÿ2~r2 ; div w ˆ

W 2 sin …7W ÿ2~r2 ‡ 4 sin2 † K…†…W ÿ2~r 2 ‡ 4 sin2 †2~r 2

…v
r† ˆ ÿ

;

…K…† cos  ‡ K 0 …† sin †~r2 @ 2K…†~r sin  @ ‡ : @~r W @ W

(29)

The rescaled equations which result are ÿ …K…† cos  ‡ K 0 …† sin †~r 2 ˆ

1 2 2

K…† ~r

‡ ~

…4W 2

2

2

sin  ‡ ~r †

@ ~ @ ~ ~ sin …7W ÿ2~r2 ‡ 4 sin2 † ‡ 2K…†~r sin  ‡  ‡ 2 @~r @ K…†…W ÿ2~r2 ‡ 4 sin2 †2~r2

;

sin …7W ÿ2~r2 ‡ 4 sin2 † K…†…W ÿ2~r 2

2

2 2

‡ 4 sin † ~r

ÿ…K…† ‡ cos  ‡ K 0 …† sin †~r 2 ˆ 0;

‡ ~ ˆ K…†2~r2 …4 sin2  ‡ W ÿ2~r2 †:

@ @ ‡ 2K…†~r sin  ‡ @~r @

ÿ…K…† cos  ‡ K 0 …†sin †~r 2

@ ~ @ ~ ‡ 2K…†~r sin  @~r @ (30)

The rescaled streamfunction is ~ ˆ K…† sin ~r2 , so that along a given streamline we have, for instance q 0 d~  @ ~ @ ~ ~…K …† sin  ‡ K…† cos † @ ~ @ ~ ~r…K 0 …† sin  ‡ K…† cos † ˆ ÿ ÿ : (31) ˆ d @ @~r @ @~r 2K…† sin  2…K…† sin †3=2

M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

21

Hence the last equation of Eq. (30) can be rewritten as 2K…†~r sin 

d~  ‡ ~ ˆ K…†2~r 2 …4 sin2  ‡ W ÿ2~r 2 †; d

or equivalently, ! q ~ ~ d~  4 sin2  ‡ W ÿ2 : 2 ~K…† sin  ‡ ~ ˆ d sin  K…† sin 

(32)

(33)

We note that the right hand side of the first equation in Eq. (30) as well as the term W ÿ2~r2 can be neglected unless sin  is small of order 1/W (which is the case in the neighborhood of the front and rear stagnation points). If we neglect these terms, then the Eq. (30) become independent of W, i.e. the ~  and ~ are of order 1) are formally scalings assumed in their derivation (which presumed that , consistent. We need to assess the effect of upstream influences, however. The stresses consist of two contributions: those generated by the local flow in the boundary layer and those generated by the memory of the upstream flow. If  is large relative to 1/W, then the stresses generated by the flow in the boundary layer are compatible with the scaling Eq. (28), but we need to consider the possibility that stresses due to upstream effects are larger. We note that the right hand side of Eq. (33) becomes infinite as sin  ! 0. The approximations leading to Eq. (33) remain valid as long as ~r < W, i.e. as long as   W ÿ2 ~. If  is of order W ÿ2 ~, then the right hand side of Eq. (33) is of order W2. Consequently, ~ starts from an upstream initial value of order W2, and the presumption underlying the derivation of Eq. (30) that ~ is of order 1 is valid only if the initial stress has had time to relax. It follows from Eq. (33) that stress relaxation occurs when s Z  1  q d  (34) ~ W ÿ2 ~ ~ reaches order of magnitude 1, i.e. when  is of order ~. Hence we have  ˆ O(Wÿ3),  ˆ O(1),  ˆ O(W3) if   ~, or equivalently, r ÿ 1  1/W. If, on the other hand r ÿ 1  1/W, then we have  ˆ O(1/W),  ˆ O(W2),  ˆ O(W5) as predicted in the analysis of the outer flow above. 6. The wake of the cylinder The preceding analysis shows that a stress concentration is generated near the cylinder in the region where is of order 1/W2. As we move downstream from the cylinder, div w becomes small, and the Eq. (5) become decoupled at leading order. Consequently, the components ,  and  simply decay to their equilibrium values on a length scale of order W, starting from their initial values near the cylinder which are given by the analysis above. Consequently, in a region with width of order W ÿ2 and length of order W downstream from the cylinder, we find stresses of order W 5. The magnitude of the stresses decreases to order W 3 when the value of the streamfunction becomes small relative to W ÿ2. This is, roughly speaking, because particles near the centerline only have memory of the stretching flow near the downstream stagnation point, while particles a little off the centerline feel the stretching from both stagnation points.

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M. Renardy / J. Non-Newtonian Fluid Mech. 90 (2000) 13±23

We note that stresses of order W 5 have not been observed in numerical simulations. Indeed, the stresses reported, for instance, in [6] appear to be of order W. (In the numerical simulations W is limited to values less than approximately 3, but the comparison of maximum stresses at different values of W in the range computed suggests they are proportional to W). This suggests that a rearrangement of the flow field takes place in order to avoid high stresses which would lead to an imbalance in the momentum equation. It would be instructive to make comparisons between the full simulation of the UCM flow past a cylinder and the stress distribution which arises if the velocity field is given as Stokes flow as in the analysis presented here. The behavior of the velocity field in the neighborhood of the front and rear stagnation points is of particularly crucial importance for the behavior of the stresses. Careful documentation of the details of the velocity field in these regions would shed significant light on the nature of the flow. Another conclusion of the analysis above is that putting the cylinder into a channel has absolutely no effect on the asymptotics of the stresses in the neighborhood of the cylinder and in the wake. Indeed, the influence of the upstream shear flow is felt only at a distance from the cylinder of order Wÿ1/5. This scaling would change if the shear rate upstream of the cylinder were different from zero, and it would be of some interest to consider the flow past a cylinder in a channel where the cylinder is located in an off center position. Acknowledgements This research was supported by the National Science Foundation under Grant DMS-9870220. References [1] P.J. Oliveira, F.T. Pinho, G.A. Pinto, Numerical simulation of non-linear elastic flows with a general collocated finitevolume method, J. Non-Newtonian Fluid Mech. 79 (1998) 1±43. [2] Y. Huang, J. Feng, Wall effects on the flow of viscoelastic fluids around a circular cylinder, J. Non-Newtonian Fluid Mech. 60 (1995) 179±198. [3] J. Sun, N. Phan-Thien, R.I. Tanner, An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG, J. Non-Newtonian Fluid Mech. 65 (1996) 75±91. [4] M.B. Bush, The stagnation flow behind a sphere, J. Non-Newtonian Fluid Mech. 49 (1993) 103±122. [5] V. Delvaux, M.J. Crochet, Numerical prediction of anomalous transport properties in viscoelastic flow, J. NonNewtonian Fluid Mech. 37 (1990) 297±315. [6] X.L. Luo, An incremental difference formulation for viscoelastic flows and high resolution FEM solutions at high Weissenberg numbers, J. Non-Newtonian Fluid Mech. 79 (1998) 57±75. [7] H. Jin, N. Phan-Thien, R.I. Tanner, A finite element analysis of the flow past a sphere in a cylindrical tube: PTT fluid model, Comput. Mech. 8 (1991) 409±422. [8] W.J. Lunsmann, I. Genieser, R.C. Armstrong, R.A. Brown, Finite element analysis of steady viscoelastic flow around a sphere in a tube: calculations with constant viscosity models, J. Non-Newtonian Fluid Mech. 48 (1993) 63±99. [9] Y. Fan, M.J. Crochet, High order finite element methods for steady viscoelastic flows, J. Non-Newtonian Fluid Mech. 57 (1995) 283±311. [10] X.L. Luo, Operator splitting algorithm for viscoelastic flow and numerical analysis for the flow around a sphere in a tube, J. Non-Newtonian Fluid Mech. 63 (1996) 121±140. [11] F.P.T. Baaijens, S.H.A. Selen, H.P.W. Baaijens, G.W.M. Peters, H.E.H. Meijer, Viscoelastic flow past a confined cylinder of a low density polyethylene melt, J. Non-Newtonian Fluid Mech. 68 (1997) 173±203.

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[12] M. Renardy, High Weissenberg number boundary layers for the upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech. 68 (1997) 125±132. [13] M. Renardy, The stresses of an upper convected Maxwell fluid in a Newtonian velocity field near a reentrant corner, J. Non-Newtonian Fluid Mech. 50 (1993) 127±134. [14] M. Renardy, A matched solution for corner flow of the upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech. 58 (1995) 83±89.