## Abstract

We discuss here the steady planar flow of the upper convected Maxwell fluid at re-entrant corners in the singular limits of small and large Weissenberg number. The Weissenberg number is a parameter representing the dimensionless relaxation time and hence the elasticity of the fluid. Its value determines the strength of the fluid memory and thus the influence of elastic effects over viscosity. The small Weissenberg limit is that in which the elastic effects are small and the fluid's memory is weak. It is an extremely singular limit in which the behaviour of a Newtonian fluid is obtained in a main core region away from the corner and walls. Elastic effects are confined to boundary layers at the walls and core regions nearer to the corner. The actual asymptotic structure comprises a complicated four-region structure. The other limit of interest is the large Weissenberg limit (or high Weissenberg number problem) in which the elastic effects now dominate in the main regions of the flow. We explain how the transition in solution from Weissenberg order 1 flows to high Weissenberg flows is achieved, with the singularity in the stress field at the corner remaining the same but its effects now extending over larger length-scales. Implicit in this analysis is the absence of a lip vortex. We also show (for the main core region) that there is a small reduction in the velocity field at the corner and walls where it becomes smoother. This high Weissenberg number limit has a six-region local asymptotic structure and comment is made on its relevance to the case in which a lip vortex is present.

## 1. Introduction

The planar contraction geometry is a common benchmark problem used in the numerical simulation of viscoelastic flows. The problem combines geometric simplicity with flow complexity (often a lip as well as a salient corner vortex is present) making it suitable for the development and assessment of numerical methods. Moreover, the situation is of practical relevance, arising in industrial processes such as extrusion and injection moulding, as well as of rheological interest to test the suitability of constitutive equations to represent polymer solutions (e.g. Quinzani *et al*. 1995). Central to the configuration is the presence of a re-entrant corner which has and still does provide a significant challenge to numerical stability and analytical understanding.

Although the upper convected Maxwell (UCM) model can only partially predict real fluid rheology, it is of sufficient importance to merit attention. This importance arises since it can often be obtained as a limiting form of more complex and realistic constitutive equations; the omitted terms often serve to modify or regularize the UCM behaviour. Further, the UCM equations pose some of the most severe numerical difficulties compared to other common constitutive differential models since it has the highest elastic stress response for high deformation rates. This makes it suitable to be used for the development of accurate and robust numerical methods that can then be used for other constitutive equations (see, for example, the discussion in Oliveira & Pinho (1999)).

In the UCM model, the influence of elasticity is measured through the Weissenberg number , which represents a dimensionless relaxation time for the flow (although the Deborah number is a common alternative). Despite the recent progress made in the development of efficient and stable numerical methods for the UCM (and related) models, many challenges remain. One such issue is the upper limit to the range of Weissenberg numbers over which converged solutions can be attained. Beyond this limit many schemes suffer from numerical instabilities with the upper Weissenberg limit decreasing as the mesh is refined. Although this numerical upper Weissenberg limit has increased as schemes have improved, it still nevertheless persists and the so-called high Weissenberg number problem continues to attract much attention. A number of factors have been proposed as the reasons for encountering difficulty in this limit, including the presence of the geometric singularity, difficulty in obtaining adequate resolution of boundary layers within the flow and the dominance of nonlinear terms in the constitutive equations for high values of the Weissenberg number (e.g. Philips & Williams 1999; Aboubacar *et al*. 2002). A key feature of experimental and numerical work is the occurrence of a lip vortex at order 1 Weissenberg numbers, which appears to grow in size as the Weissenberg number increases. The situation we consider here for the high Weissenberg problem is the simpler one in which the lip vortex feature is assumed to be absent and the flow originating at the upstream wall reaches the downstream wall. In this sense, we consider the same parallel flow structure for both the small and high Weissenberg problems with no separating streamlines originating from the corner. The solution structure of this case is still relevant to those cases in which the lip vortex is present and will be commented upon in the discussion.

The dimensionless governing equations for steady incompressible flow that we consider here are the usual continuity, momentum and UCM constitutive equations(1.1)withwhere * v* is the velocity, T is the extra-stress tensor, D is the rate-of-strain tensor and

*p*is the pressure. The dimensionless parameters are the Reynolds number (

*Re*), together with the Weissenberg number (

*We*) representing the dimensionless relaxation time. We require to solve these equations in the two-dimensional sector with

*θ*=0 representing the upstream wall and the downstream wall where

*r*is the radial distance from the corner. We are interested in describing the asymptotic structure of these equations in the neighbourhood of the corner in the Newtonian limit and the high Weissenberg limit . We may set

*Re*=0 and only consider the creeping flow equations, as the inertia terms can be shown to be uniformly subdominant in regions local to the corner, irrespective of the size of the Weissenberg number, due to the stress singularity.

Relevant to the analysis that follows, the above equations written in component form and with respect to both Cartesian and polar coordinates are given in appendix A. Since we are dealing with planar flow, it is convenient to introduce the stream function *ψ*. When considering the wall boundary layers, we will assume symmetry with respect to the upstream and downstream structures. Thus, it will be sufficient to consider one structure only and for definiteness we take Cartesian axes aligned with the upstream wall. The downstream layer can then be considered by allowing for a change in the flow direction and re-orientation of the axes by the transformation(1.2)the other variables being unchanged. This transformation leaves the governing equations invariant and relative to the upstream layer, its net effect is to simply change the sign of the stream function when considering the downstream layer.

As a preliminary observation, it is worth noting that since the problem has no natural length-scale, the Weissenberg number may be removed from the formulation. The scalings that achieve this are(1.3)with the unique value if the inertia terms are to be retained or any *q* when and creeping flow is considered. It is also worth noting that under these scalings the inertia terms are negligible at leading order when . When , this condition is satisfied for any in the limit and any in the limit . The significance of these observations is that the problem will appear as an artificial region, i.e. as part (or limit) of a larger Weissenberg number dependent region. These scalings break down (i.e. they no longer hold uniformly) in both the limits and , and it is the manner of this break down that is the subject of this paper.

The main results of this paper may be put in the context as follows. For the UCM and Oldroyd-B models, the analytical results of Hinch (1993) (with boundary layer analysis completed in Renardy 1997*a*; Rallison & Hinch 2004; Evans 2005*a*,*b*) predicted a stream function and stress singularity behaviour at the corner of the form(1.4)This has been confirmed numerically in the 270° case by Singh & Leal (1995), Baaijens (1998), Xue *et al*. (1998), Alves *et al*. (2000), Aboubacar & Webster (2001) amongst others despite initial setbacks (e.g. Lipscomb *et al*. 1987; Coates *et al*. 1992). These analytical results hold when the Weissenberg number is order 1. In §2, we show that in the small Weissenberg limit , these results generalize to(1.5)This behaviour being obtained for , where *λ* is an eigenvalue of the Newtonian flow problem and defined later in (2.5). The high Weissenberg number problem is considered in §3, where we show that(1.6)this behaviour holding for and with the absence of a lip vortex. In addition to these results, the asymptotic structure of the solution for these limits will benefit numerical simulation and we pass comment upon this in the context of past and future work. We remark that similar notation is used in both §2 and §3 in order to avoid unnecessary notational proliferation. The problems of these two sections are independent and should be treated as so.

## 2. The small Weissenberg limit

The asymptotic regions and main balances that will be described here are summarized in figure 1. In the radial direction from the corner, there are essentially two ranges for the radial distance that need to be distinguished. These ranges will be labelled as exterior and interior, respectively, as the radial distance from the corner decreases. Further, for each of these ranges we need to consider both core (away from the walls) and boundary layer (near wall) regions. We begin by considering a regular expansion in powers of , which gives a leading-order problem in which we simply set in (1.1). The region within which a solution to this problem holds will be termed the outer 1 region and forms the first of our exterior regions.

### (a) The exterior regions: core and boundary layer regions 1

In the limit , the constitutive equation gives the Newtonian balance(2.1)at leading order in the Weissenberg number . We now seek separable (self-similar) solutions of the form(2.2)where is a constant, holding for . The momentum equation in (1.1) then yields the nonlinear eigenvalue problem for the exponent *λ* and the corresponding eigenfunction as(2.3)with two-point boundary conditions(2.4)This gives the Newtonian solution(2.5)(2.6)originally derived in Dean & Montagnon (1949) and discussed further in Moffatt (1964). The constant *A* may be fixed by arbitrarily taking or simply taking *A*=1. A plot of *λ* (taken as the smallest positive root of (2.5)) against *α* is shown in figure 2 where for (and in fact *λ*<*α* over this range of *α*). Table 1 records values of *λ* and associated derived quantities for selected corner angles.

For later reference when matching with core region 2, we record the polar components of the extra-stresses in this case,(2.7)as well as the pressure (to within an additive constant)This gives the leading-order solution in core region 1 as Newtonian. As the wall (*θ*=0 without loss of generality) is approached, we have the limiting behaviours(2.8)on using and . UCM viscometric behaviour at the wall is recovered through another region, which we will denote as boundary layer region 1. For this region, (2.8) suggests the variables as(2.9)in which case the momentum and constitutive equations become(2.10)(2.11)(2.12)(2.13)(2.14)At leading order in we thus obtain(2.15)These equations possess the exact explicit solution(2.16)which matches with the core region 1 solution (see (2.8)) and gives UCM viscometric behaviour for the stresses at the wall.

We may summarize by saying that in core region 1 we have the Newtonian behaviour(2.17)Next, we consider the regions in which the Oldroyd stress derivative is retrieved within the core.

### (b) The interior region: core region 2

In this region, we introduce new variables through the scalings(2.18)where is an appropriate gauge to be determined. In we have full balance in the governing equations(2.19)(2.20)(2.21)(2.22)(2.23)where . These equations do not possess solutions in separable form and moreover they hold uniformly in *θ*. Thus, the distinction of a core and boundary layer region is artificial with any boundary layer being passive. Matching to the outer 1 region gives the behaviour(2.24)and determines the gauge as(2.25)For completeness, matching with the outer 3 region (which we will consider in §2*c*) gives(2.26)Although we cannot obtain any explicit solutions in this region since the full equations (2.19)–(2.23) hold,1 we can deduce from (2.18) and (2.25) the stream function and extra-stress scalings as(2.27)It is worth noting that the scalings (2.18) for this region are those noted in (1.3) with for removal of the Weissenberg number from the governing equations.

### (c) The artificial regions: core and boundary layer regions 3

To consider core region 3, we introduce the rescaled variables(2.28)where and will be explained below. Core region 3 will be defined as and away from the walls (or equivalently ), and in which we obtain the balance(2.29)Following Hinch (1993) (and Evans 2005*a*), we will take the self-similar solution(2.30)with and constants , . In Cartesian form, the extra-stresses areHere, and are gauges depending on the Weissenberg number that define the scalings for the stream function and the extra-stresses in the third core region. These are determined by matching the stream function and extra-stresses to their solutions in core region 2 which requires(2.31)These follow immediately by expressing the solution (2.30) in the core 2 variables (2.18). The constants , are necessarily arbitrary and assumed order 1. The parameter *n* is also introduced and determined below by matching into the wall boundary layer.

For matching purposes, we note that the outer 3 solution (2.30) in Cartesian form has the limiting wall behaviour,(2.32)where(2.33)Here, we have used and with the leading-order behaviour of the pressure *p* determined from the momentum equation.

The relaxation and rate-of-strain terms are recovered in a boundary layer region at the wall, which will be denoted as the boundary layer region 3. The scalings for this region in general form are(2.34)These scalings retain the fullest balance in the constitutive equations and here the gauge is as yet undetermined. These give the equations(2.35)(2.36)(2.37)(2.38)(2.39)We necessarily require in order for this region to be asymptotically small. Thus, in , (2.35)–(2.38) give the leading-order equations(2.40)(2.41)(2.42)(2.43)where only. Expressing (2.32) in the boundary layer region 3 variables (2.34) immediately gives(2.44)Using (2.31), we thus have(2.45)and we note that since necessarily . The leading-order matching conditions are(2.46)together with . In addition, (2.40)–(2.43) are subject to the usual wall conditions(2.47)

Determination of the gauges in (2.45) implies the following order of magnitude estimates for the variables in core region 3:(2.48)with of course *n* given by (2.45). It is straightforward to confirm from these that the relaxation and rate-of-strain terms are subdominant in the constitutive equation, leaving (2.29) as the correct balance.

For , we note that the second expression in (2.44) is self-consistent and thus remains indeterminate. It therefore acts as an artificial parameter and scales out of the problem in very much the same way as the artificial parameter used in Evans (2005*a*) for the case. In this respect, we may consider the core and boundary layer regions 3 as part of core region 2; these regions give the limiting behaviour at the corner.2 We also note that the boundary layer for this region is lost when , i.e. when radial distances are comparable to the main part of the interior region, the core region 2. Within the interior region, the stress singularity increases from its Newtonian behaviour received from core region 1 to its more singular behaviour (2.48) in core region 3.

### (d) Summary

The analysis of the small Weissenberg limit has revealed a complicated four-region structure. The Newtonian behaviour,is obtained on radial distances from the corner greater than and distances from the walls greater than *O*(*We*). For radial distances from the corner of , we obtain the full viscoelastic balance in the constitutive equations. It is only on length-scales far smaller than this, that we start to find the core similarity solution of the problem. For the 270° corner, this occurs for . Moreover, the wall boundary layer thicknesses are on even smaller scales as indicated by the sizes recorded in figure 1; these will be needed if convergence at the downstream wall is to be achieved. It is thus clear that the resolution used in Lipscomb *et al*. (1987) and Coates *et al*. (1992) is inadequate to capture the stress singularity behaviour.

For example, in Lipscomb *et al*. (1987), simulations at would necessarily need elements considerably smaller than , although the edge of the first element used is only at . Thus, their simulations are still within the predicted Newtonian region and not surprisingly the scheme used picks up results close to these together with features of a second-order fluid, which would be expected as an erroneous approximation to the full balance at the edge of core region 2, i.e. when *r* is close to . As the Weissenberg number increases, the influence of the Newtonian core region 1 diminishes and is eventually lost as core region 2 expands to hold on length-scales (see (2.25)) together with its artificial regions holding for . In comparison, it is worth noting that the successful simulations of Singh & Leal (1995) take mesh element sizes as small as radially with an element vertex angle as low as 0.06 radians and larger (effective) values. Regarding the angular resolution, it is worth noting that boundary layer 3 is of thickness (for a 270° corner) on length-scales when (the artificial regions of §2*c* still being valid). Thus, the angular refinement near the walls should be smaller than to capture the boundary layer behaviour if simulations for are to be performed. For example, if then approximately (in radians) which appears to be adequately resolved by the meshes used in Singh & Leal (1995) (cf. to the value of 0.06 radians). For smaller values of , the radial and angular resolution becomes more challenging, now requiring angular refinement smaller than on radial length-scales of that must be smaller than .

If convergence at the downstream wall is not obtained, then higher erroneous stress singularities will be obtained, as reported by Baaijens (1998). This may be the cause of the inadmissible stress behaviour results obtained in Coates *et al*. (1992). Finally, it is worth remarking that the initial work of Keunings (1986) obtained small critical values of the Weissenberg number that were mesh dependent and above which the convergence of the discretized system could not be obtained. The numerical scheme used continuation in the Weissenberg number with the Newtonian case as the initial starting point. The finest mesh gave a critical Weissenberg number of 0.112 in the 270° geometry, with a corner element size of . Whether coincidence or not, we note that at this value of the Weissenberg number, the edge of our core region 2 is , which is of the same magnitude as the corner element size. It thus appears that the scheme encountered difficulties once core region 2 grew to the same size as the corner element, with the implication that the numerics would have been capturing Newtonian behaviour up until this point.

## 3. The high Weissenberg number limit

We now consider the opposite limit of of relevance to high Weissenberg number flows. The analysis will involve results similar to those of §2*c*, which will be referred to whenever possible with the differences highlighted. We now require a six-region structure which is illustrated in figure 3. Letting in (1.1) leads us to the consideration of our first regions, core region 1 with its associated boundary layers.

### (a) Core and boundary layer regions 1

The solution for core region 1 will be the outer solution, its leading-order behaviour being given by the solution of the problem with . We thus obtain the balance (2.29) with a solution with limiting behaviour for small *r* of the form (2.30). Since the leading-order behaviour in this region must be independent of the Weissenberg number, we take . Thus, for the leading-order solution in we take(3.1)with and constants . In Cartesian form, this solution has the limiting wall behaviour,(3.2)where(3.3)Here, we have used and as with . The leading-order behaviour of the pressure *p* is determined from the momentum equation.

Viscometric behaviour is recovered through boundary layer 1 region, which is given by the scalings(3.4)where matching to core region 1 requires . In this region, we have the leading-order equations(3.5)(3.6)(3.7)(3.8)where(3.9)These equations are subject to the matching conditions(3.10)together with the usual wall conditions(3.11)We remark that there are no similarity solutions for this problem, at least in the form considered in Evans (2005*a*) since here . This boundary layer region 1 is a high Weissenberg number single-layer structure, which are considered in general in Evans (submitted *a*), while the solution for core region 1 falls within the class of high Weissenberg number solutions described by Renardy (1997*b*).

### (b) Core and boundary layer regions 2

To consider core region 2, we first introduce the rescaled variables(3.12)where will be determined below. Core region 2 will be defined as and away from the walls (or equivalently ). Matching to the solution (3.1) in core region 1 determines the scalings for the stream function, extra-stresses and pressure as(3.13)In we have the governing equations(3.14)(3.15)(3.16)(3.17)(3.18)where . We thus have the leading-order balance(3.19)within the constitutive equations, provided (which certainly holds for and will be confirmed below for the remaining range when is determined). Matching to the behaviour (3.1) in core region 1 gives(3.20)where . As the corner is approached, we expect to retrieve the self-similar solution of the problem, namely(3.21)where(3.22)The leading-order solution to (3.14)–(3.18) does not take self-similar (separable) form,3 but can do so in its extreme radial limits as shown by (3.20) and (3.21). Since the balance (3.19) holds, the solution of these equations belongs to the general class of solutions of the compressible Euler equations as discussed by Renardy (1997*b*). It is worth noting that both explicit limiting behaviours also belong to this class of solutions (and in fact are solutions of the incompressible Euler equations). We are thus postulating here that there is in fact a solution to core region 2 within the general class that links the two limiting solutions.

Viscometric behaviour is given by boundary layer region 2, which is defined by the scalings(3.23)In this region we have the leading-order equations(3.24)(3.25)(3.26)(3.27)These equations are subject to the usual wall conditions as well as the core 2 matching conditions(3.28)In these matching conditions, the functions represent the dependence of the wall behaviour for the leading-order solution of core 2. The exponent of the normal coordinate in the stream function is as yet unspecified, but this behaviour for the matching of the stream function (3.28) together with the scalings (3.23) now determines as(3.29)The appropriate choice for is in fact , the value for the problem which we will justify next by considering another region closer to the corner. However, adopting this choice then fixes as(3.30)which satisfies the requirement immediately following (3.19) for the relaxation terms to be negligible.4 This value for also ensures consistency between (3.28) and (3.21) where we obtain(3.31)with and(3.32)

Before proceeding to the final regions, it is worth elaborating upon the matching between core and boundary layer regions 1 and 2. This is best explained using Kaplun's concept of a continuum of intermediate limits between these regions (see Van Dyke 1975; Hinch 1994). We do this here by considering (3.12) and (3.13) as intermediate variables by setting(3.33)with . We obtain the equations as (3.14)–(3.18) in the intermediate core region (where the relaxation and rate-of-strain terms may be confirmed as uniformly negligible) as well as the intermediate boundary layer equations (3.24)–(3.27) under the same scalings (3.23). The matching condition (3.28) also remains the same, although we now replace with *n*. Thus, (3.29) holds with and using (3.33) we have(3.34)In this way, we can move smoothly between core and boundary layer regions 2 which occur when *s*=0 (and thus ) and core and boundary layer regions 1 when (and *n*=3). We note that the boundary layer thicknesses change size from in boundary layer region 2 to the larger width in boundary layer region 1, which is explained clearly by using (3.33), (3.34) and the intermediate variables.

### (c) Core and boundary layer regions 3

To consider core region 3, we first introduce the rescaled variables(3.35)for some gauge . Core region 3 will be defined as and away from the walls (or equivalently ), and in which we again obtain the balance (2.29). In this region we take the leading order self-similar solution(3.36)with and as given in (3.22). Matching to the solution (3.21) in core region 2 determines the gauges as(3.37)Viscometric behaviour is given by boundary layer region 3, which is defined by the scalings(3.38)The leading-order equations holding in boundary layer region 3 are again precisely (2.40)–(2.43) and subject to the no slip condition (2.47) and matching conditions (2.46) with *n* set to . The pressure is as given following (2.46).

Thus, in these two regions we recover the problem described in Evans (2005*a*) with the artificial parameter *ϵ* being replaced here with . To emphasize this, it is worth noting from (3.36) that the scalings for the stream function and extra-stresses in core region 3 can be written aswhereThus, we have the scalings of core region 2 together with the scalings of the artificial gauge now occurring as the ratio . In this sense, core and boundary layer 3 regions are not truly additional distinct Wesseinberg dependent regions, but really the limiting behaviour of the more general core and boundary layer 2 regions.

### (d) Summary

The high Weissenberg number limit thus comprises the six-region structure summarized in figure 3. In the core regions, it is worth noting that the stress singularity is and thus its effect now ranges up to distances. In the main outer core region (i.e. core region 1), the stream function vanishes to as opposed to in the case. Thus, the effect of taking the large Weissenberg limit from the flow is to reduce the velocity field in the main outer core region, but with the same stress singularity persisting. However, the problem is still present and is now contained within the regions located at distances smaller than from the corner. It is worth noting that the wall boundary layers are of thickness at radial distances and reduce to thickness at distances. It is core region 2 which accommodates the change in velocity field between the smoother stream function behaviour in core region 1 and the scaled behaviour in core region 3. Numerically, this six-region structure will be an extremely challenging problem to resolve adequately since it comprises several elements of difficulty: namely, no reduction in the stress singularity with it extending up to distances from the corner as well as wall boundary layers that change thickness with distance from the corner. This latter feature is of particular note.

## 4. Discussion

This paper deals with the local asymptotic structure for UCM fluids near a re-entrant corner in the singular limits of small and high Weissenberg number. The main Weissenberg dependent regions have been identified along with the appropriate scalings and equation balances for the dependent variables. Attention has focused upon the situation depicted in figure 4*a* where complete flow is assumed to take place around the corner with no separating streamlines present. Such a flow structure is seen in both numerical and experimental simulations for small values of the Weissenberg number. However, for large Weissenberg values (and large here seems to be ), numerical and experimental work indicate the local flow structure changes to that shown in figure 4*b* where a separating streamline is now present on the upstream side of the flow and attached to the corner. The separating streamline as drawn is concave, which is its reported shape when it first appears at moderate Weissenberg numbers, but can and does change to being convex at higher Weissenberg numbers. On larger scales than those drawn, the separating streamline is associated with a region of recirculation at the upstream wall, this being the upstream lip vortex. Works of Oliveira & Pinho (1999) and Alves *et al*. (2000) report the initial formation of a lip vortex for values near , which persists for larger Weissenberg numbers but does not appear to be present (or at least detectable) for smaller values. A maximum value of was obtained for convergence on their finest meshes. Thus, the numerical results in the literature seem to be indicating that there are two main local flow structures distinguished by the presence or absence of a separating streamline from the corner. However, an unsettling common feature of many of these numerical results is the apparent loss of the lip vortex with mesh refinement. This has been occurring since the early work of Marchal & Crochet (1987).

With these comments in mind, let us make some remarks on the type of solutions adopted here and their possible extensions:

We have adopted similarity solutions for the core regions in both the small and high Weissenberg number structures, when such solutions are available. These were taken for definiteness and to simplify the exposition, although more general and non-self-similar solutions could equally be adopted if appropriate. However, the self-similar solutions considered do appear to be the most physically relevant and those caught in full numerical simulations. A further comment is in regard to only considering leading-order terms in the outer regions, particularly for the artificial regions in both Weissenberg limits. These are sufficient to determine the scalings, but we would need to proceed to higher-order terms in order to correctly set up the boundary-value problems for the downstream boundary layers (e.g. Rallison & Hinch 2004; Evans submitted

*b*,*c*).*Initial lip vortex formations*. In addition to considering parallel flow solutions to the upstream boundary layer we could also consider reverse flow solutions in which the stream function changes sign between the upper and lower parts of the boundary layer. The scalings for the asymptotic regions would be unchanged from those of the main text in both the small and high Weissenberg limits; however, this would give a flow situation of the type shown in figure 4*b*. This would necessarily require a separating streamline at the edge of the upstream boundary layer, representing a lip vortex closely confined to the upstream wall as it occurs in its initial stages of formation. At this point, the two-point boundary-value problem for such solutions can be formulated, although their numerical solutions have yet to be demonstrated. However, a related case in which reverse flow solutions at the upstream wall have been demonstrated is the two-region boundary layer structure of Evans (2005*c*). For this case, we would anticipate that the regions and balances should be similar to that discussed here in the respective small and high Weissenberg cases.*Fully developed lip vortices*. Analysis for this situation is still to be addressed and so definitive remarks cannot be made. However, it is worth mentioning that there may be situations in which the structures that we considered here are likely to be relevant. For example, if the separating streamline makes an angle with the upstream wall, then we again have an effective re-entrant corner if . In this case, the core regions downstream of the separating streamline and the downstream boundary layer discussed in this paper would still be expected to hold for both Weissenberg limits (although this structure will probably be more relevant in the high Weissenberg limit). Now, however, the fluid entering the core regions comes from a mainstream flow for which a greater variety of conditions would be anticipated to exist than when it originates at the upstream wall. If, on the other hand, , then no effective re-entrant corner is present and the analysis here will not be directly relevant.

As regards future work, it would be interesting to see if the small Weissenberg structure can be validated in full numerical simulations. The situation for large Weissenberg number is complex and needs far more development with regard to numerical schemes before comparison with the asymptotics can be made. One final comment that is worth making purely as a conjecture is in regard to the phenomenon of the appearance and disappearance of lip vortices with mesh refinement. The local analytical solutions at the re-entrant corner for do not depend on the Weissenberg number and the flow structures of the two forms shown in figure 4 are beginning to be constructed using self-similar solutions. In principle, these flow structures are possible for all Weissenberg numbers and what determines which occurs seems to be dependent on the mainstream flow, i.e. the mainstream flow will determine which structure is to be picked out at the re-entrant corner. In this sense, global influences rather than local influences seem to determine the local flow structure. In numerical simulations, continuation in is invariably used and it appears that without adequate resolution, it is possible to change solution branches, e.g. from a solution branch with flow structure of figure 4*a* to one with that of figure 4*b*. Possible causes could be sensitivity to mainstream conditions including a stability issue with respect to the Weissenberg number.

## Footnotes

↵Equations (2.19)–(2.23) are to be solved on , which requires a numerical solution. The existence of such a solution naturally arises particularly as we additionally require the behaviours (2.26) and (2.24). As evidence to support such a solution we make the following remarks: (i) Equations (2.19)–(2.23) are precisely the problem, for which a converged numerical solution with the behaviour (2.26) has been demonstrated by workers such as Singh & Leal (1995), Baaijens (1998) (away from the walls only), Xue

*et al*. (1998), Oliveira & Pinho (1999) and Alves*et al*. (2000). (ii) The matching behaviour (2.24) with the exterior region is a consistent far-field behaviour for the equations (2.19)–(2.23). This is readily demonstrated sinceand as , we obtain the Newtonian balance consistent with that of core region 1. (iii) Converged numerical solutions have been obtained for small Weissenberg numbers at re-entrant corners (see Baaijens 1998; Xue*et al*. 1998; Oliveira & Pinho 1999; Alves*et al*. 2000). Further, even though in Lipscomb*et al*. (1987) they fail to capture the UCM stress singularity due to inadequate resolution, the numerical simulations are able to pick up the Newtonian dominated behaviour suggested by core region 1 for the small values of and 0.06. Moreover, behaviour associated with a second-order fluid is obtained, which is indicative of an approximation of the full balance in core region 2 by a leading-order Newtonian stress and could be expected to occur in the overlap region between core regions 1 and 2. Further comment is made on the results of Lipscomb*et al*. (1987) in §2*d*.↵This is further emphasized by considering the scalings that link the variables of core regions 2 and 3 directly, which arewhere the leading-order solution for and is

↵It is worth noting that a stretching solution of the formis not sufficiently general in order to match with the self-similar solution in both core regions 1 and 3 since for the constant

*n*we would require as and as . This remark also applies to more general separable forms regarding the radial function of the stream function.- Received July 10, 2005.
- Accepted May 2, 2006.

- © 2006 The Royal Society