## Abstract

We discuss here a third type of boundary-layer structure that arises in steady planar flows at re-entrant corners for the upper convected Maxwell fluid. This structure extends the class of similarity solutions that are associated with the inviscid flow equations and hold in an outer core region local to the corner. In previous work (Renardy 1995 *J. Non-Newtonian Fluid Mech.* **58**, 83–39; Evans 2005 *Proc. R. Soc. A* **461**, 117–142), single and double layer structures were used to the match outer core similarity solutions to wall boundary layers in which viscometric behaviour is obtained. Here a second double layer structure is discussed, which completes the range of validity for the core similarity solutions. This structure is fundamentally different to the single layer structure in that it only admits reverse flow solutions at the upstream wall, a situation of practical relevance to describe the initial formation of lip vortices in contraction flows.

## 1. Introduction

The asymptotic analysis of steady incompressible flows of upper convected Maxwell (UCM) fluids at re-entrant corners is continued here with the construction of solutions that represent the initial stages in the formation of lip vortices. The dimensionless governing equations are the momentum and continuity equations(1.1)together with the constitutive equations(1.2)where * v* is the velocity,

*p*is the pressure,

*is the extra-stress tensor and*

**T***is the rate of strain tensor. The Weissenberg number has been set to unity without loss of generality and the Reynolds number*

**D***Re*is assumed order one. Two-dimensional flows are considered within the sector 0<

*r*<∞, 0≤

*θ*≤

*π*/

*α*, where (

*r*,

*θ*) are polar coordinates centred at the corner. The upstream wall is given by

*θ*=0 while

*θ*=

*π*/

*α*represents the downstream wall. For re-entrant corners we are interested in the range 1/2≤

*α*<1. A more detailed explanation of the terminology, notation and problem background can be found in Evans (2005).

In Hinch (1993) and Evans (2005), it was shown that the core flow balance of the constitutive equations(1.3)possessed the self-similar solutions(1.4)where *ψ* is the stream function, *g*(*ψ*)=*c*_{1}*ψ*^{(2/n−2)} and *c*_{0}, *c*_{1} are constants. Here *n* is a free parameter, arising from the assumption of a power law variation for the stress across streamlines. It is convenient to introduce the artificial small parameter *ϵ*>0 by the scaling *r*=*ϵR*, in order to obtain order of magnitude estimates for comparison of the terms arising in equations (1.1) and (1.2) for the solution (1.4). In the region *R*=*O*(1) away from the boundaries we have(1.5)The upper convected stress derivative dominates in the constitutive equations at leading order provided 1<*n*<2/*α*. Here *α* is assumed fixed, taking values in the range 1/2≤*α*<1. When *n*=2/*α*, the upper convected stress derivative is balanced by the relaxation terms and the solution (1.4) no longer applies. The inertia terms remain subdominant as long as *n*>1. The solution (1.4) is thus valid for the range 1<*n*<2/*α*, with the stress scaling being fully determined by the angle of the corner and independent of *n*. This outer solution breaks down as the walls are approached, leading to the consideration of wall boundary layers. These have been examined in Evans (in press) for 1<*n*≤3−*α*. There, solutions could be constructed for *n*=3−*α* using a single layer structure, while a double layer structure was considered for the range 1<*n*<3−*α*. (Although the full relevance of the latter structure has yet to be explored for the re-entrant corner geometry.) We now complete the analysis by considering the range 3−*α*<*n*<2/*α*, in which we will find a second double layer structure occurs. These results are summarized in figure 1.

## 2. The boundary-layer structure for 3−*α*<*n*<2/*α*

The outer solution (1.4) does not give viscometric behaviour at the wall and we now consider regions in which the terms neglected in equation (1.2) are recovered at leading order. The upstream wall is considered without loss of generality and Cartesian axes are taken with the *x*-axis along the wall *θ*=0 and the *y*-axis along the ray *θ*=*π*/2.

The inner variables are given by the scalings(2.1)where . These scalings arise by balancing the relaxation and upper convected stress derivative terms in the constitutive relations. The component form of (1.1) and (1.2) gives for the inner region *X*=*O*(1), *Y*=*O*(1) the equations(2.2)(2.3)and for the constitutive relations(2.4)(2.5)(2.6)Matching to the outer solution (1.4) gives the conditions(2.7)where *C*_{0}=*c*_{0}*α*^{n}, . The solid boundary and no slip conditions at the wall require(2.8)

### (a) The inner region

The leading order boundary-layer equations in *X*=*O*(1), *Y*=*O*(1) are(2.9)(2.10)(2.11)(2.12)since for the range of interest 3−*α*<*n*<2/*α* with 1/2≤*α*<1. The system (2.9)–(2.12) together with equation (2.7) admits the similarity solution(2.13)whereand gives(2.14)(2.15)(2.16)(2.17)(2.18)with *p*_{0}=(1/2)*C*_{1} and ′ denoting d/d*ξ*. These equations have the exact solution(2.19)which corresponds to the exact solution(2.20)for equations (2.9)–(2.12) with (2.7), on using (2.13). To motivate such a solution, we make the following two remarks.

*The equations* *(2.10)–(2.12)* *possesses a general limiting behaviour**in separable form as the wall is approached*. *Here*, *the coefficients* *may be expressed in terms of the as yet undetermined function* *, namely*(2.21)(2.22)(2.23)where(2.24)′ denotes d/d*X* and , , are arbitrary constants of integration. The momentum equation (2.9) then givesSince we must have holding uniformly within this region, this last equation suggests the form , where equating powers and coefficients of *X* gives, respectively,(2.25)Consequently, equations (2.21)–(2.23) becomeThese expressions can recover the far-field behaviour (2.7) upon the choiceswhere equation (2.25) now gives the consistent value already noted above after equation (2.18). It is worth remarking that this choice for the coefficients satisfies the relationship , which is equivalent to the condition stated in terms of the natural stress basis in Renardy (1997) for * T* to be a rank one tensor.

*The equations* *(2.10)–(2.12)* *are of the form**As a generalization of the stretching solution* (1.4) to (1.3) *we may seek solutions in the form T*=

*h*

**vv**^{T}

*, where h now satisfies h*+

*⋅∇*

**v***h*=0.

*The general solution is h*=

*g*(

*ψ*)e

^{−s}

*for an arbitrary function g*(

*ψ*)

*, and*∂/∂

*s*≡

*⋅∇*

**v***denotes differentiation along streamlines*.

*However, such solutions are expected to eventually decay to zero as s increases, assuming that integration is taken in the direction of flow*.

*Thus we may anticipate that such solutions are not of relevance here, at least for transmitting stress information into or out of this region*.

These two remarks indicate that it is reasonable to seek solutions in which the extra-stresses maintain the same far-field form throughout this region. Assuming that the extra-stresses are given by equation (2.7) (or (2.18) for the similarity solution), results in a linear problem to determine the stream function whose solution is a superposition of the far-field and wall behaviours. Thus this region can be thought of as an extension of the core in terms of the stresses, but within which the stream function is modified.

The velocity component parallel to the wall is given by ∂*Ψ*/∂*Y*=*Xf*′(*ξ*), with the sign of *f*′ determining if flow is toward (*f*′<0) or away from (*f*′>0) the corner. At the top of the inner region we have *f*′=*nC*_{0}*ξ*^{n−1} as *ξ*→∞, while at the bottom *f*′∼1/2(2−*α*)>0 as *ξ*→0. Thus, in the downstream case where *C*_{0}>0 in the core then *f*′ remains single signed throughout the inner region with flow being uniformly away from the corner. However, in the upstream case, we have *C*_{0}<0 from the core and thus *f*′ changes sign within this region with the implication of reverse flow. In both the upstream and downstream cases, the values of *C*_{0}, *C*_{1} and *n* need to be provided by the outer core flow. We note also that since *C*_{1}=2*p*_{0}, then (1−*α*)*C*_{1} gives the coefficient of the leading order term for the pressure gradient . In §2*b*, *C*_{1} (or equivalently *p*_{0}) will be seen to be related to the coefficient of the wall shear stress.

### (b) The inner inner region

In order to recover viscometric behaviour at the wall, we consider another region in which the rate-of-strain terms are retained at leading order. This inner inner region is given by the scalings(2.26)where . The leading order equations in *X*=*O*(1), are now(2.27)(2.28)(2.29)(2.30)with as given in equation (2.13). At the wall , the solid boundary and no slip conditions can now be satisfied and matching to the inner solution (2.19) gives the conditions(2.31)The equations (2.27)–(2.31) possess the similarity solution(2.32)where(2.33)(2.34)(2.35)(2.36)(2.37)(2.38)and ′ now denotes . As , equations (2.33)–(2.36) have the regular behaviour(2.39)(2.40)(2.41)(2.42)representing viscometric behaviour with(2.43)Here, and are two free constants. The constant *a* may be interpreted as relating to the coefficient of wall shear stress, while *b* is linked to both *a* and the coefficient of the pressure gradient by equation (2.43). We note that (2.33)–(2.36) are precisely the Eqns (3.17)–(3.20) given in Evans (2005) for the similarity solution of the inner region in the viscoelastic balance case *n*=3−*α*. Consequently, the regular wall behaviour (2.39)–(2.42) is the same with the higher order terms being recorded in Evans (2005). The distinguishing feature here then is the imposition of the far-field behaviour (2.38). It is also noted that the thickness of the inner inner region is the same as that of the single boundary layer in the *n*=3−*α* case.

#### (i) Numerical solution for the inner inner region

In appendix A, an eigenmode analysis is performed for the far-field asymptotic behaviour (2.38) in order to determine its degrees of freedom for the system (2.33)–(2.36). A similar analysis for the wall behaviour (2.39)–(2.42) has already been performed in Evans (2005). This analysis of the far-field behaviour implies that specifying *C*_{1} and hence *p*_{0}=(1/2)*C*_{1} imposes three conditions on the fifth order system (2.33)–(2.36). Using the eigenmode analysis from Evans (2005), the wall behaviour (2.39)–(2.42) imposes two boundary conditions when *a*>0 is left unspecified. The constant *b* is not freely specifiable, but rather is related to both *a* and *p*_{0} through equation (2.43). Thus a well specified boundary value problem can be posed for the case *a*>0, provided only one of the constants *C*_{1}>0 and *a*>0 are specified, the other then being determined as part of the solution.

It is noted that the case *a*>0 corresponds to flow away from the corner and this will be relevant to both upstream and downstream situations. The typical flow structure is shown later in figure 3*b*. In the upstream boundary layer, a region of reverse flow occurs within the inner region. This situation is of much practical relevance as it describes the initial formation of a lip vortex in contraction flows. Such regimes have been observed experimentally for Boger fluids and are recorded in Boger & Binnington (1990) as well as being shown in figs. 3.5 and 3.6 in Boger & Walters (1993).

In contrast, the case *a*<0 gives a significantly over specified problem. Using the analysis from Evans (2005), the wall behaviour (2.39)–(2.42) now imposes four boundary conditions when *a*<0 is left unspecified. When combined with the three conditions of the far-field behaviour, this provides seven conditions on a fifth order system. Thus a solution appears unlikely, at least in principle. Physically, the case *a*<0 corresponds to reverse flow occurring within the inner inner region. We note that in particular this would give two circulating regions, one in each of the regions *Y*=*O*(1) and of the boundary layer at the upstream wall.

The case *a*=0 is degenerate and gives zero extra-stresses on the wall (i.e. ). In any non-trivial solution we expect *p*_{0}=(1/2)*C*_{1}>0 and thus equation (2.43) implies that *b*<0 corresponding to flow near to the wall being towards the corner. Such situations can be set up as initial value problems, but are not expected to possess solutions that attain the required far-field behaviour (2.38) (since this asymptotic behaviour has already been noted to be non-generic, i.e. does not contain the maximum number of degrees of freedom). We thus expect only the trivial solution *b*=*C*_{1}=0 in this case, which agrees with the numerical results presented later in figure 2*b* in the limit *a*→0^{+}.

Proceeding with the case *a*>0 and as in Evans (2005), it is convenient to make the change of variables(2.44)This gives the Eqs. (3.41)–(3.44) in Evans (2005), the wall behaviour(2.45)together with the matching conditions(2.46)For numerical implementation, it is convenient to use dependent variables that are *O*(1). As such, we use *F* with *S*_{ij} for *t*≤0 and the rescaled variables *S*_{ij}e^{−2t} for *t*>0. Explicitly,(2.47)Figure 2 illustrates the numerical solution for the case of a 270° corner (*α*=2/3). The finite difference relaxation scheme detailed in Evans (2005) was used with a 401 point uniform mesh over the *t* interval [*t*_{0},5] with *t*_{0}=−4 and a convergence tolerance of 10^{−5}. The scheme used the end point conditions of specifying the asymptotic behaviour (2.46) for *F*, *S*_{11}, *S*_{22} at *t*=5 and *S*_{11}=8*F*^{2} at *t*=*t*_{0}. A plot of *F* and the rescaled variables *S*_{ij} is shown in figure 2*a* for the particular case *C*_{1}=8. We use 2*F*e^{−t} at *t*=*t*_{0} to give estimates for *a* which are plotted in figure 2*b* for a range of *C*_{1}. Numerical values from the scheme at *t*=*t*_{0} are recorded in table 1 for selected *C*_{1}, which illustrates convergence to the wall behaviour (2.45). For comparison, we also record the values using a uniform mesh with *t*_{0}=−5.

## 3. Discussion

Figure 3 summarizes the possible flow structures for UCM fluids at re-entrant corners that have been described here and in Evans (2005). In all situations, the upper convected derivative of stress is assumed to dominate in a core region away from the walls. This gives rise to a class of self-similar solutions, conveniently parametrized by *n* which is related to the stress variation across streamlines. For a given re-entrant corner with *α* fixed, we can distinguish the three cases 1<*n*<3−*α*, *n*=3−*α* and 3−*α*<*n*<2/*α*. The case *n*=3−*α* is the unique value that gives a single layer wall structure. This single layer structure, shown in figure 3*a*, is the important case for which fluid originating at the upstream wall is able to flow fully around the corner. The flow local to the corner fully determines *n* as well as the asymptotic structure. In Evans (2005), the case 1<*n*<3−*α* was noted, with similarity solutions identified for the boundary-layer equations in the case *n*=3−(3/2)*α*, the other values *n* in this range having yet to be fully explored. However, this structure remains to be analysed and at this stage we reserve comment on both the types and applications of solutions that it can admit. In contrast, the case 3−*α*<*n*<2/*α* gives the double layer structure of figure 3*b*, which is distinguished by its reverse flow solution within the upstream boundary layer. This situation corresponds practically to the initial formation of a lip vortex. The boundary layer is now significantly thicker than that of the single layer case *n*=3−*α* and local to the corner, the fluid in the outer core flow no longer originates from the near upstream wall region but a main flow. The parameter *n* influences the boundary-layer thickness and is now determined by the core flow, i.e. global information is now required to determine it.

In principle, we note that the single layer structure for *n*=3−*α* may be capable of supporting reverse flows in the upstream boundary layer. The eigenmode analysis in Evans (2005) suggests that an appropriately specified two-point boundary value problem can be set up in the upstream context, with flow away from the corner at the wall and towards the corner at the edge of the boundary layer. However, satisfactory numerical solutions have yet to be obtained. We should remark that solutions involving an upstream lip vortex were given consideration for the Oldroyd-B fluid in Rallison & Hinch (2004). These authors conjectured a so-called elastic Kutta condition to determine the angle of separation for the upstream lip vortex, and thus implicitly assume that the lip vortex is not confined to a thin region at the upstream wall. As such it is complementary to the situations under discussion here.

These structures conveniently describe the range of *n* for which the dominance of the upper convective stress derivative holds in the core. As *n*→2/*α*^{−}, it is clear that the boundary layer of figure 3*b* breaks down as it grows in thickness, it no longer remaining thin in an asymptotic sense. In fact, at *n*=2/*α* it is noted from equation (1.5) that the relaxation terms now balance the upper convected stress derivative in the core. In the alternative limit *n*→(3−*α*)^{+}, we note that *δ*→2−*α* and *θ*→1. The inner region becomes squeezed between the outer region and the inner inner region which recovers the single boundary-layer structure.

As a final comment, we remark that future work should extend the analysis presented here to the situation of a fully developed upstream lip vortex. Assuming that the separating streamline from the corner makes an angle *π*/*α*′ with the downstream wall, then the two cases *α*′=1 and 1/2≤*α*<*α*′<1 can be distinguished. The former appears likely for maximum reduction of the corner singularity. However, if the former case is permissible, then for (*π*/*α*−*π*/*α*′)<*θ*<*π*/*α* we have an effective re-entrant corner at leading order. As the core flow no longer originates from the near upstream wall region, but rather from a main flow region, it should be capable of giving any *n* in the admissible range 1<*n*<2/*α*′. As such the three boundary-layer structures of figure 1 should then be attainable at the downstream wall. However, it remains to be seen if the UCM equations are capable of supporting such fully developed structures.

## Appendix A Eigenmode analysis

For the far-field behaviour (2.38), we consider a perturbation in the form(A1)Neglecting the forcing terms, the linearized equations for the eigenmodes are thenafter keeping only the leading order terms. These equations have the leading asymptotic behaviours(A2)withOnly the second and third modes are consistent with the behaviour (2.38), implying that this asymptotic behaviour imposes three conditions on the system (2.33)–(2.36). We note that there is no mode corresponding to small changes in *C*_{1}, which means that *C*_{1} is not a free constant. This is consistent with the requirement *C*_{1}=2*p*_{0}, since we have implicitly assumed that *p*_{0} is known and fixed for the system (2.33)–(2.36) under the perturbation (A 1).

## Footnotes

- Received September 22, 2004.
- Accepted April 26, 2005.

- © 2005 The Royal Society