Re-entrant corner flow of UCM fluids
by
J D Evans
Department of Mathematical Sciences, Univeristy of Bath, Bath, UK

Steady planar flow of the Upper Convected Maxwell (UCM) fluid is described for re-entrant corners with angles p/a where 1/2 ≤ a < 1. Local to the corner we consider a class of similarity solutions associated with the inviscid flow equations which arise from the dominance of the upper convective stress derivative in the constitutive equations. These solutions hold in an outer (core flow) region and give stress singularities of O(r^-2(1-a)) (with r the radial distance from the corner) and a stream function behaviour of O(r^na). Here n is a parameter defining distinct solutions within this similarity class. We match such solutions to inner regions at the walls (i.e. wall boundary layers), in which viscometric behaviour is retrieved and which determines n=3-a. The formulation is carried out using the natural stress basis and it is implicitly assumed that there are no regions of recirculation at the upstream wall i.e. we consider flow in the absence of a lip vortex. In this situation, a two parameter family of solutions is described for the local asymptotic behaviour of the flow and stress fields. Essential features of the analysis are the matching of the core region to the wall boundary layers, the solution (numerical and asymptotic) of the upstream and downstream boundary layers, together with the explicit determination of the essential singularity possessed by the boundary layer equations at the downstream wall.

Key words and phrases: UCM fluid, re-entrant corner, boundary layer, self-similar solutions, natural stress basis

Date received: January 26, 2008