-C ¢/~ ELSEVIER J. Non-Newtonian Fluid Mech., 68 (1997) 125-132 JNrludd H0a-~ i Short Communication High Weissenberg number boundary layers for the upper convected Maxwell fluid Michael Renardy Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA Received 26 March 1996; in revised form 22 April 1996 Abstract We formulate and analyze equations to describe elastic boundary layers at walls, which arise in the high Weissenberg number flow of an upper convected Maxwell fluid. © 1997 Elsevier Science B.V. All rights reserved. Keywords: Boundary layers; Upper convected Maxwell fluid I. Introduction Elastic boundary layers at high Weissenberg numbers have been observed in a number of flows, such as flow between eccentric cylinders [1] and flow past a sphere [2]. Recent analysis of the re-entrant corner singularity for the upper convected Maxwell fluid [3] has also revealed the existence of a boundary layer near each wall. The basic reason for the formation of such boundary layers is quite simple to understand. The convected derivatives in the constitutive relation vanish at the wall, forcing the stresses there to be viscometric. However, at high Weissenberg number, the convected derivative terms become important at a short distance from the wall, leading to the formation of a sharp boundary layer in the stresses. This feature is present already if one considers the stress integration in a given (e.g. newtonian) velocity field. In this paper, we shall derive a reduced set of equations to describe such boundary layers. Despite the thinness of the layer, there is no reduction in the dimension, and the equations in the boundary layer remain two dimensional. Therefore, an analytical solution is not generally feasible. However, it is possible to find a class of similarity solutions. The asymptotic solution for the re-entrant corner flow [3] turns out to be a special case of these. We also analyze the linearized problem for small perturbations of parallel shear flow. In this case, the boundary layer 0377-0257/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PH S0377-0257(96)01491-7