The flow of an Oldroyd fluid past a reentrant corner: the downstream boundary layer

Abstract

We reconsider the steady planar flow of an Oldroyd-B fluid within a small distance r of a sharp reentrant corner of angle π/ p, 1/2≤ p<1. Previous theoretical studies of this problem have been unable to resolve a boundary layer near the downstream wall. For a range of angles π/ p, including the benchmark problem p=2/3, and for several non-zero solvent viscosities, we have solved the equations for the downstream boundary layer numerically and additionally found asymptotic results for a frozen stress limit and for high solvent viscosities. For each angle and viscosity we find a one-parameter family of attached ‘potential’ flows, for each of which the stress is proportional to r 2 p−2 and the velocity proportional to r p(3− p)−1 as proposed by Hinch [J. Non-Newtonian Fluid Mech. 50 (1993) 161]. The limiting member of the family has zero shear stress on the upstream wall. The stress is that of a purely elastic neo-Hookean solid in the interior of the flow, with boundary layers on both the upstream and downstream walls having the similarity structure proposed by Renardy [J. Non-Newtonian Fluid Mech. 58 (1995) 83]. The behaviour of these flows is discussed. We propose a form of elastic Kutta condition to determine the separation angle for flows having an upstream lip vortex.