Three dimensional stagnation point flow into a corner

Abstract

The principal features of the three dimensional laminar motion produced when a viscous incompressible fluid impinges on a corner, formed by two infinitely long planes meeting at an angle (π 2α), are discussed mainly for the almost-planar configuration, where the slight cranking of the planes promotes flow in the third direction. On the face of it, there seem to be two quite distinct flows possible when α becomes small. One is the known two dimensional stagnation-point motion with the stagnation line at a right angle to the line of intersection. The other is in effect a three dimensional sink-flow, with fluid approaching the stagnation point radially in the cross-flow plane, which is normal to the line of intersection, while accelerating away from it, parallel to the line of intersection. (This flow can also be considered as an axisymmetric stagnation point motion with the line of intersection as the axis of symmetry and all flow direction reversed.) The explanation of this apparent non-uniqueness is that the first major alteration in the characteristics of the viscous and inviscid steady flowfields occurs while α is still small, due essentially to the interactions between the breakdown of the linearization procedure and the emergence of transverse viscous forces close to the corner. Specifically, the critical value of α is 0(l/lnR e) where Re, a characteristic Reynolds number of the motion, is assumed to be large. In that regime, for a concave corner, the pattern of the flow develops non-linearly away from the planar form, for a = 0, toward the completely different kind of motion corresponding to the sink-flow phenomenon. The flow in the corner is derived numerically and exhibits a partial reversal in the direction of the cross-plane velocity when the corner angle is sufficiently increased. New exact solutions of the Navier-Stokes equations are also proposed for the sink-flows at arbitrary positive values of α , the solution as -α > 0 + being precisely that obtained as a In Re becomes large and positive. In contrast, for the convex corner the effect of increasing the inclination ( —α ) is to compress the boundary layer substantially, and the cross-plane flow is always outward.