On the dual solutions assoclated with boundary-layer equations in a corner

Abstract

The dual solutions of two coupled third degree non-linear ordinary differential equations associated with the incompressible viscous laminar flow along a corner are considered. It is shown (through the numerical solution) that dual solutions occur in the interval βbβββ1.1211 for the Falkner-Skan parameter β with the bifurcation taking place at the regular turning point βb. In the neighbourhood of the latter it is also shown that in such a case it is appropriate to expand the solution in powers of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacIcacqaHYoGycqGHsislcqaHYoGydaWgaaWcbaGaamOyaaqa% baGccaGGPaWaaWbaaSqabeaacaGG9caaaaaa!4442!\[(\beta - \beta _b )^\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \] with the dual solutions branching out from the single solution at βb. Then, on considering a simple transient problem (which provides an exact solution of the Navier-Stokes equations when β=1.0) it is found that the branch having the greatest value of the wall shear stress (for a given β) is stable while the other is unstable, the bifurcation point being the point of exchange of stability.