Steady nonlinear waves in diverging channel flow

An infinitely diverging channel with a line source of fluid at its vertex is a natural idealization of flow in a finite channel expansion. Motivated by numerical results obtained in an associated geometry (Tutty 1996), we show in this theoretical model that for certain channel semi-angles alpha and Reynolds numbers Re := Q/2nu (Q is the volume flux per unit length and nu the kinematic viscosity) a steady, spatially periodic, two-dimensional wave exists which appears spatially stable and hence plausibly realizable in the physical system. This spatial wave (or limit cycle) is born out of a heteroclinic bifurcation across the subcritical pitchfork arms which originate out of the well known Jeffery-Hamel bifurcation point at alpha = alpha(2)(Re). These waves have been found over the range 5 less than or equal to Re less than or equal to 5000 and, significantly, exist for semi-angles a beyond the point alpha(2) where Jeffery-Hamel theory has been shown to be mute. However, the limit of alpha --> 0 at finite Re is not reached and so these waves have no relevance to plane Poiseuille flow.