On perturbations of Jeffery-Hamel flow

Abstract

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α2, which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkel's (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α2.