Asymptotic behavior of localized disturbance in a viscous fluid flow down an incline

Abstract

We analytically solve the problem of the evolution of small-amplitude waves in a uniform flow of a viscous fluid down an inclined plane. The flow is described in a hydraulic approximation. The flow is supposed to be convectively unstable, and the waves arise as a result of an instantaneous external point disturbance. The solution is presented as a Fourier integral to which the steepest descent method is applied twice. The asymptotics of the growing waves is found analytically as a function of two spatial coordinates and time. We show that the region of growing perturbations is a segment of a circle, that its linear dimensions grow linearly with time, and that it is defined by the characteristics of a system of Saint-Venant differential equations.