Stability of the Plane Bingham–Poiseuille Flow in an Inclined Channel

Abstract

We study the stability of laminar Bingham–Poiseuille flows in a sheet of fluid (open channel) down an incline with constant slope angle β∈(0,π/2). This problem has geophysical applications to the evolution of landslides. In this article, we apply to this problem recent results of Falsaperla et al. for laminar Couette and Poiseuille flows of Newtonian fluids in inclined channels. The stability of the basic motion of the generalised Navier–Stokes system for a Bingham fluid in a horizontal channel against linear perturbations has been studied. In this article, we study the flows of a Bingham fluid when the channel is oblique and we prove a stabilizing effect of the Bingham parameter B. We also study the stability of the linear system with an energy method (Lyapunov functions) and prove that the streamwise perturbations are always stable, while the spanwise perturbations are energy-stable if the Reynolds number Re is less than the critical Reynolds number Rc obtained solving a generalised Orr equation of a maximum variational problem.