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.

Stability of a Regularized Casson Flow down an Incline: Comparison with the Bingham Case

“…Therefore, our results are unexpected, highlighting that, although in a regularized formulation, models belonging to the same class of viscoplastic fluids can have stability characteristics completely different. Although we are not aware of any studies on the stability of the Casson fluid flowing down an incline, we suppose that the results of [12] can also be extended to this case. Therefore, we show that (as for the regularized Bingham fluid [10]) the regularized Casson can have stability properties that are different from the classic Casson flow.…”