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.
Highlights
The stability of laminar flows in an inclined channel is important in many geophysical and industrial applications
We study the linear stability of laminar Bingham–Poiseuille flows for an inclined sheet of fluid with constant slope angle β
The basic flow is parabolic in the shear region and is constant in the plug region
Summary
The stability of laminar flows in an inclined channel is important in many geophysical and industrial applications. They solve the eigenvalue problem with the appropriate boundary conditions using a Chebyshev collocation method, and investigate the dependence of the Reynods number on the Bingham parameter They obtain the linear stability of the basic motion and prove that the Bingham number has a stabilizing effect. We observe that Allouche et al [17] focussed on Newtonian and generalised Newtonian (in particular, Carreau) fluid film flows down an incline and compared the thresholds of oblique waves to those of two-dimensional waves for a given slope in order to reach the dominant instability in a given flow configuration In particular they consider the shear-thinning and the shear-thickening cases and performed a temporal linear stability study on the problem.
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