Stability of laminar flows in an inclined open channel

We study the stability of laminar flows in a sheet of fluid (open channel) down an incline with constant slope angle β. The basic motion is the velocity field U(z) i, where z is the coordinate of the axis orthogonal to the channel, and i is the unit vector in the direction of the flow. U(z) is a parabolic function which vanishes at the bottom of the channel and whose derivative with respect to z vanishes at the top. We study the linear stability, and prove that the basic motion is linearly stable for any Reynolds number. We also study the nonlinear Lyapunov stability by solving the Orr equation for the associated maximum problem. As in Falsaperla et al. (Phys Rev E 100(1):013113, 2019. https://doi.org/10.1103/PhysRevE.100.013113) we finally study the nonlinear stability of tilted rolls. This work is a preliminary investigation to model debris flows down an incline (Introduction to the physics of landslides. Lecture notes on the dynamics of mass wasting. Springer, Dordrecht, 2011. https://doi.org/10.1007/978-94-007-1122-8).