Stability of Hartmann shear flows in an open inclined channel

We study the stability of laminar flows in a sheet of fluid (open channel) down an incline with constant slope angle β∈(0,π/2) assuming that the fluid is electrically conducting and subjected to a magnetic field. The basic motion (the Hartmann shear flow) 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, and the magnetic field B(z)i+B0k. B0 is constant and k is the unit vector in the direction of z. U(z) and B(z) are hyperbolic functions of z: U(z) vanishes at the bottom of the channel and its derivative with respect to z vanishes at the top. By assuming that the boundaries are non-conducting (B(z) is zero on the boundaries), we study the local (linear) stability and instability, and we obtain critical Reynolds numbers for the onset of instability by solving a generalized Sommerfeld equation. We also study the nonlinear Lyapunov stability by solving the Orr equation for the associated maximum problem of the Reynolds–Orr energy equation. As in Falsaperla et al. (2019) we finally study the nonlinear stability of tilted rolls. The critical Reynolds numbers we obtain allow us to determine, for every inclination angle β, the critical velocity.