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We will study the nonlinear stability of plane Couette and Poiseuille flows with the Lyapunov second method by using the classical L 2 -energy. We will prove that the streamwise perturbations are L 2 -energy stable for any Reynolds number. This contradicts the results of Joseph (1966), Joseph and Carmi (1969) and Busse (1972), and allows us to prove, with a conjecture, that the critical nonlinear Reynolds numbers are obtained along two-dimensional perturbations, the spanwise perturbations, as Orr (1907) had supposed. This conclusion combined with some recent results by Falsaperla et al. (2019) on the stability with respect to tilted rolls, provides a possible solution to the "mismatch" between the critical values of linear stability, nonlinear monotonic energy stability and the experiments. [ABSTRACT FROM AUTHOR]

The stability of planar shear flows of an incompressible fluid is fundamental and still a debated topic. In this article we define a family of basic flows which include Couette flow and Poiseuille flow, and use analytical methods and a Chebyshev collocation method with many different approaches to investigate the monotonic behaviour of the energy along perturbations. We obtain in different ways an extension of known results for general shear flows, and we confirm the validity of some approaches that can be found in the literature. • Monotonical decrease of energy for a combination of Couette and Poiseuille flows. • Threshold computed solving a maximum problem and directly computing maximal slope. • Analytical expression of critical velocity fields for Couette flow. • Numerical approach using Chebishev collocation and spectral truncation. [ABSTRACT FROM AUTHOR]

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 + B 0 k. B 0 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. [ABSTRACT FROM AUTHOR]

Here we study the nonlinear stability of magnetohydrodynamics plane Couette and Hartmann shear flows. We prove that the streamwise perturbations are stable for any Reynolds number. This result is in a contradiction with the numerical solutions of the Euler–Lagrange equations for a maximum energy problem. We solve this contradiction with a conjecture. Then, we rigorous prove that the least stabilizing perturbations, in the energy norm, are the spanwise perturbations and give some critical Reynolds numbers for some selected Prandtl and Hartmann numbers. Similar results have been obtained by Falsaperla et al. (2021) for the classical plane Couette and Poiseuille fluid-dynamics flows. • We study stability/instability of Couette and Hartmann flows. • Streamwise perturbations are always stable. • Numerical solutions of a maximum (energy) problem show that streamwise perturbations are the least stabilizing perturbations. • To solve this contradiction we propose a conjecture: the maximum is achieved on particular admissible perturbations. • With this conjecture we prove that the least stabilizing nonlinear perturbations are the two-dimensional spanwise. This result implies a Squire theorem for nonlinear system. [ABSTRACT FROM AUTHOR]