Nonlinear energy stability of magnetohydrodynamics Couette and Hartmann shear flows: A contradiction and a conjecture. (January 2022) - Falsaperla, Paolo; Mulone, Giuseppe; Perrone, Carla
Abstract: 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. Highlights: 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.