Stability of plane shear flows in a layer with rigid and stress-free boundary conditions

We study the stability of shear flows of an incompressible fluid contained in a horizontal layer. We consider rigid-rigid, rigid-stress-free and stress-free-stress-free boundary conditions. We study (and recall some known results) linear stability/instability of the basic Couette, Poiseuille and a laminar parabolic flow with the spectral analysis by using the Chebyshev collocation method. We then use an L-2-energy with Lyapunov second method to obtain nonlinear critical Reynolds numbers, by solving a maximum problem arising from the Reynolds energy equation. We obtain this maximum (which gives the minimum Reynolds number) for streamwise perturbations Re-c = Re-y. However, this contradicts a theorem which proves that streamwise perturbations are always stabilizing, Re-y = +infinity. We solve this contradiction with a conjecture and prove that the critical nonlinear Reynolds numbers are obtained for two-dimensional perturbations, the spanwise perturbations, Re-c = Re-x, as Orr had supposed in the classic case of Couette flow between rigid planes.