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

Abstract

AbstractWe 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 $$L_2$$ 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 $$\mathrm{Re}_c={\text {Re}}^y$$ Re c = Re y . However, this contradicts a theorem which proves that streamwise perturbations are always stabilizing, $${\text {Re}}^y=+\infty $$ Re y = + ∞ . We solve this contradiction with a conjecture and prove that the critical nonlinear Reynolds numbers are obtained for two-dimensional perturbations, the spanwise perturbations, $$\mathrm{Re}_c={\text {Re}}^x$$ Re c = Re x , as Orr had supposed in the classic case of Couette flow between rigid planes.