Nonlinear stability results for plane Couette and Poiseuille flows

We prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less than Re_{Orr}(2π/(λsinθ))/sinθ when a perturbation is a tilted perturbation in the direction x^{'} which forms an angle θ∈(0,π/2] with the direction i of the basic motion and does not depend on x^{'}. Re_{Orr} is the critical Orr-Reynolds number for spanwise perturbations which is computed for wave number 2π/(λsinθ), with λ being any positive wavelength. By taking the minimum with respect to λ, we obtain the critical energy Reynolds number for a fixed inclination angle and any wavelength: for plane Couette flow, it is Re_{Orr}=44.3/sinθ, and for plane Poiseuille flow, it is Re_{Orr}=87.6/sinθ (in particular, for θ=π/2 we have the classical values Re_{Orr}=44.3 for plane Couette flow and Re_{Orr}=87.6 for plane Poiseuille flow). Here the nondimensional interval between the planes bounding the channel is [-1,1]. In particular, these results improve those obtained by Joseph, who found for streamwise perturbations a critical nonlinear value of 20.65 in the plane Couette case, and those obtained by Joseph and Carmi who found the value 49.55 for plane Poiseuille flow for streamwise perturbations. If we fix some wavelengths from the experimental data and the numerical simulations, the critical Reynolds numbers that we obtain are in a very good agreement both with the the experiments and the numerical simulation. These results partially solve the Couette-Sommerfeld paradox.