Energy stability of plane Couette and Poiseuille flows: A conjecture

Abstract

We will study the nonlinear stability of plane Couette and Poiseuille flows with the Lyapunov second method by using the classical L2-energy. We will prove that the streamwise perturbations are L2-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.