Monotone energy stability for Poiseuille flow in a porous medium

Abstract

We study the monotone energy stability of ``Poiseuille flow" in a plane-parallel channel with a saturated porous medium modeled by the Brinkman equation, on the basis of an analogy with a magneto-hydrodynamic problem (Hartmann flow) (cf. \cite{Hill.Straughan.2010}, \cite{Nield.2003}). We prove that the least stabilizing perturbations, in the energy norm, are the two-dimensional spanwise perturbations. This result implies a Squire theorem for monotone nonlinear energy stability. Moreover, for Reynolds numbers less than the critical Reynolds number $R_E $ there can be no transient energy growth.