Squire’s theorem for monotone energy stability of plane shear flows for Navier-Stokes-Voigt fluids with vertical throughflow

Abstract The Squire theorem holds for parallel shear flows governed by the linearized Navier-Stokes equations. In his paper Squire writes “For the study of the stability of flow between parallel walls it is sufficient to confine attention to disturbances of two-dimensional type”, Squire (Proc R Soc A 142:621, 1933). Conversely, for nonlinear Navier-Stokes system it is supposed that the Squire’s theorem does not hold in general (see Drazin and Reid Drazin in (Hydrodynamic Stability. Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 2004). Here we show that Squire’s theorem holds for the linear and nonlinear monotone energy stability of parallel shear flows, including Couette and Poiseuille flows between parallel walls, in the case of Navier-Stokes-Voigt fluids and in the presence of a throughflow. In particular, we prove that the longitudinal components (also called streamwise) of the velocity field are energy stabilizing for any Reynolds number and that the least energy-stabilizing perturbations are the transverse (also called spanwise) ones. The critical Reynolds numbers for nonlinear energy stability do not depend on the Voigt number or the throughflow dependent Reynolds number.