Nonlinear monotone $H^1$ stability of plane Poiseuille and Couette flows of a Navier–Stokes–Voigt fluid of order zero

The nonlinear monotone $H^1$-energy stability of laminar flows in a layer between two parallel planes filled with a Navier–Stokes–Voigt fluid is studied. It is proved that the critical Reynolds numbers for monotone $H^1$-energy stability for the Couette and Poiseuille flows of the zero-order Navier–Stokes–Voigt fluid are the same as those found by Orr for Newtonian fluids. However, the exponential decay coefficient depends on the Kelvin–Voigt parameter $\Lambda$. Furthermore, a Squire theorem holds in the nonlinear case: the least stabilizing perturbations in $H^1$-energy are the two-dimensional spanwise perturbations.