On the stability of laminar flows between plates

Abstract

Consider a two-dimensional laminar flow between two plates, so that $(x_1,x_2)\in \mathbb{R}\times [-1,1]$, given by $\mathbf{v}(x_1,x_2)=(U(x_2),0)$, where $U\in C^4([-1,1])$ satisfies $U^{\prime }\ne 0$ in $[-1,1]$. We prove that the flow is linearly stable in the large Reynolds number limit, in two different cases: $\bullet$ ${\sup }_{x\in [-1,1]}|U"(x)|+{\sup }_{x\in [-1,1]}|U"(x)|\ll {\min }_{x\in [-1,1]}|U^{\prime }(x)|$ (nearly Couette flows), $\bullet$ $U^{\prime \prime }\ne 0$ in $[-1,1]$. We assume either no-slip or fixed traction force conditions on the plates, and an arbitrary large (but much smaller than the Reynolds number) period in the $x_1$ direction.