# On the stability of laminar flows between plates

## TL;DR

This paper proves the linear stability of certain laminar flows between plates at high Reynolds numbers, under specific conditions on the flow profile and boundary conditions, extending understanding of flow stability in fluid dynamics.

## Contribution

It establishes linear stability results for laminar flows between plates in the high Reynolds number limit, covering nearly Couette flows and flows with non-zero second derivatives, under various boundary conditions.

## Key findings

- Flow is linearly stable for nearly Couette flows at high Reynolds numbers.
- Flow is linearly stable when the second derivative of the velocity profile is non-zero.
- Stability holds under no-slip or fixed traction boundary conditions.

## 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 $U^\prime\neq0$ 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), $\bullet$ $U^{\prime\prime}\neq0$ 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.