Effect of weak elasticity on Kelvin-Helmholtz instability

TL;DR

This paper analyzes how weak elasticity influences the Kelvin-Helmholtz instability in compressible elastic flows, showing it has a destabilizing effect and establishing conditions for ill-posedness.

Contribution

It provides a rigorous analysis demonstrating that weak elasticity destabilizes Kelvin-Helmholtz instability and identifies critical velocities for ill-posedness in elastic fluid flows.

Findings

Weak elasticity destabilizes Kelvin-Helmholtz instability.

Ill-posedness occurs within specific velocity bounds.

Results are valid for linear and nonlinear regimes.

Abstract

In this paper, we present an analysis of the Kelvin-Helmholtz instability in two-dimensional ideal compressible elastic flows, providing a rigorous confirmation that weak elasticity has a destabilizing effect on the Kelvin-Helmholtz instability. There are two critical velocities, $U_{\text{low}}$ and $U_{\text{upp}}$, where $U_{\text{low}}$ and $U_{\text{upp}}$ represent the lower and upper critical velocities, respectively. We demonstrate that when the magnitude of the rectilinear solutions satisfies $U_{\text{low}}+c\epsilon_{0}\le |\dot{v}^{+}_{1}| \le U_{\text{upp}}-c\epsilon_{0}$, the linear and nonlinear ill-posedness of the piecewise smooth solutions of the Kelvin-Helmholtz problem for two-dimensional ideal compressible elastic fluids is established uniformly, where $c$ is the sound speed and $\epsilon_{0}$ is some small enough positive constant.