Instability and spectrum of the linearized two-phase fluids interface problem at shear flows

TL;DR

This paper analyzes the spectral stability of linearized two-phase fluid interfaces at shear flows, extending classical theorems and identifying conditions for instability, with applications to ocean-air systems.

Contribution

It extends Howard's Semicircle Theorem to two-phase flows and characterizes eigenvalue distributions and instability conditions for shear flow interfaces.

Findings

Exactly two eigenvalues per wave number under certain conditions

Spectral instability occurs at specific boundary values of shear flow velocity

Complete eigenvalue distribution for ocean-air interface flows

Abstract

This paper is concerned with the 2-dim two-phase interface Euler equation linearized at a pair of monotone shear flows in both fluids. We extend the Howard's Semicircle Theorem and study the eigenvalue distribution of the linearized Euler system. Under certain conditions, there are exactly two eigenvalues for each fixed wave number $k\in \mathbb{R}$ in the whole complex plane. We provide sufficient conditions for spectral instability arising from some boundary values of the shear flow velocity. A typical mode is the ocean-air system in which the density ratio of the fluids is sufficiently small. We give a complete picture of eigenvalue distribution for a certain class of shear flows in the ocean-air system.