The instability of a membrane enclosed by two viscous fluids with a free surface

TL;DR

This paper investigates the stability of a membrane between two viscous fluids with a free surface, revealing novel instabilities caused by fluid-structure interactions and external shear, including membrane flutter and classical shear instabilities.

Contribution

It introduces new instability mechanisms in a viscous fluid-membrane system, including a novel instability under stable conditions and membrane flutter due to shear and wave interactions.

Findings

Discovery of a new instability caused by tangential forces and fluid-structure interaction.

Demonstration of membrane flutter driven by surface wave over-reflection or Tollmien--Schlichting waves.

Analysis of classical and novel instabilities in a viscous fluid-membrane configuration.

Abstract

This study examines the stability of a flexible material interface between two fluids of the same viscosity in interaction with a free surface. When the layers are motionless, we provide evidence for the onset of a novel instability by means of analytical and numerical solution of the associated boundary value problem in the region stable against Rayleigh--Taylor instability, i.e. when the acceleration due to gravity acts from the lighter to the heavier fluid. This destabilisation phenomenon is attributed to the non-conservative tangential forces acting at the interface and the fluid-structure interaction. Furthermore, we examine the scenario in which an external forcing mechanism induces a monotonic parallel shear flow within the upper layer. In addition to the long-established inflectional instability predicted in the inviscid limit, we demonstrate the existence of membrane flutter in the absence of density stratification. The latter is either due to an over-reflection process of surface gravity waves or to the growth of Tollmien--Schlichting waves, as outlined in the context of boundary-layer theory. This fluid-structure configuration represents a paradigmatic model for investigating the interplay between inflectional, radiation-induced and shear-induced instabilities. It also serves as a viscous counterpart to the classical Kelvin--Helmholtz instability when layers with distinct densities are assumed.