# Dynamics and stability of surface waves with bulk-soluble surfactants

## TL;DR

This paper analyzes the stability of surface waves influenced by surfactants that can transfer between the surface and bulk, demonstrating that small disturbances decay exponentially over time.

## Contribution

It introduces a model accounting for surfactant transfer between surface and bulk and proves the exponential decay of perturbations, establishing stability of equilibrium solutions.

## Key findings

- Small perturbations decay exponentially to equilibrium.
- The model captures surfactant transfer and its effect on fluid dynamics.
- Stability results are proven for the coupled surface-bulk surfactant system.

## Abstract

In this paper we study the dynamics of a layer of incompressible viscous fluid bounded below by a rigid boundary and above by a free boundary, in the presence of a uniform gravitational field. We assume that a mass of surfactant is present both at the free surface and in the bulk of fluid, and that conversion from one species to the other is possible. The surfactants couple to the fluid dynamics through the coefficient of surface tension, which depends on the the surface density of surfactants. Gradients in this concentration give rise to Marangoni stress on the free surface. In turn, the fluids advect the surfactants and distort their concentration through geometric distortions of the free surface. We model the surfactants in a way that allows absorption and desorption of surfactant between the surface and bulk. We prove that small perturbations of the equilibrium solutions give rise to global-in-time solutions that decay to equilibrium at an exponential rate. This establishes the asymptotic stability of the equilibrium solutions.