# Linear motion of multiple superposed viscous fluids

## TL;DR

This paper analyzes the linearized motion of multiple superposed viscous fluids, deriving equations for interface amplitudes, and applies them to study Rayleigh-Taylor instability and surface layer damping effects.

## Contribution

It presents a new set of equations for small-amplitude viscous fluid interface motion, including normal mode analysis, applicable to multi-layer fluid systems.

## Key findings

- Initial phase difference influences Rayleigh-Taylor instability development.
- A thin viscous surface layer damps interface motion.
- Equations can be numerically inverted for specific problems.

## Abstract

In this paper the small-amplitude motion of multiple superposed viscous fluids is studied as a linearized initial-value problem. The analysis results in a closed set of equations for the Laplace transformed amplitudes of the interfaces that can be inverted numerically. The derived equations also contain the general normal mode equations, which can be used to determine the asymptotic growth-rates of the systems directly. After derivation, the equations are used to study two different problems involving three fluid layer. The first problem is the effect of initial phase difference on the development of a Rayleigh-Taylor instability and the second is the damping effect of a thin, highly viscous, surface layer.