Sloshing of a two-layer fluid in a vertical cylinder of constant depth

TL;DR

This paper analyzes the eigenvalues and eigenfunctions of sloshing in a two-layer fluid within a vertical cylinder, revealing two families of eigenfrequencies and their dependence on physical parameters through theoretical and numerical methods.

Contribution

It provides explicit formulas for two families of sloshing eigenfrequencies in a two-layer fluid, extending understanding beyond homogeneous fluids and including numerical illustrations.

Findings

Two families of eigenfrequencies identified with explicit formulas.

Eigenfrequencies depend on interface depth and density ratio.

Numerical examples illustrate eigenmode behaviors in circular cylinders.

Abstract

Sloshing eigenvalues and eigenfunctions are studied for vertical cylinders of constant, finite depth occupied by a two-layer fluid. Two families of eigenfrequencies are obtained in the form expressing them explicitly via the eigenvalues of the Neumann Laplacian in the two-dimensional domain\, -- \,cylinder's cross-section. Eigenfrequencies belonging to one of the families behave similar to those that describe sloshing in a homogeneous fluid, whereas the other family includes a large number of sufficiently small frequencies provided the ratio of densities is close to unity. Various properties of eigenfrequencies are investigated for cylinders of arbitrary cross-section; they include the dependence on the interface depth and the ratio of densities, the asymptotics of the eigenvalue counting function. The behaviour of eigenvalues and the corresponding eigenmodes is illustrated by numerical examples for circular cylinders without and with a radial baffle.