# A Variational Characterization of Fluid Sloshing with Surface Tension

## TL;DR

This paper develops a new variational approach to analyze fluid sloshing with surface tension effects, providing existence results, domain monotonicity, and perturbation formulas for eigenvalues in a coupled potential and surface displacement framework.

## Contribution

It introduces a novel variational formulation for the coupled fluid-surface problem with surface tension and establishes key spectral properties including eigenvalue monotonicity and perturbation analysis.

## Key findings

- Established existence of solutions using calculus of variations.
- Proved domain monotonicity for the fundamental eigenvalue.
- Derived first-order perturbation formula for eigenvalues in the zero surface tension limit.

## Abstract

We consider the sloshing problem for an incompressible, inviscid, irrotational fluid in an open container, including effects due to surface tension on the free surface. We restrict ourselves to a constant contact angle and seek time-harmonic solutions of the linearized problem, which describes the time-evolution of the fluid due to a small initial disturbance of the surface at rest. As opposed to the zero surface tension case, where the problem reduces to a partial differential equation for the velocity potential, we obtain a coupled system for the velocity potential and the free surface displacement. We derive a new variational formulation of the coupled problem and establish the existence of solutions using the direct method from the calculus of variations. We prove a domain monotonicity result for the fundamental sloshing eigenvalue. In the limit of zero surface tension, we recover the variational formulation of the mixed Steklov-Neumann eigenvalue problem and give the first-order perturbation formula for a simple eigenvalue.