An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension

TL;DR

This paper extends Troesch's 1965 isoperimetric sloshing problem by incorporating surface tension effects and sinusoidal wave analysis, deriving explicit solutions for optimal container shapes and demonstrating increased sloshing frequencies with surface tension.

Contribution

It introduces a variational characterization of sloshing with surface tension and pinned contact lines, providing explicit solutions for optimal shallow container shapes under these conditions.

Findings

Maximal sloshing frequency increases with surface tension.

Optimal container shapes are non-convex for certain wave modes.

Explicit solutions show convergence to non-surface tension shapes as tension vanishes.

Abstract

In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume [doi:10.1002/cpa.3160180124]. Here, we extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber $\alpha\ge 0$ and spatial period $2\pi/\alpha$; two-dimensional sloshing corresponds to the case $\alpha = 0$. Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any $\alpha > 0$. In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given $\alpha\ge 0$ and shallow radially symmetric containers with $m$ azimuthal nodal lines, $m = 0, 1$. Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for $\alpha = 0$ and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension.