Two-dimensional sloshing: domains with interior `high spots'

TL;DR

This paper investigates the existence and construction of two-dimensional fluid domains with interior high spots, where the free surface reaches extremal elevations, using a semi-inverse method.

Contribution

It introduces a semi-inverse procedure to rigorously construct domains with interior high spots in the two-dimensional sloshing problem, including domains with multiple high spots.

Findings

Existence of interior high spots is rigorously proved for certain domains.

Constructed domains often have multiple interior high spots.

All constructed domains are bulbous on at least one side.

Abstract

Considering the two-dimensional sloshing problem, our main focus is to construct domains with interior high spots; that is, points, where the free surface elevation for the fundamental eigenmode attains its critical values. The so-called semi-inverse procedure is applied for this purpose. The existence of high spots is proved rigorously for some domains. Many of the constructed domains have multiple interior high spots and all of them are bulbous at least on one side.