Comparison of the potential energy for different equilibrium configurations of symmetric and asymmetric floating drops

TL;DR

This paper introduces a numerical approach combining Newton's method and Chebyshev spectral collocation to analyze the potential energy of various equilibrium configurations of floating drops, revealing non-uniqueness in solutions.

Contribution

It develops a novel numerical framework for solving free boundary problems related to floating drops and explores the energy landscape across multiple parameters, including asymmetric configurations.

Findings

Potential energy varies with drop configuration and position.

Multiple solutions and energy minimizers exist for the same parameters.

Numerical method effectively captures complex equilibrium states.

Abstract

We provide a numerical method for computing solutions to a free boundary problem arising from the equilibrium state of a floating drop. This numerical method is based on a Newton's method for the underlying nonlinear boundary value problems, and at each iterative step a Chebyshev spectral collocation method is employed. The problems considered here are those that can be described by using generating curves, and include problems in $\mathbb{R}^2$ and $\mathbb{R}^3$. The resulting nine-dimensional space of physical parameters is explored, and examples are given that highlight the potential energy of centrally located drops, wall-bound drops, and asymmetrical configurations in $\mathbb{R}^2$. Non-uniqueness of solutions to the corresponding Euler-Lagrange equations is displayed, and also strong evidence of non-uniqueness of energy minimizers is given.