Falling drop in an unbounded liquid reservoir: Steady-state solutions

TL;DR

This paper investigates the steady-state behavior of a liquid drop moving freely in an unbounded reservoir under gravity, establishing existence results under certain density conditions through linearization techniques.

Contribution

It provides the first rigorous proof of steady-state solutions for a freely moving liquid drop in an unbounded reservoir, using linearization and functional analysis methods.

Findings

Existence of steady-state solutions under specific density conditions

Use of linearization to analyze the governing equations

Functional analysis framework for the problem

Abstract

The equations governing the motion of a three-dimensional liquid drop moving freely in an unbounded liquid reservoir under the influence of a gravitational force are investigated. Provided the (constant) densities in the two liquids are sufficiently close, existence of a steady-state solution is shown. The proof is based on a suitable linearization of the equations. A setting of function spaces is introduced in which the corresponding linear operator acts as a homeomorphism.