Stability of equilibrium shapes in some free boundary problems involving fluids

TL;DR

This paper analyzes the stability of equilibrium shapes in free boundary problems involving two-phase viscous fluids with surface tension, across different physical scenarios, showing solutions tend to equilibrium states without singularities.

Contribution

It characterizes equilibrium states and their stability for various fluid flow cases, linking them to critical points of physical functionals under conservation constraints.

Findings

Equilibrium states correspond to critical points of physical functionals.

Solutions exist globally and converge to equilibrium states.

Stability depends on the nature of the functional and constraints.

Abstract

In this paper the motion of two-phase, incompressible, viscous fluids with surface tension is investigated. Three cases are considered: (1) the case of heat-conducting fluids, (2) the case of isothermal fluids, and (3) the case of Stokes flows. In all three situations, the equilibrium states in the absence of outer forces are characterized and their stability properties are analyzed. It is shown that the equilibrium states correspond to the critical points of a natural physical or geometric functional (entropy, available energy, surface area) constrained by the pertinent conserved quantities (total energy, phase volumes). Moreover, it is shown that solutions which do not develop singularities exist globally and converge to an equilibrium state.