# Simple model of thermal shape fluctuations of liquid interface

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Couchyam
Consider a container filled with two essentially incompressible liquids with densities ##\rho > \rho'## and (respective) volumes ##V##, ##V'##, rotated by a centrifuge in some orbit-based space lab to maintain a roughly constant (co-moving) simulated "gravitational field" g. Let's suppose that the interface between the two liquids has surface tension ##\sigma## (understood as a free energy per unit area.)

In general, the shape of the interface between the two fluids is some terribly complicated level set, which usually requires restricting attention to (infinitesimal) local coordinate patches and imposing a linearized or leading-nonlinear approximation to obtain meaningful analytical results (modeling the surface as a sort of height map.) However, this perspective ignores topologically nontrivial configurations of the fluids, which may include bubbles of various sizes and shapes.

It turns out that the effect of bubbles can be understood to some extent by discretizing the liquids and applying a variant of the lattice-gas approximation, but exact analytical results are still elusive in this model, or at least hard to obtain. Instead, let's assume for simplicity that the configuration space consists of a (more or less perfectly flat) plane separating two rectangular regions perforated by spherical bubbles of one liquid in the other of various (typically microscopic or smaller) sizes. What other information if any might you need to determine the total free energy of the interface between the two liquids in this model, or the dependence of the free energy on the surface tension, temperature, volume ratio, etc.?

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