Foams in contact with solid boundaries: equilibrium conditions and conformal invariance

TL;DR

This paper derives equilibrium conditions for 2D foams on curved solid surfaces, explores their conformal invariance properties, and relates foam configurations to conformal maps, with applications to confined foams and experimental comparisons.

Contribution

It generalizes foam equilibrium equations to curved surfaces and analyzes conformal invariance, providing new insights into foam behavior on complex geometries.

Findings

Equilibrium equations are derived for foams on curved surfaces.

Conformal invariance holds under specific conditions for foam vertices and edges.

Approximate relations between plate profiles and conformal maps are established.

Abstract

A liquid foam in contact with a solid surface forms a two-dimensional foam on the surface. We derive the equilibrium equations for this 2D foam when the solid surface is curved and smooth, generalising the standard case of flat Hele Shaw cells. The equilibrium conditions at the vertices in 2D, at the edges in 3D, are invariant by conformal transformations. Regarding the films, conformal invariance only holds with restrictions, which we explicit for 3D and flat 2D foams. Considering foams confined in thin interstices between two non parallel plates, normal incidence and Laplace's law lead to an approximate equation relating the plate profile to the conformal map. Solutions are given for the logarithm and power laws in the case of constant pressure. The paper concludes on a comparison with available experimental data.