A hyperbolic geometric flow for evolving films and foams

Abstract

Simulating the behavior of soap films and foams is a challenging task. A direct numerical simulation of films and foams via the Navier-Stokes equations is still computationally too expensive. We propose an alternative formulation inspired by geometric flow. Our model exploits the fact, according to Plateau's laws, that the steady state of a film is a union of constant mean curvature surfaces and minimal surfaces. Such surfaces are also well known as the steady state solutions of certain curvature flows. We show a link between the Navier-Stokes equations and a recent variant of mean curvature flow, called hyperbolic mean curvature flow , under the assumption of constant air pressure per enclosed region. Instead of using hyperbolic mean curvature flow as is, we propose to replace curvature by the gradient of the surface area functional. This formulation enables us to robustly handle non-manifold configurations; such junctions connecting multiple films are intractable with the traditional formulation using curvature. We also add explicit volume preservation to hyperbolic mean curvature flow, which in fact corresponds to the pressure term of the Navier-Stokes equations. Our method is simple, fast, robust, and consistent with Plateau's laws, which are all due to our reformulation of film dynamics as a geometric flow.