On Tracking Free Surfaces in Liquids

We consider a variational principle suitable for tracking boundaries of gas bubbles in liquids and its consequences. The principle is derived from a generalization of the Principle of Stationary Action applied to a Riemannian manifold of volume preserving flow maps. The dual variational principle for the markers tracking the free surfaces is induced by the Wasserstein-Kantorovich metric. This later variational principle provides means to compute the dynamics of free, implicit, surfaces without explicitly solving for the fields and quantities which define them. The presented theory suggests the following approximate evolution equation for the characteristic function of the gas, written in the Lagrangian frame of reference, ρG (∂ttχ(x, t)− λ∂tχ(x, t)) = div (χ(x, t) ((ρG − ρL)~g + α∇H(x, t))) , x ∈ S(∇χ), t > 0, where S(∇χ) represents the set of points on the gas-liquid interfaces, and H denotes the mean curvature of the interface. The constants α and λ denote the surface tension and Rayleigh’s friction dissipation constants, respectively. The density of the liquid and gas are denoted ρL and ρG. The gravitational force is denoted ~g. The regularized version of this equation is given, for some ǫ > 0, by ρG (∂ttsǫ(x, t)− λ∂tsǫ(x, t)) = div (