Tracking Free Boundaries in Fluids Using a Variational Principle

Abstract

We derive a variational principle suitable for tracking free boundaries in fluids. The variational principle is based on the Lagrangian formulation of the Navier–Stokes equations. 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 indicatrices identifying the free boundaries is based on the Wasserstein–Kantorovich metric.