Hamiltonian Dynamic Equations for Fluid Films

Abstract

Two-dimensional models for hydrodynamic systems, such as soap films, have been studied for over two centuries. Yet there has not existed a fully nonlinear system of dynamic equations analogous to the classical Euler equations. We propose the following exact system for the dynamics of a fluid film Here δ/δ t is the invariant time derivative, ρ is the two-dimensional density of the film, C is the normal component of the velocity field, Vα are the tangential components, Bαβ is the curvature tensor, and α is the covariant surface derivative. The surface energy density e(ρ) is a generalization of the common surface tension and eρ is its derivative. The Laplace model corresponds to e(ρ) =σ/ρ, where σ is the surface tension density. The proper choice of e(ρ) in paramount in capturing particular effects displayed by fluid films.The proposed system is exact in the sense that neither velocities nor deviation from the equilibrium are assumed small. The system is derived in the classical Hamiltonian framework. The assumption that e is a function of ρ alone can be relaxed in practical physical and biological applications. This leads to more complicated systems, briefly discussed in the text. © 2010 by the Massachusetts Institute of Technology.