Dynamic fluid film equations

A soap film can be formed by dipping a closed contour wire into a soapy solution as in the figure on the right.

Alternatively, a catenoid can be formed by dipping two rings in the soapy solution and subsequently separating them while maintaining the coaxial configuration.

Stationary fluid films form surfaces of minimal surface area, leading to the Plateau problem.

On the other hand, fluid films display rich dynamic properties.

They can undergo enormous deformations away from the equilibrium configuration.

Furthermore, they display several orders of magnitude variations in thickness from nanometers to millimeters.

Thus, a fluid film can simultaneously display nanoscale and macroscale phenomena.

Then the variable thickness of the film is captured by the two dimensional density

The dynamics of fluid films can be described by the following system of exact nonlinear Hamiltonian equations which, in that respect, are a complete analogue of Euler's inviscid equations of fluid dynamics.

In fact, these equations reduce to Euler's dynamic equations for flows in stationary Euclidean spaces.

The foregoing relies on the formalism of tensors, including the summation convention and the raising and lowering of tensor indices.

that spans a stationary closed contour boundary.

be the contravariant components of the tangential velocity projection.

Furthermore, let the internal energy density per unit mass function be

is the surface energy density results in Laplace's classical model for surface tension: where A is the total area of the soap film.

-derivative is the central operator, originally due to Jacques Hadamard, in The Calculus of Moving Surfaces.

The governing system above was originally formulated in reference 1.

) manifolds, the system becomes which is precisely classical Euler's equations of fluid dynamics.

If one disregards the tangential components of the velocity field, as frequently done in the study of thin fluid film, one arrives at the following simplified system with only two unknowns: the two dimensional density

Exact nonlinear equations for fluid films and proper adaptations of conservation theorems from classical hydrodynamics P. Grinfeld, J. Geom.