Fluid films, such as soap films, are commonly encountered in everyday experience. 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.
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.
In the study of the dynamics of free fluid films, such as soap films, it is common to model the film as two dimensional manifolds. Then the variable thickness of the film is captured by the two dimensional density .
where the -derivative is the central operator, originally due to Jacques Hadamard, in the The Calculus of Moving Surfaces. Note that, in compressible models, the combination is commonly identified with pressure . The governing system above was originally formulated in reference 1.
For the Laplace choice of surface tension the system becomes:
Note that on flat () stationary () 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 and the normal velocity :