Nonlinear systems | Manifolds | Differential geometry
Dynamic fluid film equations
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. 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 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 . 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. (Wikipedia).
