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 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
\rho
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.
Consider a thin fluid film
S
C
V\alpha
\nabla\alpha
B\alpha
\alpha | |
B | |
\beta |
\alpha | |
B | |
\alpha |
e\left(\rho\right)
E
E=\intS\rhoe\left(\rho\right)dS.
This choice of
e\left(\rho\right)
e\left(\rho\right)=
\sigma | |
\rho |
where
\sigma
E=\sigmaA
where A is the total area of the soap film.
The governing system reads
\begin{align}
\delta\rho | |
\deltat |
+\nabla\alpha\left(\rhoV\alpha\right)&=\rho
\alpha | |
CB | |
\alpha |
\ \ \rho\left(
\deltaC | |
\deltat |
+2V\alpha\nabla\alphaC+B\alphaV\alphaV\beta\right)&=-\rho2e\rho
\alpha | |
B | |
\alpha |
\ \ \rho\left(
\deltaV\alpha | |
\deltat |
+V\beta\nabla\betaV\alpha-C\nabla\alphaC-2CV\beta
\alpha | |
B | |
\beta |
\right)&=-\nabla\alpha\left(\rho2e\rho\right) \end{align}
where the
{\delta}/{\delta}t
\rho2e\rho
p
For the Laplace choice of surface tension
\left(e\left(\rho\right)=\sigma/\rho\right)
\begin{align}
\delta\rho | |
\deltat |
+\nabla\alpha\left(\rhoV\alpha\right)&=\rho
\alpha | |
CB | |
\alpha |
\ \ \rho\left(
\deltaC | |
\deltat |
+2V\alpha\nabla\alphaC+B\alphaV\alphaV\beta\right)&=\sigma
\alpha | |
B | |
\alpha |
\ \
\deltaV\alpha | |
\deltat |
+V\beta\nabla\betaV\alpha-C\nabla\alphaC-2V\beta
\alpha | |
B | |
\beta |
&=0\end{align}
Note that on flat (
B\alpha=0
C=0
\begin{align}
\partial\rho | |
\partialt |
+\nabla\alpha\left(\rhoV\alpha\right)&=0\ &\ \rho\left(
\partialV\alpha | |
\partialt |
+V\beta\nabla\betaV\alpha\right)&=-\nabla\alpha\left(\rho2e\rho\right)\end{align}
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
\rho
C
\begin{align} {
\delta\rho | |
\deltat |
1. Exact nonlinear equations for fluid films and proper adaptations of conservation theorems from classical hydrodynamics P. Grinfeld, J. Geom. Sym. Phys. 16, 2009