Consider a thin fluid film  that spans a stationary closed contour boundary. Let  be the normal component of the velocity field and  be the contravariant components of the tangential velocity projection. Let  be the covariant surface derivative,  be the covariant curvature tensor,  be the mixed curvature tensor and  be its trace, that is mean curvature. Furthermore, let the internal energy density per unit mass function be  so that the total potential energy  is given by
This choice of  :
where  is the surface energy density results in Laplace's classical model for surface tension:
where A is the total area of the soap film.
The governing system reads
where the -derivative is the central operator, originally
due to Jacques Hadamard, in 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.