Title: Incompressible fluid problems on embedded surfaces: Modeling and variational formulations,
Abstract: Governing equations of motion for a viscous incompressible material surface
are derived from the balance laws of continuum mechanics. The surface is
treated as a time-dependent smooth orientable manifold of codimension one in an
ambient Euclidian space. We use elementary tangential calculus to derive the
governing equations in terms of exterior differential operators in Cartesian
coordinates. The resulting equations can be seen as the Navier-Stokes equations
posed on an evolving manifold. We consider a splitting of the surface
Navier-Stokes system into coupled equations for the tangential and normal
motions of the material surface. We then restrict ourselves to the case of a
geometrically stationary manifold of codimension one embedded in $\Bbb{R}^n$.
For this case, we present new well-posedness results for the simplified surface
fluid model consisting of the surface Stokes equations. Finally, we propose and
analyze several alternative variational formulations for this surface Stokes
problem, including constrained and penalized formulations, which are convenient
for Galerkin discretization methods.