# Incompressible fluid problems on embedded surfaces: Modeling and variational formulations

## TL;DR

This paper derives and analyzes the governing equations for viscous incompressible fluids on evolving surfaces, focusing on stationary embedded manifolds, and introduces variational formulations suitable for numerical methods.

## Contribution

It presents a derivation of surface Navier-Stokes equations using exterior calculus and introduces new well-posedness results and variational formulations for the surface Stokes problem.

## Key findings

- Derived surface Navier-Stokes equations on evolving manifolds.
- Proved well-posedness for the surface Stokes equations on stationary manifolds.
- Developed variational formulations suitable for Galerkin discretization.

## 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.