An Arbitrary Lagrangian-Eulerian Finite Element Formulation for Fluid Flow on Curved and Deforming Surfaces

An arbitrary Lagrangian-Eulerian finite element formulation for two-dimensional incompressible fluids flowing on arbitrarily curved, deforming surfaces is presented here. In addition to the physical unknowns of the fluid velocity and surface tension, a mesh velocity degree of freedom is included. The mesh is prescribed to move only normal to the surface, such that in-plane shearing of the mesh is avoided. A weak formulation independent of the surface parameterization is developed and solved implicitly using a forward Euler temporal discretization and Newton-Raphson iteration within every time step. Several numerical examples are provided.