Speaker
Description
Scalar vorticity formulation for fluid equations on surfaces is computationally attractive. However, the vorticity equation is incomplete on a non-simply-connected surface. We derive a new evolution equation for the finite dimensional harmonic (cohomology) components of the flow. We also show that the vorticity equation has a curvature-dependent, vorticity production term in addition to the advection-diffusion equation. The new terms finally make the vorticity formulation complete, and reveal structures that were previously overlooked. Specifically, they unify frictional boundary conditions, the Kutta condition, conservation law associated to isometry gauge, and a new conservation law the topological linking between the vortex geometry and streamlines of harmonic flows. In the limit of inviscid point vortex configuration, this new conserved quantity can be elegantly expressed in terms of the divisor class group when the vortices on surfaces are viewed as a divisor on a Riemann surface. These mathematical structures make these terms easy to incorporate computationally. They also lead to new nontrivial analytic solutions to the Euler equations.
In this talk, we also show that Scriven's equation for viscously evolving surface can be derived by Onsager's variational principle. This variational formulation survives after discretization, leading to a simple computational framework for Scriven's flow on arbitrary triangle mesh.