Variational Stokes with Polynomial Reduced Fluid Model

Abstract

Standard fluid simulators often apply operator splitting to independently solve for pressure and viscous stresses. This decoupling, however, induces incorrect free surface boundary conditions. Such methods are unable to simulate various fluid phenomena reliant on the balance of pressure and viscous stresses, such as the liquid rope coil instability exhibited by honey. Unsteady Stokes solvers, when used as a sub-component of Navier-Stokes, retain coupling between pressure and viscosity, and are thus able to resolve these behaviours. The simultaneous application of stress and pressure terms, however, creates much larger, and thus more computationally expensive, systems than the standard decoupled approach.

To accelerate solving the unsteady Stokes problem, we propose a reduced fluid model wherein interior regions are represented with incompressible polynomial vector fields. Sets of standard grid cells are consolidated into super-cells, each of which are modelled using only 26 degrees of freedom. We demonstrate that the reduced field must necessarily be at least quadratic, with the affine model being unable to capture viscous forces. We reproduce the liquid rope coiling instability, as well as other simulated examples, to show that our reduced model provides qualitatively similar results to the full Stokes system for a smaller computational cost.