A simple finite difference method for time-dependent, variable coefficient Stokes flow on irregular domains

TL;DR

This paper introduces a simple, stable finite difference method for simulating time-dependent Stokes flow with irregular boundaries, supporting variable properties and complex geometries without remeshing.

Contribution

It presents a variational embedded boundary approach on Cartesian grids that simplifies implementation and extends to coupled mechanics, ensuring stability and efficiency in irregular domains.

Findings

Achieves first order convergence in velocity for test cases

Successfully reproduces jet buckling in 2D and 3D simulations

Uses a single sparse linear system per time step for efficiency

Abstract

We present a simple and efficient variational finite difference method for simulating time-dependent Stokes flow in the presence of irregular free surfaces and moving solid boundaries. The method uses an embedded boundary approach on staggered Cartesian grids, avoiding the need for expensive remeshing operations, and can be applied to flows in both two and three dimensions. It uses fully implicit backwards Euler integration to provide stability and supports spatially varying density and viscosity, while requiring the solution of just a single sparse, symmetric positive-definite linear system per time step. By expressing the problem in a variational form, challenging irregular domains are supported implicitly through the use of natural boundary conditions. In practice, the discretization requires only centred finite difference stencils and per-cell volume fractions, and is straightforward to implement. The variational form further permits generalizations to coupling other mechanics, all the while reducing to a sparse symmetric positive definite matrix. We demonstrate consistent first order convergence of velocity in L1 and Linf norms on a range of analytical test cases in two dimensions. Furthermore, we apply our method as part of a simple Navier-Stokes solver to illustrate that it can reproduce the characteristic jet buckling phenomenon of Newtonian liquids at moderate viscosities, in both two and three dimensions.