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We present a simple and efficient variational finite difference method forsimulating time-dependent Stokes flow in the presence of irregular freesurfaces and moving solid boundaries. The method uses an embedded boundaryapproach on staggered Cartesian grids, avoiding the need for expensiveremeshing operations, and can be applied to flows in both two and threedimensions. It uses fully implicit backwards Euler integration to providestability and supports spatially varying density and viscosity, while requiringthe solution of just a single sparse, symmetric positive-definite linear systemper time step. By expressing the problem in a variational form, challengingirregular domains are supported implicitly through the use of natural boundaryconditions. In practice, the discretization requires only centred finitedifference stencils and per-cell volume fractions, and is straightforward toimplement. The variational form further permits generalizations to couplingother mechanics, all the while reducing to a sparse symmetric positive definitematrix. We demonstrate consistent first order convergence of velocity in L1 andLinf 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 thatit can reproduce the characteristic jet buckling phenomenon of Newtonianliquids at moderate viscosities, in both two and three dimensions.

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