A Second Order Virtual Node Algorithm for Stokes Flow Problems with Interfacial Forces and Discontinuous Material Properties

Abstract

We present a numerical method for the solution of the Stokes equations that handles interfacial discontinuities due to both singular forces and discontinuous fluid properties such as viscosity and density. The discretization couples a Lagrangian representation of the material interface with an Eulerian representation of the fluid velocity and pressure. The method is efficient, easy to implement and yields discretely divergence-free velocities that are second order accurate. No knowledge of the jumps on the fluid variables and their derivatives is required along the interface. We discretize the equations using an embedded approach on a uniform MAC grid employing virtual nodes and duplicated cells at the interfaces. These additional degrees of freedom allow for accurate resolution of discontinuities in the fluid stress at the material interface but require a Lagrange multiplier term to enforce continuity of the fluid velocity. We provide a novel discretization of this term that accurately resolves the constant pressure null modes. We show that the accurate resolution of these modes accelerates the overall speed of our simulations. Interfaces are represented with a hybrid Lagrangian/level set method. The discrete coupled equations for the velocity, pressure and Lagrange multipliers are in the form of a symmetric KKT system. Numerical results indicate second order accuracy for the velocities and first order accuracy for the pressure (in L1).