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Virtual Node Algorithms for Stokes Interface Problems


We present two numerical methods for the solution of the Stokes equations designed to handle both interfacial discontinuities, geometrically irregular flow domains and discontinuous fluid properties such as viscosity and density. The methods are efficient, easy to implement and yield second order accurate, discretely divergence free velocities. We call these methods Virtual Node Algorithms. The first method handles the case in which the fluid viscosity is continuous across the interfaces, while the second method handles the case in which the fluid viscosity is discontinuous across the interfaces. In both cases, we assume the fluid viscosity to be uniformly constant over the spatial extension of each fluid. We discretize the Stokes equations using an embedded approach on a uniform MAC-grid employing virtual nodes at interfaces and boundaries. Interfaces and boundaries are represented with a hybrid Lagrangian/level set method. For the continuous viscosity case, we rewrite the Stokes equations as three Poisson equations and use the techniques developed in Bedrossian et al. (2010) [1] to impose jump and boundary conditions. We also use a final Poisson equation to enforce a discrete divergence-free condition. All four linear systems involved are symmetric positive definite with three of the four having the standard 5-point Laplace stencil everywhere. Numerical results are presented indicating second order accuracy in L∞ for both velocities and pressure. For the discontinuous viscosity case, we presented a method which requires no knowledge of the jumps on the fluid variables and their derivatives along the interface. The degrees of freedom associated with virtual nodes allow for accurate resolution of discontinuities in the fluid stress at the interfaces 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. The discrete coupled equations for the velocity, pressure and Lagrange multipliers are in the form of a symmetric KKT system. Numerical results are presented indicating second order accuracy for the velocities and first order accuracy for the pressure (in L∞).

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