Immersed Boundary Method for Variable Viscosity and Variable Density Problems Using Fast Constant-Coefficient Linear Solvers I: Numerical Method and Results

We present a general variable viscosity and variable density immersed boundary method that is first-order accurate in the variable density case and, for problems possessing sufficient regularity, second-order accurate in the constant density case. The viscosity and density are considered material properties and are defined by a dynamically updated tesselation. Empirical convergence rates are reported for a test problem of a two-dimensional viscoelastic shell with spatially varying material properties. The reduction to first-order accuracy in the variable density case can be avoided by using an iterative scheme, although this approach may not be efficient enough for practical use. In our timestepping scheme, both the inertial and viscous terms are split into two parts: a constant-coefficient part that is treated implicitly, and a variable-coefficient part that is treated explicitly. This splitting allows the resulting equations to be solved efficiently using fast constant-coefficient linear solvers, and in this work, we use solvers based on the Fast Fourier transform (FFT). As an application of this method, we perform fully three-dimensional, two-phase simulations of red blood cells accounting for variable viscosity and variable density. We study the behavior of red cells during shear flow and during capillary flow.