Eulerian numerical methods for flow through poroelastic media

Poroelasticity, the coupling of fluid flow and solid deformation within a porous medium, describes behavior seen in a wide array of settings across biology, physics, and engineering. In this talk, we introduce an Eulerian method for the simulation of flow through an incompressible poroelastic medium. The equations of motion are time-integrated using a finite-difference method, and a combined incompressibility condition is satisfied through the application of a Chorin-type projection method. This requires the solution of a global Poisson problem at each time step. We apply this approach to an active gel model---inspired by crosslinked polymer networks under the influence of motor proteins in the Eukaryotic cytoskeleton---to investigate instabilities resulting from active contractile stresses. Simulations show the instabilities drive behaviors such as solid patch condensation and macroscopic contraction. The resulting pattern formation is shown in two and three dimensions.