Incompressible flows in elastic domains: an immersed boundary method approach

This study illustrates how the immersed boundary method may be applied to perform the numerical simulation of incompressible flows in two-dimensional domains bounded by elastic boundaries. It presents the basic intermediate steps involved in the derivation of a solution methodology, from a scientific motivation to the numerical results, which can be applied for both steady and transient problems, even when the boundaries have an arbitrary shape. Its motivation, briefly presented, was borne in a bioengineering problem: the numerical simulation of the performance of ventricular assist devices. The mathematical model is composed by the Navier-Stokes equations, where the forcing term contains singular forces which arise from the elastic stresses acting on the boundaries. The incompressibility constraint is modified to introduce the inflow and outflow conditions into the problem through the use of sources and sinks. The methodology is applied to simulate two problems: the steady flow between two parallel plates, for which the exact solution is known and can be used to validate the approach, and the periodic flow in a winding channel, a transient problem in a non-trivial domain.