Mathematical models of immersed boundary complex fluids

A major research interest in recent computational rheology has been on modeling complex biological fluid and fluid-structure interaction. With considerable progress in formulating methods for Newtonian fluids and given the fact that many biological fluids are complex, it is vitally important for researchers in computational rheology to understand non-Newtonian responses in complex fluids. The objective of this work is to contribute to this knowledge by way of a new modeling approach. Within the framework of the immersed boundary method, the first chapter focuses on developing a Lagrangian mesh model and testing it in planar Poiseuille flow. The second chapter studies peristaltic pumping in both the Lagrangian mesh and Oldroyd-B models. The third chapter then investigates fluid-particle interaction in both models.;The first chapter introduces the Lagrangian mesh method and studies the model using planar Poiseuille flow as a test case. Numerical results are compared with an Oldroyd-B model and suggest that a Lagrangian mesh model with a regridding step and meshgrid refinement produces simulations that are comparable to the Oldroyd-B model.;The second chapter investigates peristaltic pumping and implements the Lagrangian mesh model. The focus here is to present a numerical method for simulating a viscoelastic fluid coupled to contractile and moving boundaries. This includes both Oldroyd-B and Lagrangian mesh approaches, both using an immersed boundary method for fluid coupling to elastic walls. The results observed suggest that peristaltic transport of a viscoelastic fluid is affected by the Weissenberg number, as well as the peristaltic wavelength and amplitude.;A complex fluid-structure interaction model for particle flow in a channel is presented in the final chapter. This test case addresses a typical problem in biological fluid dynamics that involves the interaction of an elastic structure with a surrounding fluid. It is shown here that the Lagrangian mesh model produces similar results to the Oldroyd-B model and suggests that the non-Newtonian response impedes particle transportation.