This dissertation presents a novel numerical method to study the pulsing behavior of soft corals. Evidence indicates that the pulsing behavior of soft corals in the family Xeniidae facilitates photosynthesis of their symbiotic algae. One way to investigate this complex behavior is through mathematical modeling and numerical simulations. The immersed boundary method is used to model the interaction of the coral tentacles with the surrounding fluid. The flow is then coupled with a photosynthesis model. The photosynthesis is modeled by advecting and diffusing oxygen, the byproduct of photosynthesis, where the coral tentacles act as a moving source of oxygen. This work develops a methodology for solving a system of partial differential equations with boundary conditions on a moving immersed elastic boundary. Two-dimensional numerical simulations are presented where the Reynolds and Péclet numbers are varied in the simulations to understand how these parameters affect the mixing and photosynthesis. The mixing is quantified using both the fluid flow and oxygen concentration dynamics. The results show that for the biologically relevant Péclet number, the fluid dynamics significantly affect the photosynthesis and that the biologically relevant Reynolds number is advantageous for mixing and photosynthesis. The models and methods developed have been contributed to the open-source software library implementation of the immersed boundary method, IB2d. Three-dimensional numerical simulations of soft coral pulsing are also presented using the IBAMR software library. A three-dimensional mixing analysis of the flow is presented. Further, preliminary results of the three-dimensional corals pulsing with a background oxygen concentration are presented with the methodology for modeling the three-dimensional coral tentacles as a sink or source of a concentration.

Settling marine aggregates are essential in transporting dissolved carbon dioxide from the surface ocean to the deep sea. While sinking, they may accumulate in thin layers where density stratification is present, becoming nutrient hotspots for bacteria and animal activity. We simulate settling marine aggregates in an ambient homogeneous and a density-stratified fluid to study their dynamics. We model a marine aggregate as a fractal collection of cubes. In a homogeneous fluid, the flow is computed in the Stokes regime using a boundary integral method. We focus on analyzing the forces acting on the aggregate in the presence of various background flows. We have identified an effective hydrodynamic radius that captures the settling dynamics accurately. We couple the velocity with the advection-diffusion equation to track the heat density or salt concentration in time. We use this method to quantify how the presence of stratification affects the settling speed and time of the aggregates.We also present the study of complex fluids with higher-order rheology. We introduce rate-dependent flows having non-Newtonian behaviors. We pay attention to a flow of granular material, which is the second most commonly used material in industry after water. We model a second-order stress tensor in strain rate for granular material within an Adaptive Mesh Refinement (AMR) framework, AMReX-incflo, to solve for incompressible flows. We also formulate a regularization of the yield stress terms to account for viscosity divergence at low strain rates. To incorporate a higher-order strain rate tensor with the incompressible solver numerically, we added a two-stage Runge-Kutta scheme in the AMReX-incflo code and the default solver, the explicit Euler method.