The immersed boundary method is a mathematical and computational framework for problems involving the interaction of a fluid with immersed structures. In the dissertation, we consider also the role of solutes (possibly charged) and their interactions with membranes. We propose a numerical scheme for the advection-diffusion of solutes in fluid-solute-structure interaction . The transport of solute across possibly moving boundaries is controlled by a chemical barrier along the boundary. Moreover, when the solutes are electrical ions, they generate an electrical potential according to the Poisson equation, and they drift relative to the fluid according to the gradient of the electrical potential. The results show electroneutrality except in space charge layers near membranes. We study the transmembrane ionic current as a function of the voltage across the membrane and the ionic concentration on both sides of the membrane.

The immersed boundary method with advection-electrodiffusion is enhanced by local mesh refinement. This makes it feasible to resolve the steep gradients associated with space charge layers, and the singular structure of the chemical potential. The Stokes equations are solved by a cell-centered approximate projection method. The advection-electrodiffusion equations are solved by the combination of geometric and algebraic multigrid methods with two layers of ghost cells. Implicit schemes make the time stepping flexible overcoming the constraint of the CFL condition.

Two examples of osmotic swelling and concentration-dependent contraction are illustrated by the immersed boundary method with advection-electrodiffusion . The osmotic pressure and water permeability generate relative motion between membrane and fluid in the normal direction to the membrane. The influence of internal versus external solute concentration in determining the volume of an elastic semi-permeable vesicle is studied with and without electrical effects. Next, with the water permeability, internal actin fibers are placed in a parallel alignment with the elasticity sensitive to local calcium concentration. With the high calcium concentration in cytoplasm, an active contraction is observed.