Modeling, analysis and numerical simulation of reactive solute transport problems in moving domains

In this thesis, we study mathematical models and numerical schemes for reactive transport of a soluble substance in deformable media. The medium is a cylindrical channel with compliant adsorbing walls. The solutes are dissolved in a fluid flowing through the channel. The fluid, which carries the solutes, is viscous and incompressible. The problem is modeled by a convection-diffusion adsorption-desorption equation in moving domains.

First, we present the mathematical formulation of the model in the arbitrary Lagrangian-Eulerian (ALE) framework. We study the well-posedness of the model.We then discretize the conservative variational form of the problem in the ALE framework in space, using the moving mesh ALE finite element method (ALE-FEM). In time, it is discretized using a novel Patankar linearization technique. We then prove global conditional stability for the fully discrete problem.

Next, we present a conservative, positivity preserving, high resolution linear ALE-FCT scheme for this problem in the presence of dominant convection processes and wall reactions on the moving wall. Numerical simulations are performed to show validity of the scheme under various scenarios. The grid convergence of the numerical scheme is studied for the case of fixed meshes and moving meshes in fixed domains. Then, we simulate reactive transport in moving domains under linear and nonlinear wall reactions, and show that the motion of the compliant channel wall enhances adsorption of the solute from the fluid to the channel wall.

Finally, we present a conservative, positivity preserving, high resolution nonlinear ALE-FCT scheme. The scheme is proved to be mass conservative in time, and positive at all times. Reactive transport is simulated using this scheme for its validation, to show it convergence, and to compare it against the linear ALE-FCT scheme. The nonlinear ALE-FCT is shown to perform better than the linear ALE-FCT scheme for large time steps.