A conservative, positivity preserving scheme for reactive solute transport problems in moving domains

We study the mathematical models and numerical schemes for reactive transport of a soluble substance in deformable media. The medium is a channel with compliant adsorbing walls. The solutes are dissolved in the fluid flowing through the channel. The fluid, which carries the solutes, is viscous and incompressible. The reactive process is described as a general physico-chemical process taking place on the compliant channel wall. The problem is modeled by a convection–diffusion adsorption–desorption equation in moving domains. We present a conservative, positivity preserving, high resolution ALE-FCT scheme for this problem in the presence of dominant transport processes and wall reactions on the moving wall. A Patankar type time discretization is presented, which provides conservative treatment of nonlinear reactive terms. We establish CFL-type constraints on the time step, and show the mass conservation of the time discretization scheme. Numerical simulations are performed to show validity of the schemes against effective models under various scenarios including linear adsorption–desorption, irreversible wall reaction, infinite adsorption kinetics, and nonlinear Langmuir kinetics. The grid convergence of the numerical scheme is studied for the case of fixed meshes and moving meshes in fixed domains. Finally, we simulate reactive transport in moving domains under linear and nonlinear chemical reactions at the wall, and show that the motion of the compliant channel wall enhances adsorption of the solute from the fluid to the channel wall. Consequences of this result are significant in the area of, e.g., nano-particle cancer drug delivery. Our result shows that periodic excitation of the cancerous tissue using, e.g., ultrasound, may enhance adsorption of cancer drugs carried by nano-particles via the human vasculature.