diffusionreactiontransport

Diffusion-reaction-transport (DRT) describes systems in which chemical species are carried by diffusion and advective transport while simultaneously undergoing chemical reactions. It is used to model processes across chemistry, environmental engineering, biology, and materials science where transport and transformation interact.

Mathematically, for each species i with concentration Ci(x,t), the evolution is governed by a partial differential equation of the form ∂Ci/∂t = ∇·(Di ∇Ci) - ∇·(vi Ci) + Ri(C1,...,Cn), where Di is the diffusion coefficient (possibly tensor-valued), vi is the velocity field for advection, and Ri represents net production or consumption from chemical reactions. In porous or heterogeneous domains, effective parameters account for porosity, tortuosity, and sorption or phase partitioning. Boundary and initial conditions specify conditions at domain borders and at t=0.

Diffusion follows Fick's laws; advection carries mass with flow; reactions follow kinetic laws, often nonlinear. When multiple species couple, rich spatiotemporal behaviors can emerge, including traveling waves, stationary patterns, or oscillations, as seen in reaction-diffusion systems.

Numerical solution relies on finite difference, finite element, or finite volume methods, often with operator splitting to separate diffusion, advection, and reaction steps. Challenges include stiffness from fast reactions, disparate time scales, coupling of multiphysics, and high computational cost in three-dimensional or heterogeneous media.

Applications span groundwater remediation and contaminant transport, electrochemical devices such as batteries and fuel cells, catalysis in porous catalysts, tissue-level transport of nutrients and signaling molecules, and ecological models of population spread and interaction.