Advectiondiffusionreaction

Advection-diffusion-reaction is a class of mathematical models used to describe the transport and transformation of substances in flowing media. The model combines three coupled processes: advection, diffusion, and reaction.

Advection refers to transport by the bulk motion of the fluid. In a velocity field v, the advective transport of concentration c is represented by the term v · ∇c. Diffusion accounts for spreading due to concentration gradients, described by a diffusion coefficient D (scalar or tensor) and the term ∇ · (D ∇c). Reaction encompasses local production or consumption through chemical, biological, or other transformations, with a source term R(c, x, t).

In many formulations for a scalar concentration c(x,t) in an assumed flow, the advection-diffusion-reaction equation can be written as ∂c/∂t + v · ∇c = ∇ · (D ∇c) + R(c, x, t). For more general conditions, including variable density or mass conservation, alternative but related forms such as ∂c/∂t + ∇ · (v c) = ∇ · (D ∇c) + R may be used, depending on the modeling assumptions.

Solutions require initial concentration distributions and boundary conditions, such as specified concentrations or fluxes on domain boundaries. In practice, D and R may depend on position, time, or c, making the problem nonlinear.

Applications span groundwater and environmental engineering, atmospheric and oceanic transport, chemical reactors, and biomedical contexts. Analytical solutions exist for simplified linear cases; most real-world problems rely on numerical methods such as finite difference, finite element, or finite volume schemes. Dimensional analysis using Peclet and Damköhler numbers helps identify the relative importance of transport, spreading, and reaction processes.