Advectiondiffusion

Advection-diffusion is a model for the transport of a scalar quantity, such as pollutant concentration, within a moving fluid. It combines advection, the transport by the bulk motion of the fluid, with diffusion, the spreading caused by concentration gradients. The equation expresses conservation of the scalar.

Mathematically, for a scalar concentration C(x,t) in a velocity field u(x,t) and a diffusion coefficient tensor D, the advection-diffusion equation is ∂C/∂t + u·∇C = ∇·(D∇C) + S, where S represents sources or sinks. If D is constant and isotropic, the diffusion term reduces to D∇^2C. The velocity field u is typically determined by fluid dynamics.

The relative importance of advection and diffusion is captured by the Peclet number Pe = UL/D, with U a characteristic velocity and L a length scale. Large Pe indicates advection-dominated transport, while small Pe indicates diffusion-dominated transport. Solutions require initial conditions C(x,0) and boundary conditions on the domain boundaries, which may be Dirichlet, Neumann, or Robin types.

Advection-diffusion models are widely used in environmental engineering (pollutant transport in rivers and groundwater), atmospheric and ocean sciences (aerosol and dye dispersion), chemical engineering (reactor mixing), and heat transfer, where the scalar is temperature. The equation can be extended with reaction terms or nonuniform, anisotropic diffusion.

Analytical solutions exist for simple geometries and constant coefficients, such as one-dimensional problems, but most practical problems require numerical methods. Finite difference, finite element, and finite volume methods are common, often with upwind schemes for stability in advection-dominated regimes. Realistic modeling may couple the advection-diffusion equation to the Navier–Stokes equations.