Reaction–diffusion–advection equation

The reaction–diffusion–advection equation is a partial differential equation that models the concentration of a chemical species in a classical reaction–diffusion–advection process. In that process, a chemical species undergoes a reaction, can diffuse in the solvent, and is transported by the bulk movement of the solvent (advection). The equation modeling the concentration \(u(\boldsymbol{x},t) \) (units: M), \( \boldsymbol{x} \in \mathbb{R}^n \) of the chemical species u is:

\[ \frac{\partial u}{\partial t} + \nabla \cdot \left( \boldsymbol{v} u - D\nabla u \right) = f, \]

where D is the diffusion coefficient (units: length² /time), \( \boldsymbol{v} \in \mathbb{R}^n \) is the bulk velocity (units: length/time), and \( f \) (units: M/s) is the reaction term that models the generation or decay of the species u. In general, each \( f,D, \) and \( \boldsymbol{v} \) may depend on space, time, or the concentration of u itself. The equation above is a conservation of mass in a continuum model, where accumulation at a point is the next flux (rate in minus rate out) of a point plus the generation at that point (see continuity equation).

See also

• Reaction–diffusion equation
• Advection–diffusion equation