Generic Scalar Transport Equation - Similar Equations in Other Contexts

Similar Equations in Other Contexts

The convection–diffusion equation is a relatively simple equation describing flows, or alternatively, describing a stochastically-changing system. Therefore, the same or similar equation arises in many contexts unrelated to flows through space.

• It is formally identical to the Fokker–Planck equation for the velocity of a particle.
• It is closely related to the Black–Scholes equation and other equations in financial mathematics.
• It is closely related to the Navier–Stokes equations, because the flow of momentum in a fluid is mathematically similar to the flow of mass or energy. The correspondence is clearest in the case of an incompressible Newtonian fluid, in which case the Navier–Stokes equation is:

where M is the momentum of the fluid (per unit volume) at each point (equal to the density multiplied by the velocity v), is viscosity, P is fluid pressure, and f is any other body force such as gravity. In this equation, the term on the left-hand side describes the change in momentum at a given point; the first term on the right describes viscosity, which is really the diffusion of momentum; the second term on the right describes the advective flow of momentum; and the last two terms on the right describes the external and internal forces which can act as sources or sinks of momentum.