The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation,^[1] or (generic) scalar transport equation.^[2]

The general equation is^[3][4]

where

The right-hand side of the equation is the sum of three contributions.

In a common situation, the diffusion coefficient is constant, there are no sources or sinks, and the velocity field describes an incompressible flow (i.e., it has zero divergence). Then the formula simplifies to:^[5][6][7]

In this form, the convection–diffusion equation combines both parabolic and hyperbolic partial differential equations.

Using fourier transform in time domain, its general solution can be obtained:

The stationary convection–diffusion equation describes the steady-state behavior of a convective-diffusive system. In a steady state, ∂c/∂t = 0, so the formula is:

The convection–diffusion equation can be derived in a straightforward way^[4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume:

where j is the total flux and R is a net volumetric source for c. There are two sources of flux in this situation. First, diffusive flux arises due to diffusion. This is typically approximated by Fick's first law:

i.e., the flux of the diffusing material (relative to the bulk motion) in any part of the system is proportional to the local concentration gradient. Second, when there is overall convection or flow, there is an associated flux called advective flux:

The total flux (in a stationary coordinate system) is given by the sum of these two:

Plugging into the continuity equation:

In general, D, v, and R may vary with space and time. In cases in which they depend on concentration as well, the equation becomes nonlinear, giving rise to many distinctive mixing phenomena such as Rayleigh–Bénard convection when v depends on temperature in the heat transfer formulation and reaction–diffusion pattern formation when R depends on concentration in the mass transfer formulation.

In some cases, the average velocity field v exists because of a force; for example, the equation might describe the flow of ions dissolved in a liquid, with an electric field pulling the ions in some direction (as in gel electrophoresis). In this situation, it is usually called the drift–diffusion equation or the Smoluchowski equation,^[1] after Marian Smoluchowski who described it in 1915^[8] (not to be confused with the Einstein–Smoluchowski relation or Smoluchowski coagulation equation).

Typically, the average velocity is directly proportional to the applied force, giving the equation:^[9][10]

where F is the force, and ζ characterizes the friction or viscous drag. (The inverse ζ−1 is called mobility.)

When the force is associated with a potential energy F = −∇U (see conservative force), a steady-state solution to the above equation (i.e. 0 = R = ∂c/∂t) is:

(assuming D and ζ are constant). In other words, there are more particles where the energy is lower. This concentration profile is expected to agree with the Boltzmann distribution (more precisely, the Gibbs measure). From this assumption, the Einstein relation can be proven:^[10]

The convection–diffusion equation (with no sources or drains, R = 0) can be viewed as a stochastic differential equation, describing random motion with diffusivity D and bias v. For example, the equation can describe the Brownian motion of a single particle, where the variable c describes the probability distribution for the particle to be in a given position at a given time. The reason the equation can be used that way is because there is no mathematical difference between the probability distribution of a single particle, and the concentration profile of a collection of infinitely many particles (as long as the particles do not interact with each other).

The Langevin equation describes advection, diffusion, and other phenomena in an explicitly stochastic way. One of the simplest forms of the Langevin equation is when its "noise term" is Gaussian; in this case, the Langevin equation is exactly equivalent to the convection–diffusion equation.^[10] However, the Langevin equation is more general.^[10]

The convection–diffusion equation can only rarely be solved with a pen and paper. More often, computers are used to numerically approximate the solution to the equation, typically using the finite element method. For more details and algorithms see: Numerical solution of the convection–diffusion equation.

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.

where M is the momentum of the fluid (per unit volume) at each point (equal to the density ρ multiplied by the velocity v), μ 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.

In biology, the reaction–diffusion–advection equation is used to model chemotaxis observed in bacteria, population migration, evolutionary adaptation to changing environments, and the spatiotemporal dynamics of molecular species including morphogenesis. An example is a study of VEGFC patterning in the context of lymphangiogenesis.^[11]

In semiconductor physics, this equation is called the drift–diffusion equation. The word "drift" is related to drift current and drift velocity. The equation is normally written:^[12]

where

The diffusion coefficient and mobility are related by the Einstein relation as above:

where kB is the Boltzmann constant and T is absolute temperature. The drift current and diffusion current refer separately to the two terms in the expressions for J, namely:

This equation can be solved together with Poisson's equation numerically.^[13]

An example of results of solving the drift diffusion equation is shown on the right. When light shines on the center of semiconductor, carriers are generated in the middle and diffuse towards two ends. The drift–diffusion equation is solved in this structure and electron density distribution is displayed in the figure. One can see the gradient of carrier from center towards two ends.