Convection–diffusion equation
[2] The general equation in conservative form is[3][4]
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.
For example, when methane burns, it involves not only the destruction of methane and oxygen but also the creation of carbon dioxide and water vapor.
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:
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.
The total flux (in a stationary coordinate system) is given by the sum of these two:
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).
In this case the equation can be put in the simple diffusion form:
where the derivative of the left hand side is the material derivative of the variable c. In non-interacting material, D=0 (for example, when temperature is close to absolute zero, dilute gas has almost zero mass diffusivity), hence the transport equation is simply the continuity equation:
Using Fourier transform in both temporal and spatial domain (that is, with integral kernel
This is the basis of temperature measurement for near Bose–Einstein condensate[6] via time of flight method.
[7] The stationary convection–diffusion equation describes the steady-state behavior of a convection–diffusion system.
In one spatial dimension, the equation can be written as
Which can be integrated one time in the space variable x to give:
On the other hand, in the positions x where D=0, the first-order diffusion term disappears and the solution becomes simply the ratio:
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).
Typically, the average velocity is directly proportional to the applied force, giving the equation:[10][11]
where F is the force, and ζ characterizes the friction or viscous drag.
This concentration profile is expected to agree with the Boltzmann distribution (more precisely, the Gibbs measure).
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 the diffusion of momentum by viscosity; 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.
The convection–diffusion equation (with 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.
where kB is the Boltzmann constant and T is absolute temperature.
[14] 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.
