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Concept
Convection–diffusion equation
Summary
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, or (generic) scalar transport equation.
The general equation is
where
c is the variable of interest (species concentration for mass transfer, temperature for heat transfer),
D is the diffusivity (also called diffusion coefficient), such as mass diffusivity for particle motion or thermal diffusivity for heat transport,
v is the velocity field that the quantity is moving with. It is a function of time and space. For example, in advection, c might be the concentration of salt in a river, and then v would be the velocity of the water flow as a function of time and location. Another example, c might be the concentration of small bubbles in a calm lake, and then v would be the velocity of bubbles rising towards the surface by buoyancy (see below) depending on time and location of the bubble. For multiphase flows and flows in porous media, v is the (hypothetical) superficial velocity.
R describes sources or sinks of the quantity c. For example, for a chemical species, R > 0 means that a chemical reaction is creating more of the species, and R < 0 means that a chemical reaction is destroying the species. For heat transport, R > 0 might occur if thermal energy is being generated by friction.
∇ represents gradient and ∇ ⋅ represents divergence. In this equation, ∇c represents concentration gradient.
The right-hand side of the equation is the sum of three contributions.
The first, ∇ ⋅ (D∇c), describes diffusion. Imagine that c is the concentration of a chemical. When concentration is low somewhere compared to the surrounding areas (e.g. a local minimum of concentration), the substance will diffuse in from the surroundings, so the concentration will increase.
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