Convection diffusion equation


Convection diffusion equation

The Convection Diffusion equation is a parabolic partial differential equation. It is very important for the description of physical phenomena where particles or energy (or other physical quantities) are transferedinside a physical system due to two processes: diffusion and convection. In the simples form (when the diffusion coefficient and the convection velocity are constant) the equation can has the

:igg. partial _t u = D, abla ^2 u - v, abla u igg.

The two terms on the right hand side represents different physical processes, the first correspond tonormal diffusion while the second describes the convection. D is the diffusion coefficient, anda is the convection velocity.

Derivation

The convection diffusion equation can be derived in a straightforward way from the continuity equation, which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system. Effectively, no material is created or destroyed:

:igg. partial _t u + ablacdotvec{j}=0,

where vec{j} is the flux of the diffusing material. The current can be written using the phenomenological Fick's first law, which assumes that the flux of the diffusing material in any part of the system is proportional to the local density gradient. When there is convection, the flux also depends on the convection velocity.Combining these two terms the current is written in the form

:vec{j}=-D, abla,u,+,v,u .

The combination of these two equations, gives the convection-diffusion equation.

Referencies

*Granville Sewell, "The Numerical Solution of Ordinary and Partial Differential Equations",Academic Press (1988). ISBN 0-12-637475-9


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