Convection-diffusion equation

The convection-diffusion equation is a partial differential equation from the field of statistical physics and transport phenomena. It describes the transport of particles , energy , temperature , etc. through a combination of diffusion and flow ( convection / advection ).

If a convection-diffusion equation describes the transport of probability density , it is usually referred to as the Fokker-Planck equation - if the probability density relates to particle positions, one speaks of the Smoluchowski equation. For the transport of temperature, it is closely related to the heat conduction equation. The convection-diffusion equation can be understood as an extension of the diffusion equation or the reaction-diffusion equation .

definition

The general form of the convection-diffusion equation is:

${\displaystyle {\frac {\partial c}{\partial t}}={\vec {\nabla }}\cdot \left({\underline {\underline {\mathrm {D} }}}\ {\vec {\nabla }}c\right)-{\vec {\nabla }}\cdot \left({\vec {v}}c\right)+R(c,{\vec {r}},t)}$

Where:

• ${\displaystyle c\equiv c({\vec {r}},t)}$the particle concentration in place and time . Depending on which transport processes are described, it can also represent other quantities, such as mass , energy , temperature , electrical charge , probability, etc. ${\displaystyle {\vec {r}}}$ ${\displaystyle t}$ ${\displaystyle c}$
• ${\displaystyle {\underline {\underline {\mathrm {D} }}}}$is the diffusion coefficient , which here takes the general form of a 2nd level tensor (i.e. a matrix ).
• ${\displaystyle {\vec {v}}({\vec {r}},t)}$is a generally location- and time-dependent speed field that describes the directional transport (convection).
• ${\displaystyle R(c,{\vec {r}},t)}$is an optional reaction term (see reaction-diffusion equation ).
• ${\displaystyle {\vec {\nabla }}}$is the nabla operator .

In many cases it can be assumed that the diffusion is an isotropic effect , i.e. independent of the direction. Without the reaction term and with a constant velocity field, the following simplified form is obtained:

${\displaystyle {\frac {\partial c}{\partial t}}=D\cdot {\vec {\nabla }}^{2}c-{\vec {v}}\cdot {\vec {\nabla }}c}$

Here is the Laplace operator . ${\displaystyle {\vec {\nabla }}^{2}}$

Derivation

The convection-diffusion equation can be derived from the continuity equation. This describes the conservation of the size (i.e. the number of particles ) in a volume and reads: ${\displaystyle c}$

${\displaystyle {\frac {\partial c}{\partial t}}+{\vec {\nabla }}\cdot {\vec {j}}=0}$

Here is a current density that describes the flow of the size (of the particles) through interfaces of a small volume. The amount of only changes due to inflow or outflow through the surface of the volume under consideration. The flow can now be described by two terms: ${\displaystyle {\vec {j}}}$ ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle {\vec {j}}}$

1. The first Fick's law gives the contribution that describes the transport by diffusion. ${\displaystyle {\vec {j}}_{\text{diffusion}}=-D\ {\vec {\nabla }}c}$
2. The flow field results in a convection term ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {j}}_{\text{konvektion}}={\vec {v}}c.}$

The sum of these contributions, after insertion into the continuity equation, results in the diffusion-convection equation. ${\displaystyle {\vec {j}}={\vec {j}}_{\text{diffusion}}+{\vec {j}}_{\text{konvektion}}}$

literature

• Franz Schwabl : Statistical Mechanics . Springer, 2006, ISBN 3-540-31095-9 .
• Klaus Stierstadt: Thermodynamics: From Microphysics to Macrophysics . Springer, 2010, ISBN 978-3-642-05098-5 , p. 414 ff.

Individual evidence

1. ^ S. Chandrasekhar: Stochastic Problems in Physics and Astronomy . In: Reviews of Modern Physics . tape 15 , no. 1 , January 1943, ISSN  0034-6861 , p. 1-89 , doi : 10.1103 / RevModPhys.15.1 .