# Generic scalar transport equation

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<math> \frac{\partial{\rho u}}{\partial t} + \nabla \cdot (\rho \vec u u ) =\nabla \cdot (\mu \nabla u ) - \frac {\partial p}{\partial x} + B_x + V_x </math> <br> | <math> \frac{\partial{\rho u}}{\partial t} + \nabla \cdot (\rho \vec u u ) =\nabla \cdot (\mu \nabla u ) - \frac {\partial p}{\partial x} + B_x + V_x </math> <br> | ||

where <math> B_x </math> is the body force in the x-direction and <math>V_x</math> includes the viscous terms that are not expressed by <math>\nabla \cdot (\mu \nabla u )</math> <br> | where <math> B_x </math> is the body force in the x-direction and <math>V_x</math> includes the viscous terms that are not expressed by <math>\nabla \cdot (\mu \nabla u )</math> <br> | ||

+ | <br> | ||

+ | Upon inspection of the above equations, it can be infered that all the dependent variables seem to obey a generalized conservation principle. If the dependent variable (scalar or vector) is denoted by <math>\phi</math>, the '''generic''' differential equation is <br> | ||

+ | <math> \frac{\partial{\rho \phi}}{\partial t} + \nabla \cdot (\rho \vec u \phi ) =\nabla \cdot (\Gamma \nabla \phi ) S_{\phi}</math> <br> |

## Revision as of 21:58, 13 December 2005

A differential equation epxresses a certain conservation principle. Whether be it in electromagnetics, fluid dynamics, heat transfer, radiation, electronics... conservation principles are the basis for the derivation of differential or integro-differential equations. In this respect, any differential equation addresses a certain quantity as it dependent variable and thus expresses the balance between the phenomena affecting the evolution of this quanitity. For example, the temperature of a fluid in a heated pipe is affected by convection due to the solid-fluid interface, and due to the fluid-fluid interaction. Furthermore, temperature is also diffused inside the fluid. For a state state problem, with the absence of sources, a differential equation governing the temperature will definetely express a balance between convection and diffusion.

A brief inspection of the equations governing various physical phenomena will reveal that all of these equations can be put into a generic form thus allowing a systematic approach for a computer simulation.

For example, the conservation equation of a chemical species is

where denotes the velocity field, denotes the diffusion flux the of the chemical species, and denotes the rate of generation of caused by the chemical reaction.

The **x-momentum** equation for a Newtonian fluid can be written as

where is the body force in the x-direction and includes the viscous terms that are not expressed by

Upon inspection of the above equations, it can be infered that all the dependent variables seem to obey a generalized conservation principle. If the dependent variable (scalar or vector) is denoted by , the **generic** differential equation is