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Generic scalar transport equation

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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  c_i is

 \frac{\partial{\rho c_i}}{\partial t} + \nabla \cdot (\rho \vec u c_i + \vec J) = R_i

where  \vec u denotes the velocity field,  \vec J denotes the diffusion flux the of the chemical species, and  R_i denotes the rate of generation of  R_i caused by the chemical reaction.

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

 \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

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

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 \phi, the generic differential equation is

 \underbrace{ \frac{\partial{\rho \phi}}{\partial t}}_{Transient \ term} + \underbrace{ \nabla \cdot (\rho \vec u \phi )}_{Convection \ term} =\underbrace {\nabla \cdot (\Gamma \nabla \phi )}_{Diffusion \ term} + \underbrace {S_{\phi}}_{Source \ term}

where  \Gamma is the diffusion coefficient or diffusivity.

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