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Generic scalar transport equation

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A differential equation expresses 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.

  • The transient term,  \frac{\partial{\rho \phi}}{\partial t} , accounts for the accumulation of  \phi in the concerned control volume
  • The convection term, \nabla \cdot (\rho \vec u \phi ), accounts for the transport of  \phi due to the existence of the velocity field (note the velocity  \vec u multiplying  \phi )
  • The diffusion term,  \nabla \cdot (\Gamma \nabla \phi ), accounts for the transport of  \phi due to its gradients.
  • The source term,  S_{\phi} , accounts for any sources or sinks that either create or destroy  \phi . Any extra terms that cannot be cast into the convection or diffusion terms are considered as source terms.

The objective of all discretization techniques (Finite Difference, Finite Element, Finite Volume, Boundary Element...) is to devise a mathematical formulation to transform each of these terms into an algebraic equation. Once applied to all control volumes in a given mesh, we obtain a full linear system of equations that needs to be solved.

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