Generic scalar transport equation

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Revision as of 06:20, 14 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 $c_i$ 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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-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. <br>
+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.<br>
-For example, the conservation equation of a chemical species <math> c_i </math> is <br>
+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 <math> c_i </math> is
-<math> \frac{\partial{\rho c_i}}{\partial t} + \nabla \cdot (\rho \vec u c_i + \vec J) = R_i </math>
-where <math> \vec u </math> denotes the velocity field, <math> \vec J </math> denotes the diffusion flux the of the chemical species, and <math> R_i </math> denotes the rate of generation of <math> R_i </math> caused by the chemical reaction. <br>
+:<math> \frac{\partial{\rho c_i}}{\partial t} + \nabla \cdot (\rho \vec u c_i + \vec J) = R_i </math>
-The '''x-momentum''' equation for a Newtonian fluid can be written as <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> \vec u </math> denotes the velocity field, <math> \vec J </math> denotes the diffusion flux the of the chemical species, and <math> R_i </math> denotes the rate of generation of <math> R_i </math> caused by the chemical reaction.
-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>
+The '''x-momentum''' equation for a Newtonian fluid can be written as
-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> \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}</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> \Gamma </math> is the '''diffusion''' coefficient or '''diffusivity''' <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>
+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
+:<math> \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}</math>
+where <math> \Gamma </math> is the '''diffusion''' coefficient or '''diffusivity'''.

Generic scalar transport equation

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Revision as of 06:20, 14 December 2005

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