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8.3.1 Generalization of Mathematical Approach for Derivations

In this section a general approach for the derivations for conservation of any quantity e.g. scalar, vector or tensor, are presented. Suppose that the property \(\phi\) is under a study which is a function of the time and location as \(\phi(x,y,z,t)\). The total amount of quantity that exist in arbitrary system is  
\[
    \label{dif:eq:math:phiG}
    \Phi = \int_{sys}  \phi\,\rho\,dV \tag{61}
\]
Where \(\Phi\) is the total quantity of the system which has a volume \(V\) and a surface area of \(A\) which is a function of time. A change with time is  
\[
    \label{dif:eq:math:DphiDt1}
    \dfrac{D\Phi}{Dt}  = \dfrac{D}{Dt} \int_{sys} \phi\,\rho\,dV \tag{62}
\]
Using RTT to change the system to a control volume (see equation (??)) yields
\[
    \label{dif:eq:math:DphiDt}
     \dfrac{D}{Dt} \int_{sys} \phi\,\rho\,dV =  
     \dfrac{d}{dt} \int_{cv} \phi\,\rho\,dV + \int_{A} \rho\,\phi\,\pmb{U}\cdot dA \tag{63}
\]
The last term on the RHS can be converted using the divergence theorem (see the appendix) from a surface integral into a volume integral (alternatively, the volume integral can be changed to the surface integral) as
\[
    \label{dif:math:divergenceTheorem}
    \int_{A} \rho\,\phi\,\pmb{U}\cdot dA = \int_{V} \nabla\cdot\left(\rho\,\phi\,\pmb{U} \right) dV  \tag{64}
\]
Substituting equation (64) into equation yields
\[
    \label{dif:eq:RTTextended1}
    \dfrac{D}{Dt} \int_{sys} \phi\,\rho\,dV = \dfrac{d}{dt} \int_{cv} \phi\,\rho\,dV +  
    \int_{cv} \nabla\cdot\left(\rho\,\phi\,\pmb{U} \right) dV \tag{65}
\]
Since the volume of the control volume remains independent of the time, the derivative can enter into the integral and thus combining the two integrals on the RHS results in
\[
    \label{dif:eq:RTTextended}
    \dfrac{D}{Dt} \int_{sys} \phi\,\rho\,dV =  \int_{cv} \left( \dfrac{d \left(\phi\,\rho\right)}{dt} +  
    \nabla\cdot\left(\rho\,\phi\,\pmb{U} \right) \right) dV \tag{66}
\]
The definition of equation (61) LHS can be changed to simply the derivative of \(\Phi\). The integral is carried over arbitrary system. For an infinitesimal control volume the change is   
\[
    \label{dif:eq:math:infinitesimalChange}
    \dfrac{D\,\Phi}{Dt}  \cong \left( \dfrac{d \left(\phi\,\rho\right)}{dt} +
   \nabla\cdot\left(\rho\,\phi\,\pmb{U} \right) \right) \overbrace{dx\,dy\,dz}^{dV}  \tag{67}
\]

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.