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8.3.1 Generalization of Mathematical Approach for Derivations

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  • 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  
        \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  
        \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
         \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
        \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
        \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
        \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   
        \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}


    • 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.