# 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.