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8.3.2.1: The General Mass Time Derivative

Using \(\phi=1\) is the same as dealing with the mass conservation. In that case \(\dfrac{D\,\Phi}{Dt} =  \dfrac{D\,\rho}{Dt}\) which is equal to zero as  
\[
    \label{dif:eq:math:infinitesimalChangeRho1}
    \int \left(
     \dfrac{d \left(\overbrace{1}^{\phi}\,\rho\right)}{dt} +
   \nabla\cdot\left(\rho\,\overbrace{1}^{\phi}\,\pmb{U} \right) \right) \overbrace{dx\,dy\,dz}^{dV} = 0    \tag{68}
\]
Using equation (61) leads to
\[
    \label{dif:eq:math:infinitesimalChangeRho}
     \dfrac{D\,\rho}{Dt} = 0 \longrightarrow \dfrac{\partial  \,\rho}{\partial t} +
   \nabla\cdot\left(\rho\,\pmb{U} \right) = 0    \tag{69}
\]
Equation (69) can be rearranged as
\[
    \label{dif:eq:math:mass:start}
    \dfrac{\partial \,\rho}{\partial t} + \pmb{U}\,\nabla\cdot\rho + \rho\,\nabla\cdot\pmb{U} = 0   \tag{70}
\]
Equation (70) can be further rearranged so derivative of the density is equal the divergence of velocity as  
\[
    \label{dif:eq:math:mass:startRe}
    \dfrac{1}{\rho}  
        \left( \overbrace{\dfrac{\partial \,\rho}{\partial t} +
                \pmb{U}\,\nabla\cdot\rho}
            ^{\text{substantial derivative}} \right)  
            = - \nabla\cdot\pmb{U}  \tag{71}
\]
Equation (71) relates the density rate of change or the volumetric change to the velocity divergence of the flow field. The term in the bracket LHS is referred in the literature as substantial derivative. The substantial derivative represents the change rate of the density at a point which moves with the fluid.

Acceleration Direct Derivations

One of the important points is to find the fluid particles acceleration. A fluid particle velocity is a function of the location and time. Therefore, it can be written that  
\[
    \label{diff:eq:locatonU}
    \pmb{U}(x,y,z,t) = U_x(x,y,x,t)\,\widehat{i} +
     U_y(x,y,z,t)\,\widehat{j} +
     U_z(x,y,z,t)\,\widehat{k}  \tag{72}
\]  
Therefore the acceleration will be  
\[
    \label{diff:eq:locatonA}
    \dfrac{D\pmb{U}}{Dt} = \dfrac{d\,U_x}{dt}\,\widehat{i} +
     \dfrac{d\,U_y}{dt} \,\widehat{j} + \dfrac{d\,U_z}{dt} \,\widehat{k}  \tag{73}
\]
The velocity components are a function of four variables, (\(x\), \(y\), \(z\), and \(t\)), and hence
\[
    \label{dif:eq:dUxdt}
    \dfrac{D\,U_x}{Dt} = \dfrac{\partial \,U_x}{\partial t} \overbrace{\dfrac{d\, t}{d\,t}}^{=1} +  
     \dfrac{\partial \,U_x}{\partial x} \overbrace{\dfrac{d\, x}{d\,t}}^{U_x} +  
     \dfrac{\partial \,U_x}{\partial y} \overbrace{\dfrac{d\, y}{d\,t}}^{U_y} +  
     \dfrac{\partial \,U_x}{\partial z} \overbrace{\dfrac{d\, z}{d\,t}}^{U_z}    \tag{74}
\]
The acceleration in the \(x\) can be written as
\[
    \label{dif:eq:dUxdtC1}
    \dfrac{D\,U_x}{Dt} = \dfrac{\partial \,U_x}{\partial t} +  
    {U_x} \dfrac{\partial \,U_x}{\partial x} +  
    {U_y} \dfrac{\partial \,U_x}{\partial y} +  
    {U_z} \dfrac{\partial \,U_x}{\partial z}   = \dfrac{\partial \,U_x}{\partial t} +  
    \left( \pmb{U}\cdot\nabla\right)\,U_x  \tag{75}
\]
The same can be developed to the other two coordinates which can be combined (in a vector form) as
\[
    \label{dif:eq:dUdt}
    \dfrac{d\,\pmb{U}}{dt} = \dfrac{\partial \,\pmb{U}}{\partial t} + \left( \pmb{U}\cdot\nabla\right)\,\pmb{U}  \tag{76}
\]
or in a more explicit form as
\[
    \label{dif:eq:dUxdtC}
    \dfrac{d\,\pmb{U}}{dt} = \overbrace{\dfrac{\partial \,\pmb{U}}{\partial t}}^{\text{ local acceleration } }   +   
    \overbrace{\pmb{U} \dfrac{\partial \,\pmb{U} }{\partial x} +  
    \pmb{U} \dfrac{\partial \,\pmb{U} }{\partial y} +  
    \pmb{U} \dfrac{\partial \,\pmb{U} }{\partial z} }^{\text { convective acceleration  } }   \tag{77}
\]
The time derivative referred in the literature as the local acceleration which vanishes when the flow is in a steady state. While the flow is in a steady state there is only convective acceleration of the flow. The flow in a nozzle is an example to flow at steady state but yet has acceleration which flow with a very low velocity can achieve a supersonic flow.

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.