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6.1.3: Momentum Governing Equation

The right hand side, according Reynolds Transport Theorem (RTT), is
\[
    \dfrac{D}{Dt} \int_{sys} \rho \,\pmb{U} dV =
    \dfrac{t}{dt} \int_{c.v.} \rho \,\pmb{U} dV +  
    \int_{c.v.} \rho \, \pmb{U} \pmb{U}_{rn} dA   
    \label{mom:eq:RTT}  \tag{9}
\]
The liquid velocity, \(\pmb{U}\), is measured in the frame of reference and \(\pmb{U}_{rn}\) is the liquid relative velocity to boundary of the control volume measured in the same frame of reference. Thus, the general form of the momentum equation without the external forces is

Integral Momentum Equation

\[
\label{mom:eq:gRTT}
    \begin{array}{rl}
     \int_{c.v.} \pmb{g} \, \rho\, dV -  \int _{c.v.}\pmb{P}\,dA +  \int _{c.v.} \boldsymbol{\tau\,\cdot}\,\pmb{dA} \\    
     = \dfrac{t}{dt} \int_{c.v.} \rho\, \pmb{U} dV + & \displaystyle
   \int_{c.v.} \rho \,\pmb{U}\, \pmb{U_{rn}}\, dV
    \end{array}  \tag{10}
\]

With external forces equation (10) is transformed to

Integral Momentum Equation & External Forces

\[
\label{mom:eq:gov}
\begin{array}[c]{ll}
     \sum\pmb{F}_{ext} + \int_{c.v.} \pmb{g} \,\rho\, dV - &  
            \int_{c.v.}\pmb{P}\cdot \pmb{dA} + \int_{c.v.} \boldsymbol{\tau}\cdot \pmb{dA} = \\  
    & \dfrac{t}{dt} \int_{c.v.} \rho\, \pmb{U} dV + \int_{c.v.} \rho\, \pmb{U} \,\pmb{U_{rn}} dV
\end{array} \tag{11}
\]

The external forces, Fext, are the forces resulting from support of the control volume by non–fluid elements. These external forces are commonly associated with pipe, ducts, supporting solid structures, friction (non-fluid), etc. Equation (11) is a vector equation which can be broken into its three components. In Cartesian coordinate, for example in the x coordinate, the components are

\[
    \label{mom:eq:govX}
    \sum F_x + \int_{c.v.} \left(\pmb{g}\cdot \hat{i}\right) \,\rho\, dV  \int_{c.v.} \pmb{P}\cos\theta_x\, dA +
    \int _{c.v.} \boldsymbol{\tau}_x \cdot \pmb{dA} = \\
   \dfrac{t}{dt} \int_{c.v.} \rho\,\pmb{U}_x\,dV +
   \int_{c.v.} \rho\,\pmb{U}_x\cdot\pmb{U}_{rn} dA  \tag{12}
\]
where \(\theta_x\) is the angle between \(\hat{n}\) and \(\hat{i}\) or (\(\hat{n} \cdot\hat{i}\)).

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.