Skip to main content
Engineering LibreTexts

6.1.3: Momentum Governing Equation

  • Page ID
    713
    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    The right hand side, according Reynolds Transport Theorem (RTT), is

    \[ \dfrac{D}{Dt} \int_{sys} \rho \,\pmb{U} dV =
    \dfrac{t}{dt} \int_{c.v.} \rho \,\pmb{U} dV +
    \int_{c.v.} \rho \, \pmb{U} \pmb{U}_{rn} dA
    \label{mom:eq:RTT}
    \]

    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

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

    With external forces equation (10) is transformed to

    Integral Momentum Equation & External Forces

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

    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

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

    where \(\theta_x\) is the angle between \(\hat{n}\) and \(\hat{i}\) or (\(\hat{n} \cdot\hat{i}\)).

    Contributors and Attributions

    • 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.


    This page titled 6.1.3: Momentum Governing Equation is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


    This page titled 6.1.3: Momentum Governing Equation is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Genick Bar-Meir via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

    • Was this article helpful?