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A Newton's Second Law Integral Form

  1. Sep 19, 2016 #1

    joshmccraney

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    Hi PF!

    I was reading my book and I understand the following $$\sum \vec{F} = \frac{\partial}{\partial t} \iiint_{CV} \rho \vec{u} dV +\iint_{CS} \rho \vec{u} ( \vec{u_{rel}} \cdot \hat{n}) dS$$ ##CV## is a control volume, ##CS## is control surface, ##u## is velocity, ##u_{rel}## is velocity relative to control volume, and the rest is self-spoken for. However, then the book states the following $$m_{CV} \frac{d}{dt} \vec{u_{CV}} = \frac{\partial}{\partial t} \iiint_{CV} \rho \vec{u_{CV}} dV +\iint_{CS} \rho \vec{u_{CV}} ( \vec{u_{rel}} \cdot \hat{n}) dS$$ where ##u_{CV}## is the velocity of the control volume. Any help in understanding why this is?
     
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  3. Sep 20, 2016 #2
    The first equation looks OK to me if ##u_{rel}## were replaced by just u. In the second equation, I'm not sure what ##u_{CV}## is supposed to represent.
     
  4. Sep 20, 2016 #3

    joshmccraney

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    ##\vec{u}## is the fluid velocity relative to an inertial frame, while ##\vec{u_{rel}} = \vec{u} − \vec{u_{CV}}## is the fluid velocity relative to the control volume. Then only the fluid velocity relative to the control volume boundary will induce a flux through this boundary. Does that look correct?
     
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