Newton's Second Law Integral Form

[member 428835]

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?

 

[Chestermiller]

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.

 

[member 428835]

$\vec{u}$ 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?