# A Newton's Second Law Integral Form

1. Sep 19, 2016

### joshmccraney

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?

2. Sep 20, 2016

### Staff: Mentor

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.

3. Sep 20, 2016

### joshmccraney

$\vec{u}$ 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?