A liquid flows through a flat surface with uniform vector velocity $\overrightarrow{v}$.
Let $\overrightarrow{n}$ an unit vector perpendicular to the plane.
Show that $\overrightarrow{v} \cdot \overrightarrow{n}$ is the volum of the liquid that passes through the unit surface of the plane in the unit of time.
Could you give me some hints how we could show this??
The envisaged situation is steady in time and linear in space, therefore easy to handle.
Imagine your plane is the $(x,y)$-plane with a square hole $[0,1]^2$ in it, but the hole is closed by a sluice valve. The normal to the plane is $n=(0,0,1)$. Assume the fluid is some sort of thick paint. Now open the valve for one second and close it again. During this second the paint will emerge with its velocity ${\bf v}$. The emerged paint will somehow try to keep its shape and in the end fill an oblique prism of a certain volume $V$. Now compute this volume.