Flow rate is calculated using only the parallel velocity

[davidbenari]

So flow rate is calculated using only the parallel velocity to the area vector. Why is this? How can I mathematically prove this? Namely, how do I prove any perpendicular component of the velocity vector is not contributing to any volume output? I know this is the result of the dot product; I want to know why the dot product is valid reasoning for this scenario.

thanks.

 

[Nathanael]

If the entire velocity were perpendicular, what would be the flow rate?

 

[b34n5]

Consider the definition of the volumetric flow rate

$\dot{V}$=$\frac{dV}{dt}$

say

$V=\vec{A} \cdot \vec{x}$

substitution yields

$\dot{V}=\vec{A}\cdot\frac{d\vec{x}}{dt}=\vec{A}\cdot\vec{v}$

where $\vec{v}=\frac{d\vec{x}}{dt}$ and $A$ materially conserved