Flow rate is calculated using only the parallel velocity

  1. Jul 25, 2014 #1
    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.
     
  2. jcsd
  3. Jul 25, 2014 #2

    Nathanael

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    If the entire velocity were perpendicular, what would be the flow rate?
     
  4. Jul 26, 2014 #3
    Consider the definition of the volumetric flow rate

    [itex]\dot{V}[/itex]=[itex]\frac{dV}{dt}[/itex]

    say

    [itex]V=\vec{A} \cdot \vec{x}[/itex]

    substitution yields

    [itex]\dot{V}=\vec{A}\cdot\frac{d\vec{x}}{dt}=\vec{A}\cdot\vec{v}[/itex]

    where [itex]\vec{v}=\frac{d\vec{x}}{dt}[/itex] and [itex]A[/itex] materially conserved
     
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