Flow rate is calculated using only the parallel velocity

  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. Nathanael

    Nathanael 1,445
    Homework Helper

    If the entire velocity were perpendicular, what would be the flow rate?
     
  4. 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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