Why Does Stoke's Stream Function Represent Fluid Flow?

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The discussion centers on the Stoke stream function and its representation of fluid flow. It explains that the function Ψ is defined in relation to the position vector OP, which, when rotated around a reference axis, determines the quantity of fluid crossing a surface of revolution. Specifically, it states that for closely spaced points P and P', the flow rate across the surface is quantified as 2πdΨ per unit time. The conversation highlights the importance of understanding the vector relationships and flow characteristics in this context. Overall, the thread seeks clarity on the mathematical representation of fluid dynamics using the Stoke stream function.
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Summary:: Stoke stream function

[Mentor Note -- Thread moved from the technical forums, so no Homework Template is shown]

1622692859148.png

Why the quantity of fluid that crosses the surface of revolution formed by the vector OP is
1622692955444.png
?
 
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What vector? There appears to be several here (or else the picture without context is just confusing to me). Have you tried to work this out on your own, and if so, what did you try so far?
 
Sorry about that. Here are more information.
"Let a function Ψ be defined such that if the position vector OP is rotated around the reference axis, that is, if the coordinate ω is varied through 2π while r and θ are held fixed, the quantity of fluid that crosses the surface of revolution formed by the vector OP will be 2pc. Now apply this definition to two points P and P' that are close together, as shown in the Figure. Then if the line element PP' is rotated about the reference axis, the resulting surface will have a quantity of fluid 2πdΨ crossing it per unit time. "

I am wondering why the flow rate across a unit area of the surface is
1622698976684.png
 

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