Why Does Stoke's Stream Function Represent Fluid Flow?

In summary, the Stoke stream function is a defined function Ψ where the quantity of fluid that crosses the surface of revolution formed by the vector OP is equal to 2pc when the coordinate ω is varied through 2π while r and θ are held fixed. This definition is applied to two points P and P' close together, and when the line element PP' is rotated about the reference axis, the resulting surface has a flow rate of 2πdΨ crossing it per unit time.
  • #1
chriswong
2
0
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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  • #2
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?
 
  • #3
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
 

FAQ: Why Does Stoke's Stream Function Represent Fluid Flow?

What is Stoke's stream function?

Stoke's stream function is a mathematical function used to describe the motion of a fluid in two dimensions. It is defined as the stream function divided by the fluid density.

How is Stoke's stream function related to fluid flow?

Stoke's stream function is related to fluid flow because it represents the velocity potential of a fluid. In other words, it describes the motion of a fluid without considering its density or viscosity.

Why is Stoke's stream function useful in fluid dynamics?

Stoke's stream function is useful in fluid dynamics because it simplifies the Navier-Stokes equations, making them easier to solve. It also helps to visualize and analyze fluid flow patterns.

How is Stoke's stream function different from Bernoulli's equation?

Stoke's stream function and Bernoulli's equation are different concepts. While Stoke's stream function describes the motion of a fluid in two dimensions, Bernoulli's equation describes the relationship between pressure, velocity, and height in a fluid.

What are the limitations of Stoke's stream function?

Stoke's stream function is limited to two-dimensional, incompressible, and irrotational flow. It also cannot be used to describe flows with significant turbulence or vorticity. Additionally, it does not account for boundary conditions or external forces acting on the fluid.

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