Deriving Half-Max Velocity in Pipe Flow

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SUMMARY

The average velocity in pipe flow is definitively half of the maximum velocity, derived from the Navier-Stokes equation in cylindrical coordinates. This conclusion is based on key assumptions: the flow is incompressible, the pipe pressure decreases linearly, the flow rate is finite, and the flow is unidirectional with zero velocity at the pipe walls. These principles are essential for understanding fluid dynamics in cylindrical systems.

PREREQUISITES
  • Navier-Stokes equations in cylindrical coordinates
  • Incompressible fluid dynamics
  • Linear pressure gradient analysis
  • Boundary conditions in fluid flow
NEXT STEPS
  • Study the derivation of the Navier-Stokes equation for cylindrical coordinates
  • Explore incompressible flow assumptions in fluid mechanics
  • Investigate linear pressure gradient effects on flow velocity
  • Learn about boundary layer theory and its implications in pipe flow
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Fluid dynamics engineers, mechanical engineers, and students studying advanced fluid mechanics will benefit from this discussion on deriving velocity profiles in pipe flow.

jonnygeordie
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Could someone help me with this. In pipe flow show that the average velocity is half of the maximum velocity. I have it written in my notes that this is true, but not how to derive it.
 
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You can get the answer if you solve Navier-Stokes equation in cylindrical coordinates with the following assumptions:

1) Flow is incompressible.
2) Pipe pressure decreases linearly.
3) Flow rate is finite.
4) Flow rate is static, hence, the fluid only flows in one particular direction (you can guess which one based on the symmetry of a cylindrical pipe).
5) Flow rate at the sides of the pipe = 0.
 

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