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An introduction to Electricity and Strength of Materials with Peter Eyland
Poiseuille's Law Derivation
Consider a solid cylinder of fluid, of radius r inside a hollow cylindrical pipe of radius R.
The driving force on the cylinder due to the pressure difference is: The viscous drag force opposing motion depends on the surface area of the cylinder (length L and radius r): |
In an equilibrium condition of constant speed, where the net force goes to zero.
We know empirically that the velocity gradient should look like this: |
At the centre At the edge |
From the velocity gradient equation above, and using the empirical velocity gradient limits, an integration can be made to get an expression for the velocity.
Which has a parabolic form as expected.
Now the equation of continuity giving the volume flux for a variable speed is:
Substituting the velocity profile equation and the surface area of the moving cylinder:
Poiseuille's equation
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