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 • r=0 • • v is at its maximum. At the edge • r=R • v=0

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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