Fluid Flow in Pipes General Background

The frictional force per unit area or the frictional shear stress, t, is directly proportional to velocity, U, and inversely proportional to the spacing h yielding 

Invoking the friction laws. Equation (3.2), and noting that the u decreases with y results in 

upon integration, leads to the velocity distribution 

The integration constant can be obtained by invoking the no slip condition at the wall, that is, m = 0 at y = i for C = R /4 therefore resulting in a parabolic velocity distribution over the radius as 

The volume flow rate can be evaluated recognizing that the volume of the paraboloid of revolution is 1/2 times the base times the height which leads to the Hagen-Poiseuille equation of laminar flow through pipes (Schlichting, 1979) as follows  

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