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Equation (6-11) is known as the Hagen-Poiseuille equation.

This result can also be derived by equating the shear stress for a Newtonian fluid, Eq. (6-9), to the expression obtained from the momentum balance for tube flow, Eq. (6-4), and integrating to obtain the velocity profile:

Inserting this into Eq. (6-6) and integrating over the tube cross section gives Eq. (6-11) for the volumetric flow rate.

Another approach is to use the Bernoulli equation [Eq. (6-2)] and Eq. (6-8) for the friction loss term ef. The integral in the latter equation is evaluated in a manner similar to that leading to Eq. (6-10) as follows. Eliminating ef between Eq. (6-8) and the Bernoulli equation [Eq. (6-2), i.e., pef = —AO] leads directly to

2nTw

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