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Poiseuille Flow
Steady viscous fluid flow driven by an effective pressure gradient established between the two ends of a long straight pipe of uniform
circular crosssection is generally known as Poiseuille flow, since it was first studied experimentally by
J.L.M. Poiseuille in 1838. Suppose that the pipe is of radius . Let us adopt cylindrical coordinates whose symmetry axis coincides with that of the
pipe. Thus, measures distance along the pipe, corresponds to the center of the pipe, and corresponds to the pipe wall.
Suppose that

(982) 
is the uniform effective pressure gradient along the pipe, and

(983) 
the time independent velocity profile driven by this gradient. It follows from Section 2.19 that
and .
Hence, (961) reduces to

(984) 
Taking the component of this equation, we obtain

(985) 
where use has been made of (155). The most general solution of the
above equation is

(986) 
where and are arbitrary constants. The physical constraints are that the flow velocity is
nonsingular at the center of the pipe (which implies that ), and is zero at the edge of the pipe [i.e., ], in
accordance with the no slip condition. Thus, we obtain

(987) 
The volume flux of fluid down the pipe is

(988) 
According to the above analysis, the quantity should be directly proportional to the effective
pressure gradient along the pipe. The accuracy with which experimental observations
show that this is indeed the case (at relatively low Reynolds number) is strong evidence in favor of the assumptions that there is no slip at the pipe
walls, and that the flow is nonturbulent. In fact, the result (988), which is known as Poiseuille's law, is valid experimentally provided the Reynolds number of the flow,
, remains less than about
. Here,
is the mean flow speed. On the other hand, if the
Reynolds number exceeds the critical value
then the flow in the pipe becomes turbulent, and Poiseuille's law breaks down.
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Up: Incompressible Viscous Flow
Previous: Flow Down an Inclined
Richard Fitzpatrick
20120427