Page 392 - Introduction to Continuum Mechanics
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376 Plane-Poiseuille Flow
Thus,
and
Since v must be bounded in the flow region, the integration constant b must be zero. Now, the
nonslip condition on the cylindrical wall demands that
where d is the diameter of the pipe, thus
and
The above equation states that the velocity over the cross- section is distributed in the form of
a paraboloid of revolution.
The maximum velocity is (at r=0)
The mean velocity v is
and the volume rate of flow Q is
where
As in the case of plane Poiseuille flow, if the effect of gravity is included, the velocity profile
in the pipe remains the same as that given by Eq. (6.13.3), however, the driving force now is
