Mechanics of Fluids

compensate for the reduction in velocity experienced by the fluid near the
walls, that near the centre must be accelerated. Not until the position C
is reached, however, is the full parabolic distribution established. It is this
final, fully developed form to which the equations we have derived apply.
The parabolic velocity profile is theoretically reached only after an infinite
distance; but it is usual to regard the so-called entry length as the distance
along the pipe to the point at which the maximum velocity is only 1% different
from the final value. The way in which the flow thus settles down in the
entry length is discussed in greater detail in Section 7.9. It may, however, be
said here that except in short pipes (i.e. with a length up to about 200 times
the diameter) the different conditions at the inlet to the pipe do not significantly
affect the change of pressure over the whole length. No significant
deviation from the parabolic profile occurs at the outlet end.
From eqn 6.6 the mean velocity of the fluid may be calculated.
Mean velocity = Q
Area =−πR4
8µ
dp∗ �
πR
dx
2 =− R2 dp
8µ
∗
dx
Steady laminar flow in circular pipes 195
(6.9)
From eqn 6.4 we see that the maximum velocity occurs in the centre of the
pipe, where r = 0.
∴ umax =− dp∗
�
R
dx
2
�
(6.10)
4µ
Thus, for fully developed laminar flow, the mean velocity is 1
2 umax.
The Hagen–Poiseuille formula has been amply verified by experiment and
it is interesting to note that this agreement between theory and experimental
results is perhaps the principal justification for the assumption that a fluid
continuum does not slip past a solid boundary. The integration constant in
eqn 6.3 had to be determined from the conditions at the wall: if a velocity
other than zero had been assumed, say u0, the value of Q given by eqn 6.6
would be increased by an amount πR 2 u0. It is only when u0 is zero that
agreement between theory and experiment is obtained. Newton’s hypothesis
(eqn 6.1) is also of course vindicated by the agreement with experimental
results.
Another result which the Hagen–Poiseuille formula verifies is that in laminar
flow the drop in piezometric pressure is proportional directly to the
mean velocity. Moreover, the formula is completely determined by our analysis
and does not involve any additional coefficients that have to be obtained
experimentally – or estimated – for a particular pipe. Thus we should expect
moderate roughness of the walls of the pipe not to affect laminar flow, and
this conclusion too is confirmed by experiment.
The Hagen–Poiseuille formula was developed on the assumption that
the centre-line of the pipe was straight. Slight curvature of the centre-line,
in other words, a radius of curvature large compared with the radius of
the pipe, does not appreciably affect the flow through the pipe. For smaller
radii of curvature, however, the flow is not accurately described by the
Hagen–Poiseuille formula.