18 FLUIDITY AND PLASTICITY

reasoning of Hagenbach. He showed that Hagenbach should
have reached a value which is identical with that given by the
others. The correction may be simply deduced as follows:

The kinetic energy of the fluid passing any cross-section of a
cylindrical tube per unit of time is

' 2 64Z V

where p is the density of the fluid. Since the volume of fluid
passing any cross-section per unit of time is TJK2I, the energy sup-
plied in producing the flow is irR^Ipg, hence, the energy converted
into heat within the tube must be irR*I (pg - p/2). From Eqs, (2)
and (6) we have

Thus taking into account the loss in kinetic energy, the formula
of Poiseuille becomes

mpV .-,

~wr -
in which m is a constant which according to the above derivation
is equal to unity. The formula of Hagenbach differed only in.
that the constant m is equal to 2~^ or 0.7938.
It is of historical interest in this connection to note that Ber-
nouilli's assumption that all of the particles flowing through a pipe
have the same velocity, leads one to the conclusion that the
kinetic energy of the fluid passing any cross-section per unit of
7T.B2/3
time is exactly one-half of that given above or — ^ — > and the
value of m in that case would be only 0.50. This value was actu-
ally suggested by Reynolds (1883) when the openings of the tubes
were rounded or trumpet-shaped, but m = 0.752 when the ends
are cylindrical. It may be added that Hagenbach compared his
value of 0.7938 with the observed values obtained by various
hydraulicians working with wide tubes, Hagen 0.76, Weisbach
0.815, Zeuner 0.80885, Morin 0.82, and Bossut 0.807, and he
found that his value was near the mean. But account should
have been taken of the fact that their results apply to the tur-
bulent regime, but not necessarily to the regime of linear flow.
Boussinesq (1891) while admitting the correctness of the