Mechanics of Fluids

The torque T on the cylinder of radius b
= stress × area × radius
= (µr∂ω/∂r) r=b × 2πbh × b
where h represents the height of the cylinder in contact with the fluid, and
end effects are assumed negligible.
∴ T = 2πµb 3 � �
∂ω
h = 2πµb
∂r
3 � �
A
h (from eqn 6.30)
r=b
µb 3
= 2πhA = 4πha2b2 µ�
a2 = kµ� (6.32)
− b2 where k is a constant for any given apparatus.
It may readily be shown that the torque on the rotating cylinder is the
same. Equation 6.32 applies whether the inner cylinder is stationary while
the outer one rotates, or the outer cylinder is stationary while the inner one
rotates.
In the derivation of eqn 6.32 it was assumed that h was large enough to
render negligible any special effects at the ends. In practice, however, the
cylinders are of moderate length and some account must therefore be taken
of effects produced by the ends. The end effects are very similar to that of
an additional length of the cylinder in contact with the fluid, in other words
(h + l) rather than h would arise in eqn 6.32. By using two, or preferably
more, values of the liquid depth, and therefore of h, simultaneous equations
are obtained from which l may be eliminated.
If the radii of the two cylinders are closely similar eqn 6.32 may be slightly
simplified. Putting the annular clearance c = a − b and (a + b) = 2 × (mean
radius) = mean diameter = D, we obtain
a
T = 4πhµ�
2b2 � �
D + c
2� �
D − c
2�
= 4πhµ�
Dc
(a + b)(a − b) 2 2
= πhµ�
4Dc (D2 − c 2 ) 2
Then neglecting c 2 in comparison with D 2 , we have
T = πhµ�D3
4c
The measurement of viscosity 217
(6.33)
A simple laboratory viscometer of this type is that devised by G. F. C. Searle
(1864–1954).
It is important to remember that neither eqn 6.32 nor eqn 6.33 is applicable
if the motion of the fluid is turbulent. Moreover, the assumption has been
made throughout that the two cylinders are concentric. The formulae are
therefore not applicable (except in very rare instances) to journal bearings,
for, as we shall see in Section 6.7.3, a journal bearing supports a load only
if the journal and bearing are not concentric.