Mechanics of Fluids

230 Laminar flow between solid boundaries
✷
Solution
or
(a) Since Q = qb, and (dp/dx)1 =−(dp/dx)2 we may write
� � � �
V Q 12µ V Q 12µ
− =− −
2 bh1
2 bh2
which may be written as
h 2 1
h 2 2
� �
V Q 12µ
−
2 bHh2 H2h2 � �
V Q 12µ
=− −
2 bh2
2
h2 2
Q = V (1 + H
2
2 )
(1 + H3 ) bHh2 = V (1 + H
2
2 )
(1 + H3 ) bh1 QED.
(b) Q = V (1 + H
2
2 )
(1 + H3 ) bh1
= 10 m · s−1
2
× (1 + 4)
= 139 × 10 −6 m 3 · s −1
(1 + 8) × 0.1 m × (0.5 × 10−3 m)
(c) The load F supported by the bearing is given by
� �
�
dp L L V
F =
b = 1 −
dx 1 2 2 2
(1 + H2 )
(1 + H3 �
12µ
) h2 L
1
2
4 b
= 10 m · s−1
�
�
(1 + 4) 12 × 870 kg · m−3 × (2 × 10−4 ) m2 · s−1 1 −
2 (1 + 8) (0.5 mm) 2 × (10−3 m/mm) 2
× (0.1 m)2
4
= 4640 N
× 0.1 m
6.7.3 Journal bearings
The basic wedge principle again applies in a journal bearing on a rotating
shaft. This can carry a transverse load only if the journal (i.e. the part of
the shaft within the bearing) and the bush (i.e. the lining of the bearing) are
not exactly coaxial. Under load, the journal (centre C in Fig. 6.22) takes up
an eccentric position in the bush (centre O). The distance CO between the
centres is known as the eccentricity, e, and the ratio of this to the average
clearance c (= R b − R) is termed the eccentricity ratio, ε. The thickness h
of the lubricant film (greatly exaggerated in the diagram: in practice c is of
order 0.001 R) varies with the angle θ which is measured, in the direction of
shaft rotation, from the line CO. (The magnitude and direction of CO will