Mechanics of Fluids

in eqn 6.24 we obtain
�
D
Vp
2
�
+ c = c3
�
F + W
6µl πD2 �
− ρgl
/4
Steady laminar flow between moving planes 209
(6.26)
The clearance c is normally small compared with the radius of the piston D/2,
so the left-hand side of eqn 6.26 may be simplified to VpD/2. Rearrangement
then gives
F + W − ρglπD 2 /4 = 3
4 πµl
� �3 D
Vp
(6.27)
c
The term ρglπD 2 /4 represents the buoyancy of the piston, and in some
instances this is negligible compared with F. This, however, is not always so
and the buoyancy term should not be omitted without investigation.
For upward movement of the piston the signs of F and Vp are of course
changed.
Equation 6.27 is a formula commonly used, but even for steady conditions
it is only approximate. The clearance c has been assumed small in comparison
with the radius of the piston; the shear force on the piston has been
neglected; the circumferences of piston and cylinder are not exactly equal;
end effects have been neglected. It may be shown, however, that the accuracy
is much improved if the mean diameter 2(D/2 + c/2) = (D + c) is used in
eqn 6.27 in place of the piston diameter D.
In a dashpot of this kind the piston has to be maintained concentric with
the cylinder by external means. If it is free to move laterally the piston tends
to move to one side of the cylinder and, having once touched the side of the
cylinder, is reluctant to leave it. Under such conditions the relation between
the load on the piston and the rate at which oil flows through the clearance
space is drastically altered and the effectiveness of the dashpot in restraining
the movement of the piston is greatly reduced. Three buttons, equally spaced
round the circumference of the piston, are sometimes used to maintain its
concentricity with the cylinder.
Many other forms of dashpot are in use, but an account of these is beyond
the scope of this book.
Example 6.4 A simple dashpot consists of a piston of diameter
50 mm and length 130 mm positioned concentrically in a cylinder of
50.4 mm diameter. If the dashpot contains oil of specific gravity 0.87
and kinematic viscosity 10 −4 m 2 · s −1 , determine the velocity of the
dashpot if the difference in pressures �p ∗ is 1.4 MPa.
Solution
Clearance: c = (50.4 − 50)/2 = 0.2 mm
Dynamic viscosity:
µ = ρν = (0.87 × 10 3 ) kg · m −3 × 10 −4 m 2 · s −1
= 0.087 kg · m −1 · s −1