Page 91 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

74 SLENDER STRUCTURES AND AXIAL FLOW

[Parenthetically, a more 'fluid mechanical' derivation given by Pa'idoussis & Issid
(1974) will be outlined here, in which an element of the pipe 6s is considered containing
fluid of volume W. The rate of change of momentum over W may be written as

(3.30)

where d@V is a small element within 6"lr. Then, by making the plug flow approximation,
the velocity V, may be approximated by (3.27). Therefore,

at dt

k. (3.31)

Hence, equation (3.30) yields

dM dU
-=M-&i+M (3.32)
dt dt
which corresponds to the acceleration as given by (3.28).]
A derivation in which the radial dimensions of the pipe are not ignored is given in
Section 4.2, but leads to the same form as above. Therefore, recalling that s 2: x, by using
(3.28) or (3.32) one obtains the first two of the following equations:

the last equation above is the lateral acceleration of the pipe and requires no explanation.
Hence, combining (3.19), (3.21), (3.22) and (3.33) one obtains
a
I---
E -+E
( *it ) a# ax
aw a2w
+c-++-=o. (3.34)
at at*
Also, adding equations (3.18) and (3.20) and using (3.33) yields

(3.35)
which integrated from x to L becomes
1
(T - pA)l - (T - PA) = - (M +m)g (L -x). (3.36)
x=L
If the flexible pipe discharges the fluid to atmosphere at x = L - the situation shown in
Figure 3.l(b,c) - T, which is then entirely due to fluid friction, is zero at x = L; unless
there is an externally applied tension, denoted by T - as could be the case for the system
of Figure 3.l(a). The pressure, p. at x = L will also be zero, unless the pipe does not