Page 217 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 199

coordinate along a strcamtube off the centreline (and should not be confused with the s
used in Sections 3.3.1 and 3.3.2). Hence, the sum of (4.2) and (4.3) gives

the intermediate result is obtained with the aid of Figure 4.l(c), while the last step is
reached through neglect of second-order terms.
Since llrll is small and arel is negligible, the last integral of (4.1) may be approximated
as follows:

in which it is recalled that w is the vector displacement of the pipe centreline in the
y-direction. The second term in (4.5) is obtained through the following sequence of
operations: 23 x W; = 23 x Ui(l + Q/Ui) 2: 23 x Ui = 2 [-(a2w/axat)k] x(U,i) =
2Ui(a2w/ax at)j = 2Ui(a2w/ax at), where {i, j, k) are unit vectors associated with {t, q, 0.
Throughout, the small inclination of the (6, q}-plane vis-&vis the (x, y}-plane is utilized,
subject to order-of-magnitude constraints. Hence, combining (4.4) and (4.5), the rate of
change of fluid momentum is

(4.6)

which yields components per unit length in the x- and y-direction, respectively equal to

d Ui +2Ui - ui
a2w
dx ax at
piAiUi ~ and piAi + (4.7)
The second expression may be written in the compact form piAi%’w, where 9 = [(a/%) +
u,(a/ax)], and

92w = 9[%w] = (4.8)
ax at
It is instructive to note that there are no terms involving dAi/dx in (4.7), as there would
have been if the lateral momentum change had erroneously been evaluated by a simplistic
application of the formula [(slat) + u,(a/ax)]{piAi[(aw/at) + Vi(aW/ax)l)!
Now, the next steps in the derivation of the equation of motion may be taken. Working
in a similar way as in Section 3.3.2 (cf. Figure 3.6) by considering an element 8c of the
pipe [Figure 4.2(a)], force balances in the x- and y-direction and a moment balance yield

(4.9b)

(4.9c)