Page 99 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

82 SLENDER STRUCTURES AND AXIAL FLOW

the first part of which becomes MU2 Jy w; BWL, because of the boundary conditions, and
cancels the second term of the second integral of (3.59). The expression above also makes
it clear that the centrifugal term M U2wl1 does not arise from the hnetic energy, as might
have been supposed, but from the second term in the statement of Hamilton’s principle,
equation (3.57). The second item concerns the term 2Uu in Tf, in equations (3.61). Once
the variation is taken, this leads to Jt 2U 6u 1: = 0.
For arbitrary variations Sw and with s 2: x, the term within the curly brackets in
equation (3.64) is the desired equation of motion. It is the same as (3.38), but with
E* = 0, c = 0 and dU/dt = 0, in accordance with the assumptions made here.
Consider next a pipe with clamped ends, but allowing sliding at the downstream one
[Figure 3.l(b)]. In this case the second integral of (3.59) is zero, but UL in the first
integral is not, and it is again this term rather than the kinetic energy that is responsible
for the centrifugal force term in the equation of motion. Everything else remains the
same, including the inextensibility condition. After considerable manipulation, the same
equation of motion is obtained - but only if UL is not ignored, whereas it was in Housner’s
derivation.
Consider finally the case of fully clamped ends - not allowing any sliding at x = L.
As pointed out by McIver, in this case there is no motion possible at x = L, Le. 6x~ =
SZL = 0; that is, the ‘contraction’ in the sense used by Benjamin and defined for in-
L 1 12
extensible pipes by UL = - so ?w ds is zero in equation (3.59), and hence so is u at any
location s along the deformed pipe. In fact, for lateral deformation to occur, there will be
some stretching of the pipe as shown in Figure 3.8(c), which results in its cross-sectional
shrinking. Thus, the element of the pipe 6s is stretched to 6s(l + ;wf2) and the flow
velocity relative to the pipe through the narrower flow passage, A(l - ;w’~), is increased
to U(l + id2) for continuity at each location s; hence, the x-component of the flow
velocity is [U(1 + $wf2)] (1 - ;wf2) 2: U. Therefore, in this case, at least approximately
to S(2),
L
Tf = [(W + uw’)2 + U2] dx,

as utilized by Housner - correct, but without the benefit of the refined arguments leading
to it. With this expression, i.e. with iC = 0, and with UL = 0, Hamilton’s principle (3.59)
yields the very same equation of motion as for the sliding end and the cantilevered
case - at least to the linear limit. In contrast to the previous two cases, here the centrifugal
force term in the equation of motion arises from the kinetic energy.

3.3.4 A comment on frictional forces

A remarkable feature of equations (3.38) and (3.1) is the total absence of fluid-frictional
effects, which at first sight might appear to be an idealization. However, within the
context of the other approximations implicit in this linearized equation, it may rigorously
be demonstrated that fluid-frictional effects play no role in the dynamics of the system,
a fact first shown by Benjamin (1961a,b). Consider once more the balance of forces in
the axial direction of elements of the fluid and the pipe, i.e. equations (3.18) and (3.20)
for the case where dU/dt = 0 and gravity is inoperative (Le. for motions in a horizontal