Page 317 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

298 SLENDER STRUCTURES AND AXIAL FLOW

pressure in the reservoir p* via
PO + ipUi = pgh* + p*, (5.67)
where the subscript 0 refers to quantities at the entrance to the pipe; p* and h* are defined
in Figure 5.2(a).
The same equations were derived by integration of the equations of motion of a continu-
ously flexible pipe conveying fluid by Rousselet (1975). These equations may be rendered
dimensionless through the following set of nondimensional parameters:

and it is also noted that pAf = M. These are more or less standard now [cf. Bajaj &
Sethna (1982a,b)], but they are different from Rousselet & Herrmann's.
The second set of equations given here are Bajaj & Sethna's (1982a,b), who considered
three-dimensional motions of the same system, Le. motions in both the y- and z-directions,
and a constant flow velocity [Figure 5.2(b)]. The Lagrangian procedure is utilized and
hence Benjamin's equation (3.10). The generalized coordinates are the end-displacements
of the two segments of the pipe, v1 and v1 + v2 in the y-direction and w1 and w1 + w2
in the z-direction. Hence, the kinetic and potential energies are given by

9 = i(m M)(l1 + 312)(w: + WT + b:) + t(m +M)/2(4 + w; + b;)
+ +
+ ~MU~(L~ i(m + ~)12(iili12 + ~ 1 + icli2)
12)
~
2
+MU(VIV2 + WlW2 + i(lU2) (5.69)
and

v=(m+Wg[(il1+12)(11 -~i)+$l2(12-~2)] +~(kl$~+k2&), (5.70)
where U: = 1: - (4 + w:) and ui = 1; - (v; + wi); $1 is the acute angle between the
upper pipe and the x-axis, while 42 that between the two pipes,

Furthermore, the position vector RL and the tangent vector r~, defined in Figure 3.l(d)
are given by
RL = (u1 + u2)i + (211 + w2)j + (WI + wdk,
(5.72)
ZL = (u2i + vd + w2k)/Z2,
where i, j, and k are the unit vectors along the x-, y- and z-axes, respectively.