Page 96 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 79

Hence, utilizing (3.47), (3.49) and (3.50) leads to the following form for the virtual
work equation:

p(u - 6r)d'V = 0. (3.51)

This, integrated over time from tl to t2, at which limits 6r = 0 again, gives Hamilton's
principle for the open system,

(3.52)

(3.53)

with Z0 = 7;, - K, being the Lagrangian of the open system.
This is next applied to the case of a cantilevered pipe conveying fluid. For simplicity,
the case of no dissipation and a constant flow velocity U is considered. Moreover, it is
presumed that the only forces involved in 6W are associated with the pressure p, measured
above the ambient of the surrounding medium; hence,
JH=- JJ + /L,+ye(tl (3.54)
p(u - Sr)(V - u) . ndY,
:fc (r )+Yl +4 (11 p(6r *
where Yc(t) is the surface covered by the pipe wall, and Y, and $(t) are the inlet and
exit open surfaces for the fluid. Next, it is presumed that any virtual displacement of
the pipe does not induce a virtual displacement of the fluid relative to the pipe. Thus,
virtual displacements of the fluid relative to the pipe are independent of those of the pipe.
Hence, since the fluid is incompressible, there can be no virtual change in the volume of
the system, and expression (3.54) simplifies to
p(u 6r)(V - u) e n dY. (3.55)
6~ = - //X+$(r, '(6' * n)dY + //x+z(tl
Now, if the fluid entrance conditions are prescribed and constant, the integrals over are
zero. Furthermore, the first integral over Ye(t) is zero since at the outlet p = 0. Hence,
the only part remaining is

(3.56)

in obtaining which u = r + Ut [Figure 3.8(a)], (u - V) n = U at $(t) and M = pA
have been utilized, A being the open (flow) area. Hence, Hamilton's principle for this
system becomes

which is identical to that obtained by Benjamin (1961a).+

'In Benjamin's derivation, as in Figure 3.l(d), RL is measured from the (x, z) = (L, 0) position, whereas
here rL = Li + RL is measured from the origin; however, as i~ = RL and SrL = SRL, the two expressions are
fully equivalent.