Page 55 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

38 SLENDER STRUCTURES AND AXIAL FLOW

The vibration of the shells induces oscillatory flow in the fluid, and the task is to
determine the unsteady fluid loading on the shells, resulting thereby. By further assuming
that the amplitude of shell vibrations is small, it follows that all fluid velocities are also
small and hence governed by equations (2.71)-(2.73a,b). Because the shells are very long,
end effects are negligible; also, because the mode of deformation is such that motion is
essentially two-dimensional as outlined in the previous paragraph, motion-induced flow
variations in the direction of the long axis of the shells are negligible. Hence, in this
case, ap/ax = 0 and 4 = 4(r, 8, t); the analysis is therefore carried out in the plane of
Figure 2.7(a).
The solution to the fluid flow resulting from this motion may be obtained via equations
(2.73a,b), although (2.72) could be used equally well (Gibert 1988). The eigenmodes of
=
each shell are of the form (v, w)~ {v, sin ne, w, cos where the relation between
v, and w, is dependent on the shell equations used [e.g. Fliigge (1960)], which need not
concern us here; n is the circumferential wavenumber. The cross-sectional deformation
for n = 1-4 is shown in Figure 2.7(c).
Consider first the case where the outer shell is replaced by a rigid immobile cylinder
of inner radius R,, and let v; and wi be the displacement components of the inner shell.
Furthermore, consider oscillation in the nth mode, such that

vi(& t) = vni(r) sin ne, roi(8, t) = w,,;(t) cos ne, (2.1 13)

in which it is understood that v,,; and W,Ii are harmonic functions, e.g. w,;(t) =
-
w,; exp(if2t). The corresponding velocity potential is
4 = $(r, e) eiRf. (2.114)
The boundary conditions for the fluid are

(2.1 15a)

(2.115b)

The solution of the Laplace equation for 5 = $(r, e), after separation of variables, gives

- 00
4(r, 6) = {r"[A,, cos ne + B, sin ne] + r-"[C,, cos ne + D, sin ne]}; (2.1 16)
n=l
application of the boundary conditions yields

The pressure on the inner shell and the outer cylinder in the nth circumferential mode
may be determined through equation (2.73b), yielding