Page 500 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

470 SLENDER STRUCTURES AND AXIAL FLOW

Benjamin (1963) recognized this and so considered next a system which is unbounded
in the flow direction, x, and which is disturbed by a sinusoidal wave travelling in that
direction - see also Ye0 & Dowling (1987). The motion within an interval of x may
be considered to comprise two modes q = qI(t) sin ax and q = q2(t) cos ax, in which
q1 and q2 are oscillations in quadrature. Through the action of the flow there may be
coupling between these two modes, and so q is generally taken to be complex, on the
understanding that %e(q exp(iax)) describes the physical disturbance. The equation of
motion is still of the form of (C.l), but now we insist that M < 0, so that m - M > 0
always; k and K are real, but C can now be complex, C = C,. + ici (cf. the Coriolis term
in the pipe problem).
Corresponding to (C.2), the energy transfer W averaged over x is given by the real
part of the integral of kpq, where p is the complex conjugate of Q. The term icq in
Q makes no contribution to W, and so the expressions for % and E in (C.4)-(C.6) are as
before, except that C is now replaced by C,. Thus, dE/dt takes the sign of C,. - c.
Representing the solutions of (C.l) by q exp(-ivt), where v is complex, we get

2i(c - Cr)
V= ((2.7)
2(m - M) c,

where R = [4(m - M)(k - K) + (c - Cr)’]/G2. It is recognized that instability is indi-
cated by 4m(v) > 0, where
r 1 I?-

Since m - M > 0, R may be positive or negative, depending on whether k > K or other-
wise. The following three cases may be distinguished. (a) When R > 0, 4m(v) > 0 for
both solutions if C,. > c, which from (C.6) corresponds to dE/dt > 0 and hence to class
B instability, i.e. to case (i) in the foregoing. (b) When -1 < R < 0, one solution is again
of class B, but the other one is of class A, being unstable for c > C,.. (c) The case of
R < -1 corresponds to class C instability, not considered here. Therefore, it is clear that
for -1 < R < 0 and c > C,. the physical system obeys the arguments given in (ii) in the
foregoing and is thus destabilized by damping.