Page 330 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 311

On the other hand, as u2 > n2 the system can no longer be proved to be stable; indeed,
from linear theory and the finite dimensional analysis, we know that it is not.
The stability of the first pair of nontrivial equilibria, v: and v; = -w: is assessed in
the same way. A Lyapunov function related to equation (5.88) is now chosen, say for
position TJ:, namely
IW
I
{
Hb = il+b12 + ; lw”I2 - n I2 } + $d(t~:’, w’)~ + ;&!(UT’, w’)lw’I2
2
+ $dlw’14 + u { &Jlw12 + (w, W)} . (5.94)
Proceeding in a similar way, it is possible to prove that Hb > 0 and dHb/dt 5 0 in some
neighbourhood of v: (Holmes 1978); thus, this equilibrium point (and similarly 21;) is
locally asymptotically stable for all u2 > n2.
Similar forms as Hb but with h; = j2n2, L 2, instead of hl = n2 as in the foregoing,
j
are appropriate Lyapunov functions for the other points of equilibrium; but in this case
the term -{ lw”I2 - h;lw’12} appearing in the expression for dHb/dt cannot be proved to
be negative definite. These points are unstable; in fact, they are saddle points.
The foregoing considerations, though important in the overall dynamical analysis, do not
in themselves prove the existence or nonexistence of a limit cycle for u > 2n; indeed, a
limit cycle could exist around UT (or more likely around both UT and v;), but sufficiently far
removed from it, since stability has only been proved in some neighbourhood of UT; beyond
that, it is conceivable that trajectories, also repelled by vo, could be attracted by a stable
limit cycle. The proof of nonexistence is provided by Holmes (1978) following a method
developed by Ball (1973a,b) for the dynamic buckling of beams. This proof, outlined in
what follows, makes use of the concept of a ‘weak solution’, which is introduced next.
A weak solution is a mathematical concept in functional analysis and topology [see,
e.g. Oden (1979; Chapter 5) or, for a more accessible treatment, Curtain & Pritchard
( 1977)l. It signifies a generalized, nonclassical solution, e.g. one not satisfying the usual
differentiability conditions. This concept allows the transformation of the problem from
one involving differential operators, such as equation (5.82), to an equivalent problem
involving continuous linear functionals, as in (5.95). For our purposes here, this enables
us to reach some useful conclusions without first having to obtain a classical solution to
equation (5.82).
A weak solution to equation (5.82) is a solution Q($, t) which satisfies the equation

(V”, 4”) + (u2 - ;dlV”(V”, 4) + 2B”2u(li’, 4) + o(li, 4) + (ii, 4) = 0, (5.95)

where the inner product is taken with a sufficiently differentiable function 4 (Ball 1973a;
Holmes & Marsden 1978). In (5.95), one can replace 4 by li and integrate, thus obtaining
an ‘energy equation’,

(5.96)

in which (G’, li) = 0 has been utilized and CO is a constant. The similarity of (5.96) to
(5.89) is obvious. This may be re-written as

=
E(t) ili)12 + v(~) CO - Irj(t)12 ds, (5.97)