Page 328 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 309

‘

Sinks
Saddle u +
7

Figure 5.8 A qualitative picture of the occurrence of divergence via a pitchfork bifurcation (a) for
a system with symmetry (EO = 0), and (b) when the symmetry is broken (E~ # 0). Adapted from
Holmes (1 977).

In this case the system is not discretized. As known from linear analysis of equa-
tion (5.82), Section 3.4.1 and equation (3.90a) in particular, nontrivial equilibria for the
pinned-pinned pipe arise for u; = jx. Hence, from (5.82) it is seen that any nontrivial
equilibrium point, u; # 0, is an eigenfunction satisfying

TJy + hJTJ’jl = 0, (5.87)

where lv>l denotes the norm; from this, it is clear that u, = 7t, as found before. It is clear
that no nontrivial equilibria exist for u2 5 k1 = n2, where hl is the first eigenvalue. lf u2 >
hl , however, there are 2r distinct nontrivial equilibria occurring in pairs, corresponding to
the r eigenvalues h, < u2, the stability of which was examined by Holmes (1977, 1978).
To study the stability of a particular equilibrium position u, [where it is understood that
normally there exist a uj’ and a uJ because of (5.87)], consider a perturbation w about
u; 5;. substitute 5; + w in (5.82) and then subtract the equation in Z;, thus obtaining
the equation for w:
w’//’ + (U* - ;,,,;,2)w” - .e(V>, w’),; + 2puW’

+ ow + w - $4{2(E>, w/)w’/ + ,wyq + ,w’(2w/’} = 0, (5.88)

1
where (a, b) = so a(c)b(c) de is the inner product, and where a = 0 is assumed without
loss of generality.
The stability of 5j is studied via a generalization of the Lyapunov second (direct)
method to partial differential equations (Movchan 1959; Parks 1967; Holmes & Marsden
1978) - see also Appendix F. 1.