Page 310 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 29 1

In his study, Bourri2res considers only the linear case. However, his approach, if
pursued far enough, would have led him eventually to expressions similar to those
derived in Section 5.2.3. In fact, the only difference between equations (5.47) and those
of Section 5.2.3 lies in the absence of gravity and time-varying flow terms, not considered
by Bourribres; this makes his work irreproachable.
The remaining task would be to combine all the five equations of (5.47) into one, and
to compare it with equation (5.39). This is not done here since it has effectively already
been done by Rousselet & Herrmann (1981), to be discussed next.
(b) Rousselet and Herrmann‘s work

Rousselet & Herrmann (1981) derived the equations of motion in two different ways:
by the force-balance method and the energy method. They obtain a set of equations,
fairly close to the one found in Section 5.2.3, but with some minor differences. Their first
method follows closely Bourrikres’ work. Two differences are simply due to the addition
of gravity forces and the assumption that unsteady flow velocity effects may be present.
Considering an element of the system (see Figure G.l), the application of Newton’s
law leads to
a a
-[(T - P) cos 01 - -(Q sin 0) + (m +M)g
as as
a2x M u2 d6
sin0 - 2M U - sin@,
= (m + M) - +M U cos0 - -
at2 R dt
a a (5.48)
-[(T - P)sin0] + -(Q cos@
as as
a2z d0 M u2
cos0 +M U sine,
= (m + M) - + 2M U - cos0 + -
at2 dt R
where R is the local radius of curvature. In these equations, (T - P) represents the tangen-
tial forces and Q the shear force, and sin8 and cos6 are related to x and z by
az ax
sin 0 = -, cos8 = -. (5.49)
as as
By means of the inextensibility condition and the definition of the curvature K, one can
also prove that
1 ax aZ2 1 az a2x
R as - a$’ R as - a$’ (5.50)
Substituting (5.49) and (5.50) into (5.48), one obtains

~((T-P):) -:(a:)+(m+M)g
as

aZx + a2 x a2x ax
at2 asat as2 as ’
= (m +M) - 2M U ~ +M U2 - +M U -
(5.51)
as ((T-P$) +; (e;)

a2z a2 z a2z az
= (m+M)- +2MU- +MU2 - +MU-.
at2 asat as2 as