Page 311 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

292 SLENDER STRUCTURES AND AXIAL FLOW

In this form, the similarity with BourriCres’ equations is self-evident. Note that K and the
condition of inextensibility have already been used implicitly. At this point, Rousselet &
Herrmann proceed to reduce this set of equations into one. With the different relationships
defined in Rousselet (1975), and after some manipulation, the nondimensional equation
is obtained, which differs from (5.39) only in two nonlinear terms involving the unsteady
velocity; they arise from an error in the use of the following relationship:

This relationship is exact, but in the order analysis, if F is of order 0, then tan 8 must be
approximated to the third order; this is not done by Rousselet & Herrmann. As explained
in Section 5.2.1, this relationship [derived in Section 5.2.3, equation (5.24)] has to be
rigorous to order 0(c4).
Rousselet & Herrmann also consider the effects of fluid friction or of the related pressure
drop, and derive a flow equation,

L
Po - aMU2 + 1 (M g x‘ - M U) ds - I‘ M(X x’ + 2 2‘) ds = 0, (5.53)

where PO is the compressive force acting on the fluid cross-section at s = 0, and aMU2
is the sum of the friction forces between the fluid and the pipe (a is a constant which
depends on the roughness of the pipe). The two partial differential equations are coupled
through the nonlinear terms. Thus, instead of considering the flow velocity as constant,
the upstream pressure (in a large reservoir) is assumed constant instead, as first proposed
by Roth (1964).

(cl Sethna, Bajaj and Lundgren’s work
Lundgren, Sethna & Bajaj (1979) and Bajaj et al. (1980) derived equations of motion by
the Newtonian (force balance) method. The assumptions made are the same as in other
work, but, from a mathematical point of view, every effort has been made to be as rigorous
as possible. Their equations appear to be exact. They use the condition of inextensibility
and the exact expression for curvature; in their derivation, all the nonlinearities come
from the terms (To - P) and EZ K’.
Lundgren et al. stopped their derivation at an early stage, without taking further advan-
tage of the inextensibility condition. In their subsequent paper (Bajaj et al. 1980), some
nonlinear terms are apparently missing, especially nonlinear velocity-dependent terms. In
the form of an integrodifferential set of equations and neglecting, for the moment, the
unsteady flow velocity, one may read [equation (5) in Bajaj et at. (1980)l

EZZ”” + 2M UZ’ + M U2 z’’ + (rn + M)z = NL, (5)’ + (2)’ = 1, (5.54)

where
a a L
+
NL = - 4 EZ - [~’(x’’~ z’I2)] - (rn + M) - (z’ 1 (x’ X + Z’ i) ds) .
as as