Page 447 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CURVED PIPES CONVEYING FLUID 419

around the centreline; the angle of rotation is denoted here by @ and, since 8 is here
assumed to be small (small deformation assumption), $ = + 7. Hence, the deformed
state of the pipe is determined by the generalized coordinates u, v, w and $.
The relative orientation of the two systems (x, y, z) and (xo, yo, 20) is given by the
following transformation (see Appendix J. 1):

where s is the curvilinear coordinate along the deformed centreline.
The components of curvature (around the XO- and yo-axes) and the twist (around the
zo-axis) in the undeformed state are given by

(6.2)

while those in the deformed state, assuming small deformations of the pipe, may be
expressed in terms of the generalized coordinates as follows (see Appendix J.2):

To complete the kinematic development, the velocity and acceleration of the internal
fluid is derived next. The fluid flow is assumed to be a plug flow (Chapter 3), the fluid
being essentially an infinitely flexible rod travelling through the pipe. The radius of
curvature is assumed to be very large compared with the pipe radius, and hence the
effects of secondary flow are neglected.
The displacement vector of the deformed centreline, expressed in the inertial reference
system that coincides with the system (xg, yo, ZO), is given by

r = ue,, +we, +we,,. (6.4)
By differentiating equation (6.4), the velocity and acceleration of the pipe may be
written as
au av aw
V - -e,, + - ey0 + -e,,
y- at at at
aZu aZv a2w
ap = - ex0 + -e + -e,.
at2 at2 yo at2
The absolute velocity of the internal fluid is