Page 453 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CURVED PIPES CONVEYING FLUID 425

(6.37)
at2

(6.38)

The gravity terms in equations (6.35)-(6.38) are given by

(6.39)

and similar ones for the yo and zo components; axe, ay0 and azo are the direction cosines
of the gravity vector with respect to (XO, yo, ZO), and pe is the density of the external fluid.
These equations are, of course, coupled but, similarly to the shell equations (Chapter 7),
may each be identified as being principally related to motion in one particular direction;
thus, the first is related to in-plane deformations, the second to out-of-plane deformations,
the third to deformations along the pipe, and the last to twist of the pipe. Hence, in-
plane motions are governed by equations (6.35) and (6.37), and out-of-plane motions by
equations (6.36) and (6.38). Note that, if the radius of curvature R, is made equal to
infinity, the curved pipe becomes a straight pipe. Moreover, if the pipe is vertical and the
axial motion is ignored, equations (6.35)-(6.38) reduce to

(6.40)

(6.41)

(6.42)

which, as may be verified, are identical to those in Sections 4.2 and 4.3 for the motions
of a uniform straight pipe conveying fluid and fully submerged in a quiescent fluid.
Equation (6.40) is identical to (6.41) because, for a straight pipe, motions in the XO- and
yo-directions are uncoupled and identical. Finally, if the surrounding fluid has negligible
effect on the dynamics of the system, setting Ma = 0, c = 0, and pe = 0 vis-&vis the
atmospheric pressure, these equations reduce to a version of equation (3.34).