Page 218 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

200 SLENDER STRUCTURES AND AXIAL FLOW

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Figure 4.2 (a) An element of the pipe showing forces and moments acting on it; (b) an element of
the contained fluid showing forces acting on it. Note that Q + (aQ/ax) Sx and piAi + (a/ax)(p,Ai) Sx
on the lower surfaces have been omitted in these diagrams (Hannoyer 1977).

in which F,, and En are the tangential and normal components of the fluid-pipe inter-
action forces associated with the internal flow (equivalent to qS and F, respectively, in
Figure 3.6), and F,, and En are the corresponding terms associated with the external,
stagnant fluid;+ T, Q, E, E*, I, and in are the same as before, for uniform pipes: the
tension, transverse shear force, modulus of elasticity, Kelvin-Voigt dissipation constant,
area-moment of inertia, and mass per unit length, respectively. The term is the fluid-
related moment due to both internal flow and external fluid, which for a pipe of nonuniform
cross-section may not tacitly be assumed to be zero.
Similarly, utilizing equations (4.7) and (4.8), x- and y-direction force balances
on an element of the fluid [cf. equations (3.18) and (3.19) of Section 3.3.2 and
Figure 4.2(b)] give

(4. loa)

(4. lob)

where Ai = Ai(x), and piAi is what was previously called M, and 9'w has been defined
in (4.8).
The external fluid, being stagnant, contributes only hydrostatic, inertial (added
mass) and damping terms: respectively equal to the buoyancy force, -peA,g, and to
-ppA,(a2w/at2) and -p,D,U,(aw/at), where Vu = (p,CD/p,D,) has the dimensions of

+The formal manner in which the external fluid forces are taken into account here is useful for later analysis,
where F,, and &, will be associated more generally with external Jlow (Chapter 8).