Page 451 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CURVED PIPES CONVEYING FLUID 423

Consider now an infinitesimal element of the fluid and forces acting on it, as shown
in Figure 6.3. Let the internal cross-sectional area of the pipe be Ai, and the internal
pressure pi(s). Components of the pressure force -A;pi in the directions (XO, yo, ZO) can
be written in terms of the components of the unit vector e,, referred to the reference frame
(XO, yo, ZO), as follows - cf. equation (6.1):

YO

Figure 6.3 Forces acting on an element of the enclosed fluid, expressed in the reference system
(xo, YO? zo).

By comparing Figure 6.2 with 6.3, and the forces (Pw, P,, PlO) with (exO, QyO, QzO),
and then using the first three of equations (6.16), the equations of motion of the fluid can
be written in the form

apX, @yo + KLP," - Rxo + Gjxo = Majxo,
as
--
as - K,P~ + toPxO - R, + GjyO = MafYo, (6.26)
apzo
~- Kip,, + KoPyo - Rzo + GfYO = Mafzo,
as
where Gjxo, GfuO, Gjz0 are components of the gravity force per unit length on the fluid.
Combination of equations (6.25) and (6.26) yields

a
as - RyO + GfyO - M afyO = 0, (6.28)
a 1
- [ -Ai.;] + %A;pi (E + E) - R, + GfzO - Mafzo = 0. (6.29)
as