Page 448 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

420 SLENDER STRUCTURES AND AXIAL FLOW

where e, is the unit vector along the z-axis, which is tangential to the strained centreline.
From equation (6.1), one can obtain

Combining equations (6.5), (6.7) and (6.8) yields

To obtain the acceleration of the fluid, we differentiate Vf, yielding

(6.10)

cf. equations (2.63) and (3.30). Substituting equation (6.7) into (6.10), the acceleration of
the fluid may be rewritten as follows:

Now the last term on the right-hand side of this equation may be written as

+--+--
(” at ay0 a aw azo a)
(vp 47)Vf = - - vf; (6.12)
at axo a
at
av
by combining equations (6.9) and (6.12) we can see that this term is of higher order and
can be neglected. In addition, it is noted that
a a a
-
e,.V=- and - _- (6.13)
az az as’
the latter because of the assumption of small motions. Hence, we obtain

avf avf
af=-+u--. (6.14)
at as
Substituting equation (6.9) into (6.14), we can write the fluid acceleration in the XO-,
yo-, and zo-directions (see Appendix J.3), as follows:

aw
+
+
s

;
;
Ufxo = - +2u ( - $ $) + u2 (e - - + - l),
a2u
at2 as2 R, as R,
a2u a2v a2U
Ufyo = +2u- at as + u2 - (6.15)
ax2 ’
Ufzo = 2 + u (2 - at)-E (%+E).
u2 au
1 au
a2w
-
R,
-