Page 219 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 20 1

velocity, De is the external diameter of the pipe, pc is the dynamic viscosity, and CD an
empirical coefficient dependent on Stokes' number - see Section 2.2.l(g) and 2.2.3 and,
for the viscous component, also Pa'idoussis (1973b) and Hannoyer & Pa'idoussis (1978).
Hence, a balance of forces due to the external fluid gives

(4.1 la)

(4.11b)

The form of the pressure forces in equations (4.1 la) and (4.1 lb) is clarified in Chapter 8;
here one may simply accept it by similarity to the internal flow terms in equations
(4.1Oa,b).
The evaluation of the aMf/ax term in (4.9~) is quite tedious and will not be reproduced
here. Suffice it to say that careful study (Hannoyer 1977) has shown that
aMf
-~ - PIA, dA, PeAe dAe a2w (4.12)
-
--c&w+---
ax 2~ dx 2~ dx at2 '
Equations (4.9a). (4.10a) and (4.1 la) may be combined to give
a
-[T + peAe - piAi - p,(AiUi)uiI = (PeAe - PIA, - m)g. (4.13)
ax
in which the fact that A, U, is constant has been recognized. Then, by combining (4.9b,c L
(4.1 Ob) and (4.1 1 b) and utilizing (4.13), the equation of lateral motion becomes

(4.14)

it is important to note that, in the dominant term plA,[912 - Ul(dU,/dx)](aw/ax), the
U,(dUl/dx)(~/~x) component cancels out once 9'w is expanded - and this is true
irrespective of magnitude considerations.
We next proceed to evaluate the only unspecified quantity in (4.14), namely that related
to T + p,A, - plAl. By integrating (4.13),
1:

T(x) = T(L) - (Ai G )ui - JI' (peAe - piA, - m)gh (4.15)

is obtained, in which
T(x) = (T + peAe - pjAi); (4.16)
it is recalled that T, p.A, U and m, unless otherwise denoted, are functions of x. TWO
cases will be analysed, separately, as follows.