Page 305 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

286 SLENDER STRUCTURES AND AXIAL FLOW

This shows that the velocity of the fluid with respect to the pipe is no longer constant.
Hence, the absolute velocity is

where the prime denotes the derivative with respect to XO. Consequently,

(5.30)

Relationship (5.17) derived for a cantilevered pipe still holds, with the difference that
the inextensibility condition is not valid here, so that U2 terms in this case survive in the
kinetic energy and are therefore not associated with the right-hand side of (5.10), which
is zero! The total kinetic energy is given by
1" I'
%= Lrn (U2+W2)dx0+ $A4 [(u+U(l +~'))~+(W++w')~]dx~. (5.31)

For the case of non-rubber-like materials (u # OS), some additional words are neces-
sary. The change of volume is no longer equal to zero, and &/SI = 1/(1 - ~uE). The
fluid being incompressible, one obtains

U(X0) = uo (1 + 2 u E) = Uo(1 + E) + uo E (2 u - 1) = u1 (xo) + uo E (2 u - l),
i.e.

(5.32)

To fourth order, the strain E is given by

(5.33)
E = u' + $ w'2 + 0(€4),
so that for a pipe of length L = 1, with (uI - 0.01, and IwI - 0.1, one obtains IE~ -
1.5 x lop2. For u = 0.4 and 0.3, the error in the flow velocity associated with taking
u 2: 0.5 is 0.3% and 0.6%, respectively, which is of same order of magnitude as the error
made by assuming the velocity profile to be uniform. Hence, equation (5.31) may still be
considered valid.
The potential energy is considered next. To derive the strain energy, the axial strain is
itself decomposed into two components: a steady-state strain due to an externally applied
tension To and pressurization P, and an oscillatory strain due to pipe oscillation. By
reference to equation (5.13, this strain energy may be expressed as

By using (5.3, this is simplified to

(5.34)