Page 95 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

78 SLENDER STRUCTURES AND AXIAL FLOW

Figure 3.7 Definition of the control volume of the open system under consideration, T, and
of a fictitious closed system, coincident with rS, at time t. The control surfaces Yo and Yc are
associated with the open and closed parts of the open system. (a) The system at time t, and (b) at
time t + dt.

u = Dr/Dt. Then, Hamilton's principle may be obtained from (3.47) by integrating it
between two instants, tl and t2; in accordance with normal variational procedure, the
system configuration is prescribed at tl and t2, i.e. 6r = 0 so that the last term vanishes,
and this leads to the familiar form (cf. Section 2.1)
6 1 ZCdt+[ 6Wdt=0. (3.48)

The extension to open systems is effected by considering a portion Yo(t) of the surface
of the control volume %(t) (Figure 3.7) to be capable of movement with a velocity V . n
normal to the surface, across which mass may be transported; n is the outward normal.
Thus, Yc(t) is associated with the closed part of the system and Yo(t) with the open
part. Figure 3.7(a) shows the system at. time t, and Figure 3.7(b) at time t+dt. This
open system does not necessarily have a constant mass or, if it does, the mass does
not necessarily comprise the same particles. On the closed part of the control volume,
bounded by Yc(t), V n = u - n.
If, at time t, x(t) coincides with %(t) as shown in Figure 3.7(a), Reynolds' general
transport equation [e.g. Shames (1992; Chapter 4)]+ reads

(3.50)

may be used since D{ }/Dt makes it clear that a closed control volume is to be employed.

+Equation (3.49) simply states that the total rate of change in [ ) is equal to the rate of change in the
volume plus that due to infludefflux through the boundaries.