Page 64 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 47

For small motions, this reduces to
a
- v2+ = vv+. (2.144)
at
The boundary conditions match the fluid velocity on the solid surfaces to those of the
two cylinders. In polar coordinates, u, = -(I/r)(a+/aO) and ue = a+/ar; hence,

=
uil =a cos 8eia', ~01 = -a sin ,eia', ~ ~ 1 %us1 = 0, (2.145)
R, R, R"
where a is the velocity amplitude of the inner cylinder in the 8 = 0 plane.
This problem was solved by Wambsganss et al. (1974) - see also Chen et a1. (1976).
It may be verified that if
v2+ = 0 (2.146a)
is satisfied, so is equation (2.144); similarly if
1 a+
2
v+---=o. (2.146b)
v at
Hence, a general solution in the form + = + +2 is sought, with and $2 satisfying
(2.146a) and (2.146b), respectively. The form of the boundary conditions suggests
$1 = Fl(r) sin dein', $2 = F2(r) sin 8e'a', (2.147)

and hence FI and F2 must satisfy
d2FI 1 dF1 1
-++---F I =o,
dr2 r dr r2
(2.148)
d2F2 1 dF2 F2=0,
-+---
dr2 r dr
Each of these equations provides two independent solutions, hence four in total, as required
and sufficient for the solution of equation (2.144), namely

9 = +I + +2 = a[Alr-' +A21 +A3Il(hr) +A4Kl(Ar)] sin 8ein', (2.149)
in which the constants AI to A4 are determined via the boundary conditions. Once + is
determined, the flow field is completely known and hence the stresses on the cylinders may
be evaluated through equations (2.64) and (2.65). The force per unit length is given by
F = -p&a~[Rr(~) sin at + Sitir(H) COS at] (2.150)

(Chen et al. 1976), where

H = {2a2[Io(a>Ko(B) - Io(B>Ko(a)l - 4a[Ii(a)Ko(B) + Io(B)Ki(a)l
+ 'WIO(~)KI (B) + 11 (B)Ko(a)I - WI1 (a)Ki (B) - 11 (B)KI (a>l)
+ (a2(1 - ~~>[b(a)Ko(B> Io(B)Ko(a)l + 2ay[Io(a)Ki(B)
-
- 11 (B)Ko(B) + 11 (B)Ko(a) - IO(B>KI (/%I+ 2a~~[Io(B)Ki (a)
- b(a)KI (a> + 11 (a)Ko(B) - 11 (a)Ko(a)lI - 1. (2.151)