Page 67 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

50 SLENDER STRUCTURES AND AXIAL FLOW

(a) for large a! and B, wide or narrow annuli [Le. S > 500 and g 2 0.005 (or y 5
0.9931,

[a2(1 + y2) - 8yl sinh(b - a) + 242 - y + y2) cosh(b -a) - 2y2@ - 2ayfi.
H=
a2( 1 - y2) sinh(/? - a) - 2aAl + y) cosh(p - a) + 2y2m + 2aya
(2.156a)
(b) for very wide annuli and large S [S > 300 and g > 40 (or y < 0.025)],

(2.156b)

(c) for the same range of S and g as in (b), an easier approximation is also valid,
namely
4
H=l+-; (2.156~)
a!

(d) for moderately wide annuli and large S (S > lo4 and g > 0.1, or S > 2 x IO3 and
g > 0.21,

(2.156d)

(e) for fairly narrow annuli (g > 0.05) and S > lo4,

a(1 + y2) sinh(ga!) + 2(2 - y + y2) cosh(ga!) - 4yfi
H= , (2.156e)
41 - y2) sinh(ga) - 2y(l + y) cosh(gcy) + 4yfi
although approximation (2.156a) is superior and almost as easy to compute;
(f) for very narrow gap and very large S (g << 1, S >> 1, g2S >> 1; e.g. g < 0.05,
s > io7, g2s > I@),

(2.156f)

In order to utilize these expressions it is recalled that = 1 + i), a complex
quantity, arising because of the form of a! and B in equations (2.155); hence, sin(A + Bi) =
sin A cosh B + i cos A sinh B, etc.
Another set of approximations were derived by Sinyavskii et al. (1980), based on the
boundary-layer approximation and valid for S >> 1, namely

(2.157)

For zero confinement (y = 0), Cd = 22/2/6 corresponds exactly to the expression
derived by Batchelor (1967; section 5.1 3).