Continuing the previous work I (J. Phys. Soc. Japan
9 (1954) 611), the author discusses the frictional force on an arbitrary cylinder which is started to move suddenly from rest with uniform velocity
W in the direction of its length. By using the WKB method, a new asymptotic expansion formula in powers of \sqrtν
t⁄
R is derived for the frictional drag τ on the cylinder with no corners;
\fracτμ
W=\frac1\sqrtπν
t+\frac12
R−\frac\sqrtν
t4\sqrtπ
R2+\fracν
t8\left[\frac1
R3+\frac∂
2∂
s2\left(\frac1
R\
ight)\
ight]+O((ν
t)
3⁄2)
where ν is the kinematic viscosity,
t is the time measured from the start,
R is the local radius of curvature of the surface measured inwards, and
ds is the line element along the surface. The first approximation gives the naturally expected value μ
W⁄\sqrtπν
t for the frictional drag per unit area, which is the same as that for a flat plate of infinite breadth. As far as the second approximation the total frictional drag on the cylinder is the same as that on a circular cylinder with the same perimeter, and is in accord with the results recently obtained by Batchelor. The effect of the variation of the curvature along the surface on the local friction first appears in the fourth order of approximation. The correction due to the presence of corners is made in the second approximation, and some errors in Batchelor’s paper are corrected. For the intermediate value of ν
t the drag calculated by these formulae shows a good agreement with that calculated by the formulae in powers of (ν
t)
−1 as given in I.
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