Rayleigh's Problem for a Cylinder of Arbitrary Shape

Abstract

The motion of an incompressible viscous fluid generated by a cylinder of arbitrary cross-sectional form which is started to move suddenly from rest with uniform velocity in the direction of its length is considered. By using Laplace transform and complex variables, asymptotic expansion formulae in powers of a 2 /(ν t ) are derived for the velocity distribution (valid in the vicinity of the cylinder) and for the frictional drag on the cylinder, correct to the order of a 2 /(ν t ), where a is the characteristic length of the cross section, ν is the kinematic viscosity, and t is the time. These formulae are given in terms of only the analytic function which maps conformally the region outside the cross section of the cylinder onto the region outside the unit circle, and of certain integrals \({E_{n}}^{m}(a/\sqrt{{\nu}}t \) which are common to any arbitrary. cylinder. In particular, when \(a/\sqrt{{\nu}t}\) is sufficiently small, the total frictional drag on the cylinder per unit length is expressed as \(4\pi{\mu}WE_{0}(a/\sqrt{{\nu}t}){\sim}4\pi{\mu}W(\beta^{-1}-\gamma\beta^{-2}{\cdots})\), irrespective of the cross-sectional form, where \(\beta{=}-2[\gamma+\log(a/\sqrt{4{\nu}t})]\), and γ=0.5772… (Euler's constant).