On the asymptotic behaviour of viscous fluid flow at a great distance from a cylindrical body, with special reference to Filon’s paradox

The asymptotic behaviour of flow at a considerable distance from an arbitrary cylindrical obstacle immersed in an otherwise uniform flow of an incompressible viscous fluid is considered on the basis of the Navier-Stokes equations. Carrying out the Oseen type of successive approximation to the third stage, the expression for the stream function is exactly determined to the order of r-1, where r is the distance from some fixed point in or near the cylinder. Then, by considering the conservation of linear and angular momenta of the fluid enclosed between the cylinder and a large contour, exact analytical formulae for the lift, drag and moment acting on the cylinder are obtained. Thus Filon’s well-known paradoxical result that the moment of a cylinder immersed in a viscous flow comes out to be logarithmically infinite with increasing extent of the flow region is given a complete explanation, and the usefulness of the Oseen type of successive approximation in dealing with the Navier-Stokes equations is confirmed.