X. The resistance of a cylinder moving in a viscous fluid

Abstract

An earlier Paper by the same authors dealt with one aspect of the same problem and in its conclusions indicated the possibility of the present development. In order to deal with the equations of motion for a viscous fluid in two dimensions, the approximation for slow motion due to Stokes was used, with a consequent need for the introduction of a boundary limiting the expanse of the fluid. Whilst keeping the analysis as general as possible, the example given related to a circular cylinder centrally placed in a parallel-walled channel. The extension now to be described follows generally similar lines; the form of equation has been changed from that of Stokes to one proposed by Oseen, the change representing a closer approach to the full equations of motion by the introduction of terms dependent on the inertia of the fluid. A consideration of the differential equations by earlier writers has indicated a close agreement between the motions near a small sphere in the two cases, but a marked difference in the more remote parts of the fluid. Oseen has shown, for the sphere, that the resistance formulæ for the two cases are identical to the first order of small quantities. In the case of the two-dimensional motion of a cylinder the differences are rather more striking. In the Stores’ form of approximation it is not possible to satisfy all the essential conditions when the expanse of fluid is infinite, whilst with Oseen’ s type of equation this particular difficulty disappears. Having seen Oseen’ s solution for the motion of a sphere in a viscous fluid, Lamb applied a similar method to the circular cylinder, and an account of his analysis is given in the ‘Philosophical Magazine’ 'and his treatise on Hydrodynamics' (p. 605); a resistance formula is deduced which is applicable at low velocities. There are two approximations in this solution, one physical and implied in the original differential equation, and the second mathematical and introduced in the solution. There is a certain degree of inter-relation between the two approximations, but it has been found that the second of them may be removed. An estimate of the degree to which Oseen’ s approximation represents the complete equation of viscous fluid motion can be obtained by comparing the results of the new calculations with those of experiment. In the result it appears that the amount still to be accounted for by the remaining inertia terms is less than that already dealt with, in the case of both the resistance of circular cylinders and the skin friction of flat plates.