The Steady Flow of Viscous Fluid Past a Sphere and Circular Cylinder at Small Reynolds Numbers

In Part I of this paper the flow patterns around a sphere are first computed in detail on the basis of Goldstein's exact analytical solution of Oseen's linearized equations of motion for the steady flow of an incompressible viscous fluid past the sphere, and it is found that, in accordance with the results of observation, a stationary vortex-ring is formed behind the sphere. Further, by carrying out numerical calculations for small Reynolds numbers, it is found that, even when the Reynolds number assumes very small values such as 0.1, the stationary vortex-ring, though of very weak strength, is still formed behind the sphere. Next, discussions on the drag on the sphere are made, with special reference to the pressure drag and the frictional drag separately. It is thus found that, as far as the calculations are made on the basis of Oseen's equations, the pressure and frictional drags contribute to the total drag on the sphere in the ratio 1: 2 for any value of the Reynolds number. Similar discussions are made in Part II for the case of steady flow of a viscous fluid past a circular cylinder, and it is thus found that, in accordance with the experimental results, the two standing eddies are always formed behind the cylinder. Further, it is found that such standing eddies are still formed even when the Reynolds number assumes very small values such as 0.05, 0.1 and 0.25, in contradiction to the common view that for very small Reynolds numbers, in the neighbourhood of unity, the two standing eddies are not formed. The drag experienced by the circular cylinder is also discussed, with special reference to the pressure drag and the frictional drag separately, and it is found that, as far as the calculations are based upon Oseen's equations of motion, the total drag on the circular cylinder is divided, in exactly the same proportion, into the pressure and frictional drags for any value of the Reynolds number.