Successive Approximations for the Navier-Stokes Equations for Flows Past a Circular Cylinder and a Sphere

Abstract

This paper attempts to find higher approximate solutions of the Navier-Stokes equations by successive approximations to a flow past a circular cylinder and a sphere at small Reynolds numbers. As the preparation of successive approximations, Oseen’s formal solutions for the vorticity ζ and the stream function Ψ are first expanded in powers of the Reynolds number R. Expansions hold the same forms as the Oseen expansions. Substitution of these expansions in the Navier-Stokes equations yields a set of differential equations for the coefficient of Rⁿ. The leading terms of expansions are given by order 1 of the Oseen expansions. The successive solutions of ζ and Ψ for a circular cylinder and a sphere are obtained up to the third and the first power of R, respectively. For the case of a circular cylinder the flow pattern is drawn at R=0.8 and the drag is derived up to the second approximation.