An analytic solution of Navier–Stokes flow past a sphere in the region of intermediate Reynolds number

Abstract

Abstract We study an analytical solution of steady, laminar, and incompressible flow past a sphere in the region of intermediate Reynolds number. The flow is governed by the Navier–Stokes (N–S) equation and the continuity equation. By applying a simple perturbation method to solve the equations, a second-order approximation cannot be obtained, as well-known (Whitehead’s paradox). Many analytical studies, such as Oseen approximation, matching technique, the homotopy analysis method, etc, have been conducted to resolve the paradox. The drag coefficients of these solutions are valid in the region of Reynolds number R d < 30 (R d is the diameter-based Reynolds number) However, the solution cannot express the flow separation behind a sphere observed in experiments. We also develop a perturbation technique to construct a solution of the N–S equation asymptotically to solve the paradox. The solution consists of power series of R d h , where h is an arbitrary constant ( 0 < h ⩽ 1 ) . By setting the value of h to avoid divergence of the solution in the all-region where steady flow exists, the solution expresses the flow separation behind a sphere, coinciding with experiments in the region of intermediate Reynolds number ( R d < 130 ), although the existing analytical solutions could not express. Also, the present solution gives the drag coefficient which agrees with experimental and numerical values in the region of R d < 30 .