Uniform Flow past a Solid Sphere at Small, but Nonzero, Reynolds Number

UNIFORM FLOW PAST A SOLID SPHERE AT SMALL, BUT NONZERO, REYNOLDS NUMBER 

let us return to the fluid mechanics problem of streaming flow past a solid sphere at small, but nonzero, Reynolds numbers. The objective is to see how the steady-state creeping-flow solution is modified if we consider the nonlinear terms in (7-2) for Re 1. As we shall see, there is essentially the same problem of trying to obtain a solution of (7-2) in the form of a regular perturbation expansion as was encountered when we tried to obtain such a solution of the thermal problem for IP P I. Indeed, this problem and its resolution are very 

Thus we begin by considering the full Navier-Stokes equation expressed in terms of the streamfunction // and nondimensionalized by use of the undisturbed velocity of the fluid relative to the sphere [/ as a characteristic velocity and the sphere radius a as a characteristic length scale. Using spherical coordinates, with ij = cos 9, this equation is 

Following Whitehead s approach, we seek a solution of (9-75) and (9-76) by means of a regular perturbation expansion of the form 

The last condition follows from the fact that the first approximation, //o, satisfies the non-homogeneousboundary condition // —r2Qi(i]) asr oo (that is, uo i as r — oo), 

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D. Uniform Flow Past a Solid Sphere at Small, but Nonzero, Reynolds Number... 

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