The Rotating Sphere in a Quiescent Fluid

We begin with some simple problems to illustrate the basic ideas. The simplest is the flow induced by rotation of a sphere of radius a with angular velocity f2 in an unbounded, quiescent fluid. Although we could formulate this problem in dimensionless terms by using ila as a characteristic velocity (where il = f2 ) and the sphere radius as a characteristic length scale, it is more convenient because the solution representation was carried out in terms of dimensional variables to simply solve it in dimensional form. 

The starting point is the pressure p. The pressure is a harmonic function and a true scalar. Further, it must be a decaying function of distance from the origin because the flow decays at infinity. Finally, a key observation in the present problem is that the pressure must be a linear function off 2. This is because the governing equations are linear and Cl appears linearly in the boundary condition. Hence both p and u must be linear functions of Cl. 

the solution for p must be constructed solely from Cl and the general position vector x in the form of the vector harmonic functions. Now, there is only a single scalar function that can be constructed from Cl and the decaying harmonics, (8-5), that is linear in Cl, namely, 

It follows, therefore, from (8-3) that u = u(ff) must be a decaying harmonic function, linear in Cl and a true vector. The only combination of Cl and the vector harmonic functions that satisfies these conditions is 

To determine C2, and thus complete our solution of the rotating sphere problem, we apply the boundary condition 

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