Magnus Effect and Force Due to Rotation of a Sphere

Consider a steady motion of a sphere of radius a, velocity Vp, and angular velocity f2 in an incompressible fluid at rest, as shown in Fig. 3.3. By selecting the Cartesian coordinates with the origin attached to the center of the sphere, with the x-axis pointing to the opposite direction of Up, and with the x-y plane containing fl, we have 

Assume that the solutions of u and p can be expressed by the Stokes expansion in the form 

The zero-order Stokes approximation thus satisfies 

It is noted that the Stokes expansion is not uniformly valid in the neighborhood of infinity. Therefore, another expansion, i.e., the Oseen expansion, is introduced to satisfy the boundary conditions at infinity. By using the matching technique, the final results can be obtained. 

The Oseen approximation satisfies Eqs. (3.51) through (3.53) in stretched coordinates 

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