Problem S10.2. Consider a rough hard sphere of radius R immersed in an incompressible fluid of point particles. Assume the sphere is free to rotate about its diameter but cannot translate. Also, assume that the sphere has angular velocity, (t)=(t).

(a) What is the velocity distribution of the fluid?

(b) Let r() be the torque on the sphere due to friction between the sphere and the medium, and let (w) be its Fourier transform. Show thatdiameter but cannot translate. Also, assume that the sphere has angular velocity, (t)=(t).

(a) What is the velocity distribution of the fluid?

(b) Let r() be the torque on the sphere due to friction between the sphere and the medium, and let (w) be its Fourier transform. Show that

where is the Fourier transform of (t). [Hint: The velocity is a polar vector and 2(t) is an axial vector, so assume the velocity has the form v = V x (g(r)oz), where g(r) is to be determined by solving Eq. (10.391). Assume stick boundary conditions on the surface of the sphere and assume the fluid velocity and all its derivatives go to zero at infinity.]

Transcribed Image Text:

F(w) = 8m2unk(1 kR 3(1+ikR),