Suppose that a small sphere of radius R rotates at angular velocity in a large container

Suppose that a small sphere of radius R rotates at angular velocity ω in a large container of otherwise static fluid. The radius and velocity are such that 

With the sphere rotating about the z axis, the resulting fluid motion is in the ϕ direction.

(a) Show that there is a solution to the ϕ component of Stokes’ equation of the form v_ϕ(r, θ) = f(r)sin θ and determine f(r).

(b) Show that all the other equations are satisfied if v_r = v_θ = 0 and ℘ is constant. This confirms that this creeping flow is unidirectional (i.e., purely rotational).

(c) Show that the torque on the sphere is

Transcribed Image Text:

ReoR²/v << 1