Question

From Acheson: Elementary Fluid Dynamics

7.2. A rigid sphere of radius a is immersed in an infinite expanse of viscous fluid. The sphere rotates with constant angular velocity Ω. The Reynolds number R-Ωα2/v is small, so that the slow flow equations apply (see eqns (7.3) and (6.12)). Using spherical polar coordinates (r, θ, φ) with θ-0 as the rotation axis, show that a purely rotary flow u us(r, e, s possible provided that O2 (Ho sin θ) 1-0 (7.72) (This is, of course, just eqn (7.71) written in terms of different coordinates; иф here means the same thing as u in Exercise 7.1.) write down the boundary conditions which иф must satisfy at r a and as r-°, and hence seek an appropriate solution to eqn (7.72), thus finding S2a3 Show that the φ-component of stress on r a is 4--3μΩ sin θ, and deduce that the torque needed to maintain the rotation of the sphere is

Equation 7.3 and 6.12 (Slow Flow Equations)

Answer 1