Avery small circular cylinder of radius R i is rotating at angular velocity i inside a

A very small circular cylinder of radius R_i is rotating at angular velocity ω_i inside a much larger concentric cylinder of radius R₀ that is rotating at angular velocity ω_o. A liquid of density ρ and viscosity μ is confined between the two cylinders, as in Fig. P4–45. Gravitational and end effects can be neglected (the flow is two-dimensional into the page).

If ω_i = ω_o and a long time has passed, generate an expression for the tangential velocity profile, u_θ as a function of (at most) r, v, R_i, R_o, r, and m, where ω = ω_i = ω_o. Also, calculate the torque exerted by the fluid on the inner cylinder and on the outer cylinder.

FIGURE P4–45

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Liquid: p. p. Ro R₂ Outer cylinder Wi Inner cylinder Wo.