Consider the same two concentric cylinders of Prob. 4–45. This time, however, the inner cylinder is rotating, but the outer cylinder is stationary. In the limit, as the outer cylinder is very large compared to the inner cylinder (imagine the inner cylinder spinning very fast while its radius gets very small), what kind of flow does this approximate? Explain. After a long time has passed, generate an expression for the tangential velocity profile, namely u_θ as a function of (at most) r, ω_i, R_i, R_o, ρ, and μ.

Data from Problem 45

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

Transcribed Image Text:

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