Chapter 4, Problem 45P

Consider the same two concentric cylinders of Prob. 4-44. 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 $u_{\theta }$ as a function of (at most) ${\omega }_i,R_j,R_o,\rho$ , and $\mu$ .. Hint: Your answer may contain an (unknown) constant, which can be obtained by specifying a boundary condition at the inner cylinder surface.