# Analyze and discuss a limiting case of Prob. 999 in which there is no inner cylinder (R i =

## Question:

Analyze and discuss a limiting case of Prob. 9–99 in which there is no inner cylinder (Ri = ωi = 0). Generate an expression for uθ as a function of r. What kind of flow is this? Describe how this flow could be set up experimentally.

Data from Problem 99

Repeat Prob. 9–96 for the more general case. Namely, let the inner cylinder rotate at angular velocity ωi and let the outer cylinder rotate at angular velocity ωo. All else is the same as Prob. 9–96. Generate an exact expression for velocity component uθ as a function of radius r and the other parameters in the problem. Verify that when ωo = 0 your result simplifies to that of Prob. 9–96.

Data from Problem 96

An incompressible Newtonian liquid is confined between two concentric circular cylinders of infinite length— a solid inner cylinder of radius Ri and a hollow, stationary outer cylinder of radius Ro (Fig. P9–96; the z-axis is out of the page). The inner cylinder rotates at angular velocity ωi. The flow is steady, laminar, and two-dimensional in the rθ-plane. The flow is also rotationally symmetric, meaning that nothing is a function of coordinate θ (uθ and P are functions of radius r only). The flow is also circular, meaning that velocity component ur = 0 everywhere. Generate an exact expression for velocity component uθ as a function of radius r and the other parameters in the problem. You may ignore gravity.

FIGURE P9–96