# Analyze and discuss two limiting cases of Prob. 996: (a) The gap is very small. Show that the velocity profile,

## Question:

Analyze and discuss two limiting cases of Prob. 9–96:

(a) The gap is very small. Show that the velocity profile, approaches linear from the outer cylinder wall to the inner cylinder wall. In other words, for a very tiny gap the velocity profile reduces to that of simple two-dimensional Couette flow. (Define y = R_{o} – r, h = gap thickness = R_{o} – R_{i}, and V = speed of the “upper plate” 5 R_{i}ω_{i}.)

(b) The outer cylinder radius approaches infinity, while the inner cylinder radius is very small. What kind of flow does this approach?

**Data from Problem 96**

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

**FIGURE P9–96**

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**Related Book For**

## Fluid Mechanics Fundamentals And Applications

**ISBN:** 9780073380322

3rd Edition

**Authors:** Yunus Cengel, John Cimbala

**Question Details**

**9**- DIFFERENTIAL ANALYSIS OF FLUID FLOW

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