A cylindrical drum of lengthl and radius r can rotate inside a fixed concentric cylindrical casing, the clearance space c between the drum and the casing being very small and filled with liquid of dynamic viscosity u. To rotate the drum with angular velocity o requires the same power as to pump the liquid axially through the clearance space while the drum is stationary, and the pressure difference between the ends of the drum is p. The motion in both cases is laminar. Neglecting end effects, show that 2µlro/3 p = c2

A cylindrical drum of lengthl and radius r can rotate inside a fixed concentric cylindrical casing, the clearance space c between the drum and the casing being very small and filled with liquid of dynamic viscosity u. To rotate the drum with angular velocity o requires the same power as to pump the liquid axially through the clearance space while the drum is stationary, and the pressure difference between the ends of the drum is p. The motion in both cases is laminar. Neglecting end effects, show that 2µlro/3 p = c2

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Question

**Topic: Fluid Mechanics**

A cylindrical drum of length \( l \) and radius \( r \) can rotate inside a fixed concentric cylindrical casing. The clearance space \( c \) between the drum and the casing is very small and is filled with a liquid of dynamic viscosity \( \mu \). To rotate the drum with angular velocity \( \omega \), the same power is required as that needed to pump the liquid axially through the clearance space while the drum is stationary. In this scenario, the pressure difference between the ends of the drum is \( p \). The motion in both cases is laminar. By neglecting end effects, it can be shown that:

\[
p = \frac{2 \mu l r \omega \sqrt{3}}{c^2}
\]

This equation establishes a relationship between the pressure difference and the dynamics of fluid movement in the system.

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