**Title: Analysis of Viscous Torque in Rotating Cylinders****Overview**When the inner cylinder is rotating and the outer cylinder remains stationary, viscous torque is generated both from the annular gap around the inner cylinder (\(M_3\)) and from the flat bottom of the inner cylinder as it rotates above the flat bottom of the outer cylinder (\(M_4\)).**Tasks:**i. **Expression Derivation:** - Obtain an expression for \(M_3\) and \(M_4\).ii. **Plot Requirement:** - If the value of \(M_4\) needs to be 1% (or less) of \(M_3\), create a plot of \( (b/a) \) as a function of \( (R_i/h) \).iii. **Calculation:** - For \(M_4\) to be 1% (or less) of \(M_3\), find the value of \(b\) given: - \(a = 0.20 \, \text{mm}\) - \(R_i = 50 \, \text{mm}\) - \(h = 80 \, \text{mm}\) - \(M_3 = 0.0245 \, \text{N}\) - \(\Omega = 1.2 \, \text{s}^{-1}\)iv. **Design Recommendations:** - What other design modifications would you recommend for this setup?**Diagram Explanation:**This document does not include diagrams, graphs, or charts. However, creating plots as described in task ii could involve graphing software to visualize the relationship between the geometric parameters, ensuring \(M_4\) remains a small percentage of \(M_3\). **Figure 1. Schematic of Coaxial Cylinder Viscometer**This diagram illustrates the structure of a coaxial cylinder viscometer, an instrument used to measure the viscosity of a fluid.- **a and b**: These arrows indicate the forces or flow direction in the system. - **Fluid**: The space between the rotating bob and the stationary cup is filled with the fluid whose viscosity is being measured.- **Rotating bob**: The inner cylinder, or bob, rotates at a specific angular velocity. This rotation creates a shear force in the fluid.- **Stationary cup**: The outer cylinder, or cup, remains fixed and encloses the rotating bob. - **Rᵢ and Rₒ**: These represent the inner radius of the rotating bob and the outer radius of the stationary cup, respectively.- **h**: This denotes the height of the fluid column between the rotating bob and the stationary cup.- **M**: This symbol indicates the torque applied to the rotating bob to maintain its motion.- **Ω**: Represents the angular velocity of the rotating bob.The viscometer's design allows for the controlled application and measurement of shear forces, providing precise data on the fluid's resistance to flow.

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When the inner cylinder is rotating and the outer cylinder is stationery, in reality, viscous torque is produced both from the annular gap around the inner cylinder (M3) as well as from the flat bottom of the inner cylinder as it rotates above the flat bottom of the outer cylinder (M4). i. Obtain an expression for M3 and M4. ii. If the values of M4 needs to be 1% of M3 (or less than 1%) - obtain a plot of (b/a) as a function of (R; / h). If the values of M, needs to be 1% of M3 (or less than 1%), obtain the value of b when a = 0.20 [mm], R; = 50 [mm], h = 80 [mm], M3 =0.0245 [N] and 2 = 1.2 [s']. iii. %3D

When the inner cylinder is rotating and the outer cylinder is stationery, in reality, viscous torque is produced both from the annular gap around the inner cylinder (M3) as well as from the flat bottom of the inner cylinder as it rotates above the flat bottom of the outer cylinder (M4). i. Obtain an expression for M3 and M4. ii. If the values of M4 needs to be 1% of M3 (or less than 1%) - obtain a plot of (b/a) as a function of (R; / h). If the values of M, needs to be 1% of M3 (or less than 1%), obtain the value of b when a = 0.20 [mm], R; = 50 [mm], h = 80 [mm], M3 =0.0245 [N] and 2 = 1.2 [s']. iii. %3D

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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