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Exams - CENG 501 Examination Problem Estimation of...

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CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer You are assigned to design a falling- cylinder viscometer to measure the viscosity of Newtonian liquids. A schematic diagram of the system is shown here. A solid cylinder of radius R 1 and length L falls under the influence of gravity through the test liquid in a cylindrical vessel of radius R 2 that is closed at the bottom. List the restrictions on the design of the apparatus and derive an expression for the viscosity as a function of the system parameters. Include the following steps: 1. List your assumptions. Explain consequence of assumption and how the system should be designed to satisfy the assumption. 2. Draw a schematic of the velocity profile in the annular gap between the inner and outer cylinders for the following coordinate reference frames: L U R 1 R 2 a) Coordinates fixed with respect to the outer cylinder. b) Coordinates moving with the inner cylinder. 3. What does the continuity equation reduce to as a result of your assumptions? What statement can be made of the velocity profile from your assumptions and the continuity equation? 4. What does the equations of motion reduce to as a result of your assumptions? 5. Express the velocity profile as a function of position in the annular gap with the potential gradient as a parameter. Let 12 / R R κ = . 6. Derive the relationship between the dimensionless pressure gradient and κ that results from the condition that the bottom of the vessel is closed. 7. Derive the expression for the shear stress on the inner cylinder. 8. Derive the expressions for the macroscopic forces acting on the inner cylinder. 9. Derive the expression for the dimensionless velocity of the inner cylinder as a function of κ . 10. Express the viscosity as a function of the velocity of the falling cylinder and the system parameters.

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CENG 501 Final Examination , December 13-20, 2000 Instructions: The examination is closed book except for Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Bird, Stewart and Lightfoot, Transport Phenomena , and your CENG 501 class notes and homework assignments. The examination problem is not to be discussed with anyone during the examination period. The time limit for the examination is three hours. It is due in Professor Hirasaki's office by the end of the examination period, December 13-20, 2000. Time examination opened: ________________________ Time examination finished and sealed: _______________ Date: __________________ I pledge that I have followed the instructions: ____________________(signature)
CENG 501 Final Examination Problem, 2001 Flow of a fluid with a suddenly applied constant wall stress. This problem is similar to that of flow ( ) ,0 xy t −∞< <∞ > > near a wall suddenly set into motion, except that the shear stress at the wall is constant rather than the velocity. Let the fluid be at rest before t=0 . At time t=0 a constant force is applied to the fluid at the wall, so that the shear stress ( −∞ < < ∞ = yx ) τ takes on a new constant value o at y=0 for t>0 .

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Exams - CENG 501 Examination Problem Estimation of...

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