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EML 5709 Homework 3 Spring 1997

1. Some people claim that the bathtub vortices on the northern hemisphere rotate counter-clockwise and in the southern hemisphere clockwise. Is this true? Are there no bathtub vortices on the equator? Base your answer on Kelvin's theorem and common sense. Solution.
2. Estimate the maximum likely rotational speed of the bathtub vortex created when you step out of the tub and pull the plug. Assume typical bathtub dimensions. Use rough estimates where needed. Solution.
3. If athmospheric air rises vertically, what general effect would you expect on the wind speeds? Solution.
4. The Lamb-Gromeko form of the momentum equations for incompressible flow is
   
   Restrict this equation to steady, inviscid flow. Then find two families of lines along which the Bernoulli law holds. How do the two families work out in two-dimensional flow? Solution.
5. You and a friend visit the fluids lab and are shown a running water tunnel. Your friend claims that the surface velocity increases as the square root of distance as it speeds up through the narrowing duct. Certainly you can see the water speed up downstream, but how can he see that it is a one-half power of distance?? Explain. The flow is inviscid and steady. Solution.
6. A circular cylinder of unit radius at ,
   
   moves towards the right with speed . The fluid flow velocity potential around this cylinder is given by
   
   Give the fluid flow velocity components on the cylinder. Solution.
7. For the flow of the previous question, show that the inviscid flow boundary condition is satisfied. Solution.
8. For the flow of the previous two questions, compare the pressure on the cylinder for the case that the cylinder is moving with unit velocity to the case that the cylinder is accelerating with speed . Use the Bernoulli law, ignoring gravity, and assuming the pressure vanishes at large distances. Solution.
9. Find the drag forces on the cylinder for the flow of the previous three questions. Solution.
10. Consider incompressible flow through a pipe with constant cross-section, except for one severe constriction. Assume that both before and after the constriction we have laminar Poiseuille flow with a parabolic velocity profile, . , with R the radius of the pipe. What can you say about the average velocity in the pipe after the constriction compared to the average velocity before the constriction? What can you say about the difference between the any two velocities before and after the constriction? So what does the Bernoulli say about the difference in pressure between the two? Does this mean that you can put in any constriction when you design pipe systems without any effect? Solution.

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