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EML 5709 Homework 3 Spring 1996

1. As a model for the formation of a tornado, assume that in cylindrical coordinates , the radial and axial velocity components near the axis are are

   

   Find the radial position of the fluid particles as a function of time. If the initial azimuthal velocity is

   

   find the azimuthal velocity at later times t using only Kelvin's theorem for rings of fluid around the axis. (Answer: .) Draw the pathlines in the plane z=0. Why does the tornado become stronger?

2. Show that the flow of the previous question satisfies the continuity equation for incompressible flow.
3. Show that the flow of the previous questions satisfies the three momentum equations of incompressible inviscid flow. Note: the momentum equations for incompressible flow may be derived using Appendix A; or you may want to study the book somewhat more closely.
4. Which of the two forms of the Bernoulli equation, if any, applies to the flow of the previous questions?
5. Water is flowing through an open channel. Explain why the water surface goes down when the fluid speeds up using the Bernoulli law. Also explain it directly from energy considerations.
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. Show that the inviscid flow boundary condition is satisfied.

7. For the flow of the previous question, 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.
8. Find the drag forces on the cylinder for the flows of the previous question.

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