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EML 5709 Homework 4 Spring 1996

1. The radius of a circular cylinder is expanding in time. The cylinder is surrounded by an incompressible fluid extending to infinity in all directions. Write the partial differential equation and boundary conditions for the velocity potential in the incompressible fluid in terms of and its derivatives.
2. Solve the partial differential equation of the previous question using separation of variables. Compare with the basic solutions.
3. Use the result of the previous question to find the pressure and velocity components in the incompressible fluid. What can you say about the pressure at large distances?
4. Is the solution derived in the previous questions an irrotational flow? Is it a solution of the Euler equations satisfying the right boundary conditions? Is it a solution of the Navier-Stokes equations satisfying the right boundary conditions?
5. Using real variables only, describe the velocity field corresponding to a vortex of strength located at , and one of opposite strength at . In particular, find and and sketch the streamlines. In what direction is the velocity on the x-axis (y=0)? What can you say about the flow corresponding to a single vortex located a distance h above a solid wall? Mirroring an image of the features of a flow into a solid region is called the mirror method. Give the pressure on the wall. Explain why the effects of wingtip vortices are much less pronounced when the plane is flying close to the ground.
6. Repeat question 5, but now use complex variables everywhere: Describe the velocity field corresponding to a vortex of strength located at , and one of opposite strength at . In particular, find F and from it and sketch the streamlines. Show that F is real on the x-axis. Find the complex conjugate velocity and show that it is horizontal on the x-axis. Find the pressure on the wall from W.
7. Using complex variables, write down the combined potential of a uniform flow in the x-direction of unit velocity, a strong sink of strength at the origin, and a strong source of strength at . Show that for small , we get potential flow past a circular cylinder. Draw the streamlines. Where does the fluid coming out of the source at the (-epsilon,0) end up?
8. Using complex variables, write down the combined potential of a uniform flow in the x-direction of unit velocity and a source of strength at the origin. Draw the streamlines. What is the velocity at large distances? Where does the fluid coming out of the source at the origin end up?
9. Make question 4.3 in the book.
10. Make question 4.7 in the book.
11. Repeat question 4.7 in the book, but now use the Blasius theorem to find the force. Compare the results.
12. A complex velocity potential in a -plane is given by

    

    where . Show that F is real on the unit circle, so that it is a streamline. Find on the unit circle in terms of the angular coordinate . Show that stagnation points occur at and . Identify F at large distances. Now draw the streamlines and describe the flow.

13. For the -plane of the previous question, assume that the corresponding physical (x,y)-plane is found from

    

    Identify the critical points where . Locate these also in the z-plane. Now sketch and describe the curve in the z-plane corresponding to the unit circle in the -plane. Show that for small , this curve is approximately equal to . Describe this approximate curve. Draw the streamlines in the z-plane.

14. Using the results of the previous questions, show that the lift coefficient of a thick airfoil can be more than .
15. Using the results of the previous questions, find the complex conjugate velocity W= dF/dz on the unit circle . Approximate this for small values of . Rewrite this in terms of to get

    

16. Using the result of the previous question to draw the pressure distribution on the top and bottom of the plate . Comment on the result.

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