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# Exercises

1. Liquid is led steadily through a pipeline that passes over a hill of height into the valley below, the speed at the crest being . Show that, by properly adjusting the ratio of the cross-sectional areas of the pipe at the crest and in the valley, the pressure may be equalized at these two places.

2. For the case of the two-dimensional motion of an incompressible fluid, determine the condition that the velocity components

satisfy the equation of continuity. Show that the magnitude of the vorticity is .

3. For the case of the two-dimensional motion of an incompressible fluid, show that

are the velocity components of a possible flow pattern. Determine the stream function and sketch the streamlines. Prove that the motion is irrotational, and find the velocity potential.

4. A cylindrical vortex in an incompressible fluid is co-axial with the -axis, and such that takes the constant value for , and is zero for , where is a cylindrical coordinate. Show that

where is the pressure at radius inside the vortex, and the circulation of the fluid outside the vortex is . Deduce that

where is the pressure at the center of the vortex.

5. Consider the cylindrical vortex discussed in Exercise 5.4. If is the pressure at radius external to the vortex, demonstrate that

where is the pressure at infinity.

6. Show that the stream function for the cylindrical vortex discussed in Exercises 5.4 and 5.5 is for , and for .

7. Consider a volume whose boundary is the surface . Suppose that contains an incompressible fluid whose motion is irrotational. Let the velocity potential be constant over . Prove that has the same constant value throughout . [Hint: Consider the identity .]

8. In Exercise 5.7, suppose that, instead of taking a constant value on the boundary, the normal velocity is everywhere zero on the boundary. Show that is constant throughout .

9. Prove that in the two-dimensional motion of a liquid the mean tangential fluid velocity around any small circle of radius is , where is the value of

at the center of the circle. Neglect terms of order .

10. Show that the equation of continuity for the two-dimensional motion of an incompressible fluid can be written

where , are cylindrical coordinates. Demonstrate that this equation is satisfied when and . Determine the stream function, and show that the fluid speed at any point is

where is the stream function at that point (defined such that at ).

11. Demonstrate that streamlines cross at right-angles at a stagnation point in two-dimensional, incompressible, irrotational flow.

12. Consider two-dimensional, incompressible, inviscid flow. Demonstrate that the fluid motion is governed by the following equations:

where , , and .

13. For irrotational, incompressible, inviscid motion in two-dimensions show that

where .

Next: 2D Potential Flow Up: Incompressible Inviscid Fluid Dynamics Previous: Velocity Potentials and Stream
Richard Fitzpatrick 2012-04-27