   Next: Two-Dimensional Potential Flow Up: Two-Dimensional Incompressible Inviscid Flow Previous: Two-Dimensional Jets

# Exercises

1. 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 .

2. 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.

3. 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.

4. Consider the cylindrical vortex discussed in Exercise 3. If is the pressure at radius external to the vortex, demonstrate that where is the pressure at infinity.

5. Show that the stream function for the cylindrical vortex discussed in Exercises 3 and 4 is for , and for .

6. 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 .

7. 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 ).

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

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

10. For irrotational, incompressible, inviscid motion in two dimensions show that where .   Next: Two-Dimensional Potential Flow Up: Two-Dimensional Incompressible Inviscid Flow Previous: Two-Dimensional Jets
Richard Fitzpatrick 2016-03-31