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# Exercises

1. Demonstrate that a line source of strength (running along the -axis) situated in a uniform flow of (unperturbed) velocity (lying in the - plane) and density experiences a force per unit length

2. Demonstrate that a vortex filament of intensity (running along the -axis) situated in a uniform flow of (unperturbed) velocity (lying in the - plane) and density experiences a force per unit length

3. Show that two parallel line sources of strengths and , located a perpendicular distance apart, exert a radial force per unit length on one another, the force being attractive if , and repulsive if .

4. Show that two parallel vortex filaments of intensities and , located a perpendicular distance apart, exert a radial force per unit length on one another, the force being repulsive if , and attractive if .

5. A vortex filament of intensity runs parallel to, and lies a perpendicular distance from, a rigid planar boundary. Demonstrate that the boundary experiences a net force per unit length directed toward the filament.

6. Two rigid planar boundaries meet at right-angles. A line source of strength runs parallel to the line of intersection of the planes, and is situated a perpendicular distance from each. Demonstrate that the source is subject to a force per unit length

directed towards the line of intersection of the planes.

7. A line source of strength is located a distance from an impenetrable circular cylinder of radius (the axis of the cylinder being parallel to the source). Demonstrate that the cylinder experiences a net force per unit length

directed toward the source.

8. A dipole line source consists of a line source of strength , running parallel to the -axis, and intersecting the - plane at , , and a parallel source of strength that intersects the - plane at , . Show that, in the limit , and , the complex velocity potential of the source is

Here, is termed the complex dipole strength.

9. A dipole line source of complex strength is placed in a uniformly flowing fluid of speed whose direction of motion subtends a (counter-clockwise) angle with the -axis. Show that, while no net force acts on the source, it is subject to a moment (per unit length) about the -axis.

10. Consider a dipole line source of complex strength running along the -axis, and a second parallel source of complex strength that intersects the - plane at , . Demonstrate that the first source is subject to a moment (per unit length) about the -axis of

as well as a force (per unit length) whose - and -components are

respectively. Show that the second source is subject to the same moment, but an equal and opposite force.

11. A dipole line source of complex strength runs parallel to, and is located a perpendicular distance from, a rigid planar boundary. Show that the boundary experiences a force per unit length

acting toward the source.

12. Demonstrate that a conformal map converts a line source into a line source of the same strength, and a vortex filament into a vortex filament of the same intensity.

13. Consider the conformal map

where , , and is real and positive. Show that

Demonstrate that , where , maps to a circular arc of center , , and radius , that connects the points , , and lies in the region . Demonstrate that maps to the continuation of this arc in the region . In particular, show that maps to the region on the -axis, whereas maps to the region . Finally, show that maps to a circle of center , , and radius .

14. Consider the complex velocity potential

where

Here, , , and are real and positive. Show that

Hence, deduce that the flow at is uniform, parallel to the -axis, and of speed . Demonstrate that

Hence, deduce that the streamline runs along the -axis for , but along a circular arc connecting the points , for . Furthermore, show that if then this arc lies above the -axis, and is of maximum height

but if then the arc lies below the -axis, and is of maximum depth

Hence, deduce that if then the complex velocity potential under investigation corresponds to uniform flow of speed , parallel to a planar boundary that possesses a cylindrical bump (whose axis is normal to the flow) of height and width , but if then the potential corresponds to flow parallel to a planar boundary that possesses a cylindrical depression of depth and width . Show, in particular, that if then the bump is a half-cylinder, and if then the depression is a half-cylinder. Finally, demonstrate that the flow speed at the top of the bump (in the case ), or the bottom of the depression (in the case ) is

15. Show that maps the semi-infinite strip , in the -plane onto the upper half ( ) of the -plane. Hence, show that the stream function due to a line source of strength placed at , , in the rectangular region , bounded by the rigid planes , , and , is

16. Show that the complex velocity potential

can be interpreted as that due to uniform flow of speed over a cylindrical log of radius lying on the flat bed of a deep stream (the axis of the log being normal to the flow). Demonstrate that the flow speed at the top of the log is . Finally, show that the pressure difference between the top and bottom of the log is .

17. Show that the complex potential

where

( ) represents uniform flow of unperturbed speed , whose direction subtends a (counter-clockwise) angle with the -axis, around an impenetrable elliptic cylinder of major radius , aligned along the -axis, and minor radius , aligned along the -axis. Demonstrate that the moment per unit length (about the -axis) exerted on the cylinder by the flow is

Hence, deduce that the moment acts to turn the cylinder broadside-on to the flow (i.e., is a dynamically stable equilibrium state), and that the equilibrium state in which the cylinder is aligned with the flow (i.e., ) is dynamically unstable.

18. Consider a simply-connected region of a two-dimensional flow pattern bounded on the inside by the closed curve (lying in the - plane), and on the outside by the closed curve . Here, and do not necessarily correspond to streamlines of the flow. Demonstrate that the kinetic energy per unit length (in the -direction) of the fluid lying between the two curves is

where is the fluid mass density, the velocity potential, and the stream function. Here, is a curve that runs from to , and denotes the amount by which the velocity potential increases as the argument of increases by .

19. Show that the complex potential

where

( ) represents the flow pattern around an impenetrable elliptic cylinder of major radius , aligned along the -axis, and minor radius , aligned along the -axis, moving with speed , in a direction that makes a counter-clockwise angle with the -axis, through a fluid that is at rest far from the cylinder. Demonstrate that the kinetic energy per unit length of the flow pattern is

where is the fluid mass density. Hence, show that the cylinder's added mass per unit length is

20. Demonstrate from Equation (6.110) that the equation of the free streamline , in the case of a liquid jet emerging from a two-dimensional orifice of semi-width formed by a gap between two semi-infinite plane walls that subtend an angle , can be written parametrically as:

where . Here, the orifice corresponds to the plane , and the flow a long way from the orifice is in the -direction. Show that for the case of a two-dimensional Borda mouthpiece, , the previous equations reduce to

Finally, show that the previous equations predict that the free streamline is re-entrant, with attaining its minimum value when .

Next: Axisymmetric Incompressible Inviscid Flow Up: Two-Dimensional Potential Flow Previous: Blasius Theorem
Richard Fitzpatrick 2016-03-31