As in three dimensions, the dipole can be formed by placing two simple line sources of 

 equal and opposite strength close together and allowing them to approach each other while 

 their strength increases without limit. It can also be formed by distributing infinitesimal 

 three-dimensional point dipoles uniformly along the fixed line, with their axes all parallel and 

 perpendicular to the line. The constant pi then represents twice tne sum of the three- 

 dimensional dipole moments per unit length along the line. 



In Lamb's Hydrodynamics,^ Section 60, m/1ji is written for A and \i/2v for \i. 

 A third type of flow in which a line singularity occurs is one in which, again, 



?=4 [15d] 



but in which the streamlines are circles having a common axis, like the magnetic lines around 

 a long straight current. In this case, along any one of the closed streamlines there is 

 obviously circulation of magnitude F = Iva {AfcE) - Itt A\ and it can be shown that F has the 

 same value around any closed curve tliat encircles the axis. Around a curve that does not 

 encircle the axis, on the other hand, F = 0. Thus the motion is irrotational everywhere except 

 at points on the axis, where the velocity becomes infinite and is undefined. 



Because of the resemblance of this type of flow to the motion in actual vortices, an 

 ideal line vortex is said to exist on the axis. Its strength is measured by the circulation F 

 around it. In actual "vortices" the central portion either is missing or is rotating more or less 

 like a rigid body. 



The corresponding velocity potential is discussed in Section 40. 



The line dipole itself can also be interpreted as a vortex dipole, since it can be pro- 

 duced by allowing two vortices with equal and opposite circulations to approach coincidence 

 while their circulations increase without limit. The axis of the resulting dipole is perpendi- 

 cular to the line joining the vortices. 



16. AXISYMMETRIC THREE-DIMENSIONAL FLOW 



Another important case is axisymnietric flow, in which axial symm^etry exists. Each 

 particle of the fluid is confined to one of a set of fixed planes intersecting along the axis; 

 and, at every point of any circle whose axis is the axis of symmetry, the pressure and the 

 magnitude of the velocity have the same values and the direction of the velocity is equally 

 inclined to the axis. 



In this case, also, a stream function exists, but it is somewhat different from that for 

 two-dimensional flow. 



In any plane through the axis of symmetry, take an arbitrary but fixed point A on the axis, 

 and any other point P joined to A by any curve AP ^ as in Figure 15. Consider the surface of 

 revolution generated by the rotation of this curve about the axis. It is evident that the volume 

 of fluid crossing this surface per second, taken positive toward the assumed negative direction 

 along the axis, is a function only of the coordinates of P; let it be represented by 27ri/r. The 



27 



