in lb upon going positively around any closed curve that encircles the source. The decrease 

 in \lj is also equal to the volume of fluid that flows outward across the cylinder represented 

 by such a curve, between two planes of flow unit distance apart. 



It may be noted that, if the flow due to a line source located at a point P is superposed 

 upon another flow in which the velocity at P is finite, the resultant streamlines approximate 

 more and more to those due to the source alone as P is approached, since in the flow due to 

 the source 17 -» oo at P. 



The equations could be balanced dimensionally by writing w = - A In {z/a) where a is 

 a constant having the dimension of length. Then r is replaced by r/a in 0, which merely adds 

 a constant to all values of ^. 



Line Vortex 



For the conjugate flow the circles become the streamlines and the radii the equipoten- 

 tials. The potential t^ ', stream function xh', and velocity are given by 



0'= - Ae, i!j'= A\n r [40h,i] 



w'=-iisin0, v' = ^cos (9, q =— [40j,k,l] 



T T r 



The corresponding complex potential is 



w = t^' + zi/;'= iA In 3 [40rn] 



It is now the potential oS' that is many-valued; in gqing counterclockwise once around the origin, 

 <;z!>' changes by -277 4. The velocity, however, is single-valued, as is dw/dz, except at the 

 origin; for (^ + 2nn has the same space derivatives at any point as has (jS itself. The circula- 

 tion, taken around any closed curve encircling the origin once, is F = 2it A. Treated as an 

 ideal case, the flow may be regarded as due to a line vortex at the origin, as described in 

 Section 15. 



In this type of flow, the singular point can be excluded by inserting a cylindrical 

 boundary along any one of the circular streamlines. Then c6'and t/^' represent a piiysically 

 possible irrotational circulatory motion about this cylinder. The circulation vanishes taken 

 around any closed path that does not enclose the cylinder. If the path goes positively once 

 around the cylinder, however, the circulation F around it is 2jtA. Thus the constant 



a = j: 



2n 



If the flow is steady, the pressure is given as usual by the Bernoulli equation. 



The velocity increases without limit as the vortex is approached, and is undefined at 

 the location of the vortex itself. Hence, if the flow due to a line vortex at P is superposed 

 upon another flow in which the velocity is finite at P, the resultant streamlines near P 



83 



