(i) Source or dipole on vertex of an angle (or sharply bent lamina): Agostinelli^^; 



(5) Lamina, plane or bent along the median line, with a centered line source on one or 

 both sides, and immersed in a stream: Colombo.^^ The velocity may be finite at the edges. 



For a similar combination including circulation or a vortex, see Section 98. 



(6) Sources or sinks in a stream. Boundaries of many other shaf)es then those described 

 in Sections 53 and 54 can be obtained by inserting various combinations of sources and sinks 

 into a uniform stream. There will always be a dividing surface along which a cylinder may be 

 introduced. This surface is closed if the total strengths of sources and sinks are equal; oth- 

 erwise it extends to infinity, in the direction of the stream if the sources are in excess, or in 

 the opposite direction if sinks predominate. Although it is not always possible in this manner 

 to match exactly the shape of an arbitrarily given cylinder, sufficiently close matches may 

 often be secured. Uniform sheets of sources, as described in the next section, may be 

 employed. 



57. SHEETS OF LINE SOURCES OR VORTICES 



Let sources be distributed uniformly over an infinite plane strip of width c. Let tne 

 source strength per unit area be o, so that a volume of fluid 2ffais emitted per second from 

 each unit area of the strip. Draw the a?y-plane so that the given strip cuts it perpendicularly 

 along the segment of the a;-axis from x = a to x = b; thus c = b - a. The resulting flow will 

 then be parallel to -the a;y-plane. 



On the strip, let x be replaced by x', and consider the sources on a substrip of width 

 (^a;' extending from a; 'to a;'+ dx'; see Figure 88. Since the volume of fluid emitted by these 

 sources is 2nadx per second, per unit of length perpendicular to the planes of flow, their 

 contribution to the complex potential is, from [40a], in which now A = adx and 2 is to be re- 

 placed by 8 - x', 



dw = - aln (z - z") dx' 



Figure 88 — A plane sheet bf line sources. 



130 



