4:ab^ X = T^r^ [(a;+o) r^ + {x-d) r^] [122i] 



It is then easily seen that the surface passes through both stagnation points. It is broadest 

 in the middle; its half width h is the value of oT when x = and is given by 



A2 = 2 ah'^/sl^^Tl? [122j] 



since cos ^2 " ~ '^°^ ^l " a/\Ja'^^h'^ when x = 0. 

 This equation and [112f] can also be written 



which shows that the shape, being fixed when h/a and l/a are known, depends only on the 

 constant b/a. 



The surface S acts again as a dividing surface. The fluid brought up by the stream 

 remains outside of S; the space inside it is occupied by fluid that is on its way from the 

 source to the sink. The streamline i/r = follows the a?-axis to Q^, divides into a sheaf of 

 lines which pass around S to reunite at Q^, and continues along the a;-axis. 



The formulas may represent streaming flow past a solid whose surface is S. Solids 

 having such shapes are called Rankine solids. Given the length 2^ and the maximum breadth 

 2h of the solid, a and h can be found from [122f] and [122j]. The velocity is most conveniently 

 found by adding vectorially the component velocities due to the stream and the two sources. 



An example of the streamlines is shown in Figure 204, for b^/a?' = 0.7, h/a = 0.97, 

 l/a = 1.58. Streamlines are drawn for equally spaced values of i/f/oT. 



The formulas could also be used for the flow inside a shell having the shape of S, 

 caused by a source and a sink at the proper points. 



If f > 0, there is a positive source a.t x = a, and a sink at a; = - a. If U < 0, all 

 velocities are reversed and the source and sink are interchanged, but the solid is unaffected. 

 (See Reference 1, Article 97; Reference 2, Section 15.27.) 



123. LINE DISTRIBUTIONS OF POINT SOURCES 



For some purposes it is useful to imagine point sources distributed continuously along 

 a line or curve. Let the algebraic strength of the sources per unit of length along the curve 

 be a , so that, from a length ds, in ads units of volume of fluid are emitted per second. Then 

 from [119d], in which ads replaces A, the potential due to the sources on ds at a distance r 

 from ds will be ads/r, and the total potential at any point (x,y, z) due to all sources on the 

 curve will be . . " ' , . ., 



310 



