Flow from a Disturbed Area. 435 



Ret", fig. 1, let w = correspond to the corner B. 

 In general 12 will have a branch point at w = ; e.g. for a 

 free stream-line 12 behaves like iv l/2 . 

 Four cases are possible : — 



(i.) Discontinuity of velocity on opposite sides of the 

 stream-line i^ = 0. 



This includes the case of discontinuous flow where the 

 fluid is at rest on one side of yjr = 0. 



The objections to this theory, in problems such as the flow 

 past a plate, are well known and will not be repeated here *. 



(ii.) The stream-line -^ = is an analytical boundary 

 across which it is impossible to continue the 

 expressions representing the motion on one side 

 of it. 



This implies that O has an infinite number of essential 

 singularities along yjr = Q. 



In the neighbourhood of these points 12 takes all possible 

 values, and therefore q, the velocity, takes all possible 

 values. 



(iii.) The electrical case : the stream-line ^ = wraps 

 itself round the boundary. 



This implies an infinite velocity at the corner B. 



(iv.) No discontinuity of velocity on opposite sides of 

 <\Jr = for which <£<0. 



In such a case the solution for yjr < must be that obtained 

 by analytic continuation, across i|r = 0, in a w-plane so 



dz 

 modified by cuts that -j— is a single-valued function of iv. 



In a practical problem these cuts must coincide with the 

 part of the w-plane which represents the rigid boundary. 



If not, every stream-line, for which the corresponding 

 line i/r = >|r in the w-plane meets the cut, will experience a 

 finite discontinuity. 



In what follows case (iv.) is considered. 



The object is to obtain an analytic expression giving an 

 irrotational motion on both sides of the stream-line -^ = 

 such that the velocity is continuous on opposite sides of AB 

 in fig. 1. 



In addition the velocity of the fluid at any point in the 

 finite part of the c-plane is to remain finite. 



* Kelvin, Collected Papers, vol. iv. 

 2 H 2 



