434 Lieut. -Col. A. R. Richardson on Stream-line 



(ii.) and (iv.) require no further explanation but are con- 

 sequences of (i.) . 



Reasons for (iii.) are also suggested. 



The following is a short account of the results of the 

 analysis. 



The boundary conditions, and (i.) above, show that there 

 is irrotational stream-line motion everywhere except in a 

 definite area, behind the plate, inside which two-dimensional 

 non-viscous irrotational flow is impossible. 



Fig. 2 shows the shape of a stream-line as it enters this 

 disturbed area. 



It curls very rapidly indeed about a point P, then unwinds 

 and, after cutting itself a finite number of times, passes away 

 to infinity *. 



This suggests that the pressure distribution over the 

 boundary of the disturbed area B, necessary to produce 

 the stream-line flow, is the same as would be produced by a 

 system of eddies at places, such as P, inside the area. 



The disturbed area does not extend up to the plate nor to 

 the bounding stream-line. 



Its actual position, and shape, vary with the postulated 

 velocity boundary conditions behind the plate. 



The first part of the analysis discusses the general problem 

 and shows that the existence of a disturbed area is a necessary 

 consequence of conditions (i.). 



The second part deals in detail with the flow past a plate 

 and with the disturbed flow from a semi-infinite pipe such as 

 is met with in sensitive jets. 



In both cases the solutions may be made to represent an 

 unsteady state, and not merely the uniform motion. 



An examination of the pressure shows that a turning value 

 occurs on the underside but not on the upperside of the* 

 plate. 



I. Flow past a corner. 



Adopt the usual notation : 



z = aj + iy j 



w = (f> + iyfr, 



fl=logf = Iog(-g)= -logq + to, 



where q and 6 refer to the velocity. 



* The curve is not completed near P as \dz\ is exceedingly small — 

 the maximum negative angle 6 being- about 27r and the maximum 

 positive angle || 7r/2. In the vicinity of P \dz\ lies between 'OOOGe— 1& 

 and *002 corresponding- to .dcj)\ ='01. 



