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



IV. Application of the solutions to physical problems. 



Exclude the essential singularity d by a domain B in the 

 ^-plane as in III. (iii.) above. 



Let r be the curve in the ^-plane corresponding to the 

 boundary of B in the w-plane. 



Postulate the pressure distribution over T which will give 

 the stream-line motion outside T. 



Inside T is a region in which stream-line irrotational flow 

 is impossible and where a disturbed eddying motion is to be 

 expected. 



When the fluid consists of a thin film, the solution may be 

 regarded as an approximation to a three-dimensional flow in 

 which movement perpendicular to the z- plane is small. 



In such a case the domain B may be restricted to a small 

 area round d, and the branches in the r-plane looked upon 

 as defining flow in the various folds of the film resembling 

 motion on a Riemann surface. 



These solutions indicate regions in which even a small 

 amount of viscosity will allow vortices to form ; for as T is 

 approached, the stream-lines begin to curl in such a way as 

 to give a reason for the formation of circular vortices. 



It must be remembered that, in problems such as the flow 

 past a plate, there will be an infinite number of solutions of 

 the type here considered, for assumptions must be made as 

 to the nature of the flow over the boundary T. 



These remarks will be illustrated by examples. 



V. Flow past a plate. (Fig. 1.) 

 Let , V 7 . i/ox c ( w - d) 



* ~ dw~ (w-l)V* ' ' " * K '' 



where 0<rf<l, b>l, c>0. 



To ensure that f is single-valued the w-plane is cut along 

 the positive part of the real axis. 



The factor -, -^-- n is the Schwarzian factor at C. 



(to — lyP 



The term b + w 1/2 is introduced to make the velocity finite 

 at infinity and to ensure the termination of the thin rigid 

 boundary at B. 



As previously explained, an essential singularity is 

 introduced to avoid having an infinite velocity near BC 



The nature of this singularity depends on the assumptions 

 as to the flow behind the plate. 



