196 On Two-Dimensional Fields of Flow. 



The angular range which the (z, f) transformation must 

 have is known from inspection of the fixed boundary ; it is 

 generally — ir. As the angular range is ir times the order 

 at infinity of dz/d^ there can be no uncertainty about the 

 power of ^ which must be introduced into <^in order to 

 give the desired angular range. Thus the synthesis of ^in 

 terms of $?q, ^, ^, and corner factors, is determined. 



When there is flow to infinity between two free stream- 

 lines which tend to parallelism, one factor is employed to 

 represent the double curve. The linear range is taken from 

 %=c through f=-f-oo and £'=— c© to f = — c. Two forms 

 of curve-factor, namely, 



> 52 = (?+F) sinh 7 -i(? 2 -c 2 ) 1/2 , & = (*-£) smhr / -i(?-c>) 1, \ 

 where k>c, can be adapted to counterbalance the variability 

 of modulus of real or pairs of imaginary factors in YdQdw ; 

 it is unnecessary to go into details of the algebra. 



As an example consider the case in which the fixed 

 boundary consists of a semi-infinite and a finite straight line 

 meeting at right angles, and the motion is due to a single 

 source. The transformations are 



^= ( ^?P- *"=-|> g {(r-«) 2 +/3 2 },. . (i5) 



whence 



dw/d£=-(m/27r) (?-*){(?- *Y + P}-\ 



The rules of synthesis indicate that 



if=(c 1/2 +? 1/2 )(* 1 ' 2 +?" 2 ) 2 r 5 - l 2 or(c 1/2 +r ,/2 )fr-«)r s - 1 2 , (iej 



according as a is positive or negative. 



9. Free stream-line problems in wliicli part of the fixed 

 boundary is curved. — The possibility of specifying solvable 

 problems of this type may be illustrated by the case of flow 

 through a semi-infinite straight pipe with a symmetrical 

 curved nozzle. Fig. 2 shows half the configuration (which 



Fisr. 2. 



v U 



is all that need be considered), and the values assigned to 

 £ at the various points. The transformations are of the 

 form 



