Figure 45 — Part of flow net for a 

 line quadrupole: w = A/z^ ^ 



39. FLOW IN AN ANGLE 



w =^ Az 



Ti/a 



[39a] 



where A and a. are real constants and a is positive. If s = re'^, 



+ i ,A = A{re'^)'"'^= Ar^f'^ (cos v -^ ^ i s\r. „ f ) 



<^ = 4;.7r/a cos ^— , '^ = At''^'^ 



[39b,c] 



dz ' 



,"S- 1 



9 = /I JI f« 



[39d,e] 



The origin is a singular point, unless n/^ is an integer. If ei > n, dw/dz becomes 

 infinite at s = 0. In any case, as z goes round the point 2 = 0, the amplitude of w increases by 

 {2TT)n/a and that of dw/dz by (277-) {tt/a - 1). Hence, if jr/a is not an integer, both w aad dw/dz 

 are multiple-valued in the neighborhood of 2 = or a; = y = 0. In applications, therefore, a 

 boundary must be introduced excluding the origin and also extending to infinity, in order to 

 make dw/dz and the components of the velocity single-valued. 



The diagram of the equipotentials is the same as that of the streamlines but rotated 

 through an angle {*/2. 



The principal application is to represent the flow between two planes meeting at an 

 angle of a radians. On one plane let = and on the other = a ; then = on both planes, 

 and they cut the a;y-plane along a streamline. On these planes = ^ Ar'"^'* . 



One sector of the streamline diagram for et = n-/3 is shown in Figure 46; that for 

 * = 3n-/2 is shown in Figure 47. 



79 



