Changine the sign of V merely reverses all velocities, the source becoming a sink. S 

 may be reversed in space by drawing the a;-axis in the opposite direction. (For notation and 

 method; see Section 34; Reference 2, Section 8.21.) 



54. LINE SOURCE AND SINK IN UNIFORM STREAM. 



Let a line source be located at {a,0) and an equal sink at {-a,0), and superpose uniform 

 flow at velocity U toward negative x; see Figure 83. From Equations [35a] and [40a] the re- 

 sultant potential and stream functions can be written, in terms of a real positive constant g, 



Figure 83 — Cylinder whose cross-section 



S is a Rankine oval, obtained from a line 



source at {a,0) and a sink at {—a,0). 



See Section 54. 



where 



w = U \ 3 + g [In (z + a) - In (2 - a )]\ 

 iji = U [x + gln~-] = U [x +-ln — 



<P ^ U[y - g (d^ - d^)] - V ly-g- tan-^ ^ J 



^ x^ + y^ - a?-' 



[54a] 

 [54b] 



[54c] 

 [54c] 



r^ = fa; - aj^ + y"^ , r^ = (x + a)^ + y^ 

 tan d^ = y./(x - a), tan ^j = y/(x + a) ■ ,- 



and all angles may be supposed to lie between -tt and n and to have the sign of y. Also 



. r /x-ax + a\'\ , , 



u = U\-l+g(-— jl [54d] 



124 



