w = U\(z-3.) e-'y + e'y + — In , [69i] 



L 3 - 2j J 277 a 



I a\ r I a^\ Ft 



4>=U[r+ — cos(6)-y) (9, i/- = f/ r sin (6» - y) + — In—, [69k,l] 



\ T I 2n \ r f 2n a 



f a^\ { o}\ r , 



?r = f/ I- 1 + — jcos (0 - y), qQ = V\A+ — JBin(0-y) + , [69m,n] 



where s-Sj = re' ^, r = [(a; - a;^)^ + ^y - yji^l'-^S 



and is a polar angle about the point 2 , or 6 = tan~^ [(y - J',)/(a; - a;.)]. For, it is obvious 

 that the entire flow is displaced in the desired manner; and at infinity the term {/se"'^ pre- 

 dominates in w and represents uniform flow at the angle y, as in Section 35. 



(For notation and method; see Section 34; Reference 1, Article 69; Reference 2, 

 Section 7.12.) 



70. TRANSLATION OF A CIRCULAR CYLINDER WITH CIRCULATION 



By viewing the situation discussed in the last section from a frame of reference that 

 moves with the fluid at infinity, as in Section 68, formulas are obtained for the case of a cyl- 

 inder that has circulation around it but is moving toward positive x at velocity U, while the 

 fluid is at rest at infinity. For this case 



a^U ir z , , 



w= + In—, [70a] 



2 2n a 



„ cos BY „ sin 6 V t ^ , 



(f,=a^U 9y il,=-a^U + — In — , [70b, c] 



r 2rr r 2n a 



o cos 9 T sin d r r , •> 



q^a^U , qf^^a^IJ + — [70d,e] 



J.2 ^ ^2 2nr 



The axes of coordinates move with the cylinder. 



A stagnation point can now occur only where cos ^ = and also 



tT 



= sin (9 = ± 1. - [70f] 



2na^U 

 For r/V > 0, = - 90°; for V/V <Q, 6 = 90°; and in either case 



r = 27ra2|p-. % [70g] 



156 



