Wu 



origin of the yz plane. (The acceleration associated with such a moving origin 

 is small, since (q-v)q has been assumed to be negligible under the present 

 condition.) The free- stream velocity of the cross flow is 



^^'''^^ ' "li^ ^ "bTJ^^'''^^ ' ^'^^''^ (85) 



pointing in the direction of y increasing. The cross-flow velocity in the yz 

 plane will be denoted by v = (0, V2, V3), which is required to tend to (0, v(x,t), 



0) as y2 + z2 — 00. The vorticity of the cross flow, l, = Bvg/By - Bvj/^z, satis- 

 fies the equation 



i^ -- V^^l-- Vayy^^.^) ■ (86) 



In terms of the stream function 41 of the cross flow, defined by V2 = B>/^/3z, 



V3 = -Bi/;/By, ^ may be written as 



i = -A20 .' ' ,, . /.-. (87) 



Noting that has a time factor exp(if^t), we obtain for the equation 



(A2-/32)A20 =0, (/3= (ia;A)^'2) . (88) 



The solution satisfying the condition at infinity and no- slip conditions at the 

 cylinder is found to be 



i// = V(x,t) f (r) sin (9 , ( r > a) (89) 



with ,, ; r.J - ^, ,, 



f (r) = r + AaKj(^r) -B^ , (90) 



2 2 Ki(/^a) 

 A = , B = 1 + ■ /qi\ 



/3a Ko(/3a) /3a Ko(/3a) ^^^> 



where (r,^) are the polar coordinates defined by y = r cos 6, z = v sin 6, 

 and Kn(/3r) denote the modified Bessel functions of the second kind. The instan- 

 taneous lift acting on a section of length dx at x, L(x,t)dx, positive in the direc- 

 tion of y increasing, due to the forces of the cross flow, is 



L(x,t) = 77/xVa2W(c7-) , (92) 



where 



W(a) = i (2B- 1) 



4 Ki(/3a)' 

 1 + 



= F(cr) + iG((T) , (93) 



/3a Ko(/3a)_ 

 a = (a2a;/v)^/2 , j^^j _ (94) 



1198 



