Since the velocity of a point rotating with the walls has components 



M = - coy, V - cox, 



the velocity of the fluid relative to the walls has components, taken in the instantaneous 

 directions of the rotating axes, 



m'=0, v' = - 2 CO X + a. 



Thus in the relative motion the lines of flow are straight and parallel to the walls. 



The constant A is connected with the total flow through the channel; the volume passing 

 per second relatively to the walls, per unit of length perpendicular to the planes of flow, is 



°2 



Q' - I v'dx = (^2 - a^) [A - CO {a^ + a2)^- 

 °l 



The relative velocity varies linearly from one wall to the other; it may vanish on an 

 intermediate plane and have opposite directions near the two walls; see Reference 10, page 79. 



101. ROTATING ANGLE 



Consider the irrotational two-dimensional motion of the fluid in an angular space formed 

 by two semi-infinite plane walls meeting at a fixed angle 2a. Let the walls rotate at constant 

 velocity co about their line of intersection. With the origin taken on this line and the a!-axis 

 drawn along the bisector of the angle, the walls will be represented on the a;y-plane by two 

 radii drawn from the origin at 6 = -a, where 6 = tan~ (y/x). 



The following assumption, suggested by the conjugate flow of Section 36, will be found 

 to satisfy Equation [99b] on the walls, with C - 0, since here S = r: 



1 „ sin 2 6* 1 „ cos 2 



(fj ^ -— cor , ip = — (oT^ . [101a, b] 



2 cos 2ct 2 cos 2 a 



Then 



sin 2 6 cos 2d \(o\t 



g ^r — , qn^ COT -— , y = — . - [101c, d,e] 



cos 2 01 cos 2 a cos 2 a 



For the velocity relative to the walls, by Equations [99e,f], 



sin 2 (9 /cos 2 ^ \ 



q;=cor — , qQ = coT[ — -1 . [101f,g] 



cos 2 a " \ cos 2 a I 



248 



