100. ROTATING CHANNEL 



Consider the channel between two parallel plane walls of infinite extent which are 

 rotating at velocity oj about an axis drawn parallel to the walls. Let there be no component 

 of the fluid velocity in the direction of the axis. The walls will be represented on the a^y-plane 

 by two parallel lines; let these lie at distances a, , a from the axis of rotation. Take the 

 origin on the axis of rotation, and draw the jc-axis perpendicular to the walls, as in Figure 167. 





y 



\ 



(Fluid) 







^ 





X 



V, 



J 

 / 



flj 



'h 



Figure 167 — A channel or infinite box in rotation. See Section 100. 



Then the equation of either wall will be of the form, x - constant. It follows that in 

 Equation [99c] y must cancel out. This condition is met if 



"/- = - J (a;^ - /) + Ax 



[100a] 



where A is an arbitrary constant. This is a permissible form for (/«, since the last term repre- 

 sents uniform motion at velocity A toward positive y, and the first term is adapted from 

 Equation [36e]. Using also Equation [36d], the corresponding potential and components of 

 velocity are 



(ji - <jj xy ~ Ay, u - - CO y, v = - o) x + A. 



[100b, c,d] 



247 



