11. THE PRESSURE EQUATION FOR ROTATING BOUNDARIES 



The following special case may be noted for reference. Suppose that incompressible 

 fluid is set into irrotational motion by the steady rotation of a solid boundary, internal or 

 external, the fluid being otherwise unbounded. Then the flow pattern will obviously be always 

 the same relative to the boundary but at any point fixed in space variations will occur. The 

 distribution of values of the potential <p can be imagined to rotate with the boundary, but 

 otherwise to remain unchanged. Since the motion is not steady in space, the Bernoulli equa- 

 tion cannot be used. Let the density p be uniform. 



Let 6 denote an angle of position about the axis of rotation and let the angular velocity 

 of rotation be a>. Then that value (ft^ oi (/) which, at time t, is at a point P is carried forward 

 by the rotation during an interval dt to a. point P'at which is greater by dd = odt. At time t, 

 on the other hand, the value of (/> at P'was 



^1 dd ^ dd 



Thus, during dt, changes at P'by 



/ <50 \ <50 



d(f) =01 - l0j + CO — dt\ = - CO — dt 



Hence, at any fixed point in space, 



•50 '^0 rin 1 



— = - CO — =co'S>'qa lllaj 



dt 36 ^ 



where oTdenotes the distance of the point from the axis of rotation, and q^ = -(l/To) dcji/3d 



and represents the transverse component of the particle velocity; see Equation [6r]. 



Thus Equation [9e] for the pressure p can be written, when the boundary rotates steadily, 



p = p {coZ qQ-^q^ -^) +J)q [lib] 



or, if n = 0, 



p = p{coo^qQ-^q^) + j)Q [lie] 



where p is either a constant or at most a function of the time. 



These equations can also be written in terms of velocities relative to axes that rotate 

 with the boundary. The radial component of velocity q/ is the same as q^, the same component 

 taken relative to fixed axes, but the transverse component is qa' ■= q^- coco. Writing 

 q^ = q^ + qa and then substituting for q^. 



lp{co^%^-q'^)^p^ ■; _ [lid] 



where q'^ = q'^ + 



18 



