﻿Stead>/ Motion of a Viscous Fluid. 463 



We see that to this approximation the velocity across any 

 -cross section follows the same parabolic law as in the flow 

 between two parallel planes. 



Solutions I., II., and IV. lead to some interesting sets of 

 stream-lines. They cannot, however, be realized physically, 

 and they seem to be of little importance. We pass to the 

 consideration of solution III. 



^L centrifugal pump. — We have 



^ = - hv6 + Ar b + 2 + Br 2 + (J log r, 

 and the components of velocity are given by 

 1 "dyfr hv 

 r "d@ "" r 



or r 



From the equations of motion we can determine the mean 

 pressure. If there are no external forces 



p=-ib P vB0 + pf(r), 



where f(r) can readily be determined if necessary. If we 

 include the whole of the space round the origin, ^> rrmst be 

 single valued, and hence B = 0. In this case the solution 

 corresponds to the motion generated by the rotation of a 

 perforated cylinder, which, as it rotates, either sucks in or 

 ejects fluid uniformly over its surface. The fluid may either 

 flow away to infinity, or it may be absorbed by a coaxial 

 porous cylinder, which may be at rest or may be rotating 

 with any angular velocity. The total flux of fluid will 

 determine b, while the angular velocities of the cylinders 

 will determine A and 0. Such an arrangement will be in fact 

 a centrifugal pump. When the fluid is viscous vanes are 

 not absolutely necessary, although, of course, they mav 

 increase the efficiency of the machine. If there is no second 

 cylinder so that the fluid extends to infinity, then we must 

 have A = if the fluid is flowing outwards, for in that case 

 Z>>0. Let the radius of the cylinder be a, and let it rotate 

 with angular velocity Q, and eject a volume Q of fluid in 

 unit time; then 



C = a 2 H, 6 = Q/2ttv, 



and we find without difficulty, 



where p Q is the pressure at infinity. Thus we have the 



