298-299] DIFFUSION OF ANGULAR VELOCITY. 541 



The corresponding result for air is 



These results indicate how rapidly a surface of discontinuity, if it could 

 ever be formed, would be obliterated in a viscous fluid. 



The angular velocity (<>) of the fluid is given by 



du U 



T ........................ (xi). 



fy (in*)* 



This represents the diffusion of the angular velocity, which is initially 

 confined to a vortex-sheet coincident with the plane y=0, into the fluid on 

 either side. 



299. When the fluid does not extend to infinity, but is 

 bounded by a fixed rigid plane y = h, then in determining the 

 motion due to a forced oscillation of the plane y = both terms of 

 (3) are required, and the boundary conditions give 



A + B = a, 



whence u = a 7 . * ., (11), 



sm 



as is easily verified. This gives for the retarding force per unit 

 area on the oscillating plane 



acoth(l + i)ph.e i & + * ) ...(12). 



The real part of this may be reduced to the form 

 /o _ sinh 2j3h cos (o~t + e -f ITT) + sin 2/3h sin (at -\- e + -TTT) 



A/ AjjjfcjCL 



cosh 2/3h cos 2/S/i 



(13). 



When @h is moderately large this is equivalent to (9) above ; 

 whilst for small values of (3h it reduces to 



as might have been foreseen. 



This example contains the theory of the modification introduced by 

 Maxwell* into Coulomb's method t of investigating the viscosity of liquids by 

 the rotational oscillation of a circular disk in its own (horizontal) plane. The 



* 1. c. ante p. 513. 



t Mem. de Vlnst., t. Hi. (1800). 



