298-299] DIFFUSION OF ANGULAR ^ 7 ELOC^TY. 541 



The corresponding result for air is 



t = 8-3 y 2 . 



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 



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 



, sinh(l + i)@(h y) .. . , /11N 



whence u = a . \ /.. ^ ^ - . e t(&amp;lt;rt+g) ............ (11), 



sm 



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

 area on the oscillating plane 



_ p = p (i + i) fa coth (l+j)0h. e*^ . . . (12). 



The real part of this may be reduced to the form 

 smh 2/3A cos (o-Z + e -I- ITT) + sin 2/3A, sin (oi 



cosh 2A- cos 2/3h 



......... (13). 



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

 whilst for small values of f3h it reduces to 



fjLa/h.cos((rt+e) ..................... (14), 



as might have been foreseen. 



This example contains the theory of the modification introduced by 

 Maxwell* into Coulomb s method f 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. iii. (1800). 



