562 VISCOSITY. [CHAP, xi 



307. The general formulae being once established, the applica- 

 tion to special problems is easy. 



1. We may first investigate the decay of the motion of a viscous fluid 

 contained in a spherical vessel which is at rest. 



The boundary conditions are that 



for r = a, the radius of the vessel. In the modes of the First Class, represented 

 by (8) above, these conditions are satisfied by 



The roots of this are all real, and the corresponding values of the modulus of 

 decay (r) are then given by 



r=-X- 1 = -(A)- 2 .............................. (iii). 



The modes n = l are of a rotatory character; the equation (ii) then 

 becomes 



t&uha=ha .................................... (iv), 



the lowest root of which is Aa=4'493. Hence 



r = -0495-. 

 v 



In the case of water, we have v = '018 c. s., and 



T=2'75a 2 seconds, 

 if a be expressed in centimetres. 



The modes of the Second Class are given by (10). The surface conditions 

 may be expressed by saying that the following three functions of x, y, z 



-5 



must severally vanish when r=a. Now these functions as they stand satisfy 

 the equations 



V 2 u = 0, V 2 V = 0, v 2 W = ........................ (vi), 



and since they are finite throughout the sphere, and vanish at the boundary, 

 they must everywhere vanish, by Art. 40. Hence, forming the equation 



du 



s + 



we find . 



