562 VISCOSITY. [CHAP, xi 



307. The general formulas 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 



u = t v = 0, w = .............................. (i), 



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

 by (8) above, these conditions are satisfied by 



*(A) = .................................... (ii). 



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

 decay (r) are then given by 



.............................. (iii). 



The modes ?i = 1 are of a rotatory character ; the equation (ii) then 

 becomes 



tan ha = ha .................................... (iv), 



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



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



r = 2-75 a 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 #, y, z 



dy 



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 



dv dv L dw n , ... 



-j- + -7- + -r- =0 (vn), 



dx dy dz 



we find ^ n + 1 (Aa) = (viii). 



