[60] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



Hence, in the case of an incompressible fluid, unless the whole mass comprised within the 

 surface S move together like a solid, there cannot fail to be a certain portion of vis viva 

 lost by internal friction. In the case of an elastic fluid, the motion which may take place 

 without causing a loss of vis viva in consequence of friction is somewhat more general, and 

 corresponds to velocities » + A», u + A«, m + Aic, where u, v, w are the same as in 



(139), and 



Au = $x + 2(ax + (Zy + yx) x - a (a? + y* + * 2 ), 



with similar expressions for Av and Aw. In these expressions a, /3, y are three constants 

 symmetrically related to x, y, %, and $ is a constant which has the same relation to each of the 

 co-ordinates *. 



51. By means of the expression given in Art. 49, for the loss of vis viva due to internal 

 friction, we may readily obtain a very approximate solution of the problem : To determine the 

 rate at which the motion subsides,- in consequence of internal friction, in the case of a series of 

 oscillatory waves propagated along the surface of a liquid. 



Let the vertical plane of xy be parallel to the plane of motion, and let y be measured ver- 

 tically downwards from the mean surface ; and for simplicity's sake suppose the depth of the 

 fluid very great compared with the length of a wave, and the motion so small that the square 

 of the velocity may be neglected. In the case of motion which we are considering, udx + vdy 

 is an exact differential d<f> when friction is neglected, and 



cp = ce~ my $m.(mx - nt), (140) 



where c, m, n are three constants, of which the last two are connected by a relation which it is 

 not necessary to write down. We may continue to employ this equation as a near approxi- 

 mation when friction is taken into account, provided we suppose c, instead of being constant, 

 to be a parameter which varies slowly with the time. Let V be the vis viva of a given portion 

 of the fluid at the end of the time t, then 



V= pc*m 2 fff €-*"•* dxdydx (141) 



But by means of the expression given in Art. 49, we get for the loss of vis viva during the 

 time dt, observing that in the present case ft. is constant, w — 0, 5 = 0, and udx + vdy = deb, 

 where is independent of x, 



which becomes, on substituting for (p its value, , 



8^m* dt fffe- Sm y dm dy dx. 



But we get from (141) for the decrement of vis viva of the same mass arising from the 



variation of the parameter c 



dc 

 - 2pm"c — dtfffe-' im y dx dy dx. 

 at 



* (See Note C at the end.) 



