518 VISCOSITY. [CHAP, xi 



If we integrate over the whole volume of the fluid, we find, for 

 the total rate of dissipation, 



(6), 



where 



f du dv dw\ 2 

 ^dx dy dz ) 



du dw 2 dv 



+ ..... < 7 >* 



If we write this in the form 



it appears that F cannot vanish unless 



a = 6 = c, and f=g = h=Q, 



at every point of the fluid. In the case of an incompressible fluid it is 

 necessary that the quantities a, b, c, f, g, h should all vanish. It easily 

 follows, on reference to Art. 31, that the only condition under which a liquid 

 can be in motion without dissipation of energy by viscosity is that there must 

 be nowhere any extension or contraction of linear elements ; in other words, 

 the motion must be composed of a translation and a pure rotation, as in the 

 case of a rigid body. In the case of a gas there may be superposed on this an 

 expansion or contraction which is the same in all directions. 



We now consider specially the case when the fluid is incompressible, so that 



If we subtract from this the expression 



du dv dw\ 2 



which is zero, we obtain 



dw dv\ 2 fdu dw\ 2 (dv du\ 2 } 



dv dw dv dw dw du dw du dudv du dv 



* Stokes, " On the Effect of the Internal Friction of Fluids on the Motion of 

 Pendulums," Gamb. Trans., t. ix., p. [58] (1851). 



