286-287] INTERPRETATION. 517 



287. To compute the rate of dissipation of energy, due to 

 viscosity, we consider first the portion of fluid which at time t 

 occupies a rectangular element xyz having its centre at (x, y, z). 

 Calculating the differences of the rates at which work is being done 

 by the tractions on the pairs of opposite faces, we obtain 



\dx ( pxxU +pxyV + p ** w ) + dy ( pyxU +PyyV + Pyz W) 



............... (1). 



The terms 



'zu 



ax dy dz ] \ dx dy dz 



express, by Art. 286 (1), the rate at which the tractions on the faces 

 are doing work on the element as a whole, in increasing its kinetic 

 energy and in compensating the work done against the extraneous 

 forces X, F, Z. The remaining terms express the rate at which 

 work is being done in changing the volume and shape of the 

 element. They may be written 



where a, b, c, f, g, h have the same meanings as in Arts. 31, 284. 

 Substituting from Art. 284 (2), (3), we get 



+ }- J/A (a + b + c) 2 + 2/A (a 2 + 6 2 + c 2 + 2/ 2 + 2 



...... W- 



If p be a function of p only, the first line of this is equal to 



DE/Dt.p&sBySz, 



provided 



(3), 



i.e. E denotes, as in Art. 11, the intrinsic energy per unit mass. 

 Hence the second line of (4) represents the rate at which energy 

 is being dissipated. On the principles established by Joule, the 

 mechanical energy thus lost takes the form of heat, developed in 

 the element. 



