HYDRODYNAMICAL RELATIONS 17 



The work done on the other faces is obtained in the same way, and 

 equating the total to the increase in energy gives 



P I [^ + i(u' + v' + w')] = -\~ (Pu) + ~ (Pv) + ~ (Pw)] 

 at \_dx dy dz J 



which in vector notation is 



p|[£^ + i(v-v)] = -div(Pv) 

 at 



The energy equation may be transformed into a more useful expression 

 by combining it with the equations of continuity and motion. Solving 

 for variation in internal energy gives 



(2.5) p-— = — Pdivv — v-gradP — pv — 



at dt 



But from Eq. (2.2) 



J. 1 dp . \ , ,. \ dp 



-divv = --^ + -(v-grad)p = - -f 



P ot p p at 



and from Eq. (2.4) 



gradP = -p— - 

 dt 



which gives on substitution 



(2.6) ,dE^Pdp 



dt p dt 



D. Pressure-density relation. In our derivation of the fundamental 

 hydrodynamic equations, the effect of dissipation processes has been 

 neglected. If the properties of a specified small element of the fluid are 

 described by these equations, a further condition on the state of this 

 element is implied which has not been explicitly stated. If no dissi- 

 pative processes take place in a given period of the motion, no element 

 moving Avith the fluid can exchange heat with any other element or its 

 surroundings during this time. The changes in the physical state of the 

 element must therefore take place at constant entropy, a situation which 

 can be expressed by the relation 



dt 



