16 HYDRODYNAMICAL RELATIONS 



riP 



[Px — Px+dx] dydz = ——- dxdydz ■ 

 dx 



Equating the force and inertia terms, we have 



du _ _dP 

 dt dx 



together with similar terms for the other two components. If the total 

 acceleration is resolved into its two parts we have 



/o ON ^^ > ^^1 ^^ . du dP 



(2.3) P^ + P^T '^ P^^ '^ P^-^ = ~-^ 



dt dx dy dz dx 



dv ^ dv , dv , dv dP 



P— + pu— + pv— -^ pw~ = - — 



dt dx dy dz dy 



dw , dw , dw , dw dP 



P-^ + P^^^ + P^-T '^ P^^ = - — 

 Of dx dy dz dz 



These equations are equivalent to the single vector equation 



(2.4) PU = P^ + P (v-grad)v = -grad P 



at dt 



C. Conservation of energy. As in the derivation of the equation of 

 motion, it is convenient to consider an element of volume moving with 

 the fluid and enclosing a fixed mass of fluid. The total energy per unit 

 mass of the fluid consists of kinetic energy and internal energy E, which 

 is the sum of thermal and any chemical energy. The change in time dt 

 for the element of volume dxdydz is 



P T [^ + K^' + i^' + W)] dt dxdydz 

 dt 



where the total time derivative is again used to account for the displace- 

 ment of the element during the interval. This change in energy must 

 equal the work done on the faces of the element. The work done in 

 time dt on an area dydz in motion along x is the product of force and 

 displacement, or Pu-dtdydz, and the net amount of work done on the 

 two faces of the volume element is 



[{Pu)x — {Pu)x+dx] dt dydz = (Pu) dt dxdydz 



dx 



