HYDRODYNAMICAL RELATIONS 15 



two components of motion and the conservation of mass requires that 

 their sum equal the increase of mass in the volume, hence 



(2.1) ^ + f (pu) + ^ ipv) + ~ (pw) = 

 dt dx dy dz 



which is the equation of continuity. This result is more concisely ex- 

 pressed in vector notation as 



(2.2) ^ + div (pv) = 



where the velocity vector v has the components u, v, w in Cartesian 

 coordinates. 



The eciuation of continuity could have been obtained equally well 

 by considering a small volume moving with the fluid and containing 

 the same definite mass of fluid at all times. Changes in density of the 

 fluid as it moves therefore require compensating changes in dimensions 

 of the element. The straightforward result obtained from this ap- 

 proach, originally due to Euler, comes out to be 



^ + pdivv = 

 dt 



if the symbol d/cU is understood to mean differentiation at a point mov- 

 ing with the fluid rather than a fixed point, i.e., 



d d , dx d , dy d , dz d d , d , d . d 



— = h— = \- u h^ [- w — 



dt dt dt dx dt dy dt dz dt dx dy dz 



With this meaning, it is evident that the results of the two approaches 

 are identical, as they must be. 



B. Conservation of momentum. In order to express Newton's second 

 law, or conservation of momentum, it is convenient to consider the 

 forces acting on element of volume dxdydz moving with the fluid rather 

 than an element fixed in space. Considering the x-component of mo- 

 tion, the acceleration of the element is given by the total time derivative 

 du/dt of the particle velocity, the total derivative being correct as it 

 expresses the total change in velocity resulting from the changes at a 

 fixed point in space and from displacement of the element in space. 

 The product of this acceleration and the mass p dxdydz of the moving 

 element must, by Newton's second law, equal the net force acting on 

 the element in the x-direction. This force is supposed due only to dif- 

 ferences in the pressure P at the two faces of area dydz and is given by 



