10 12.] FLOW OF MOMENTUM. 



portion of matter as it moves alorg. In another method, which 

 is indeed more consistent with the Eulerian notation, we fix our 

 attention on a certain region of space, and investigate the change 

 in its properties produced as well by the flow of matter inwards 

 and outwards across the boundary as by the action of external 

 forces on the included mass*. 



Let Q denote the measure, estimated per unit volume, of any 

 quantity connected with the properties of a fluid, and let us cal 

 culate the rate of increase of Q in a rectangular space dxdydz 

 having its centre at (x, y, z). This is expressed by 



(14). 



Now the amount of Q which enters per unit time the specified region 

 across the y^-face nearest the origin is t Qu \ ~ dx J dydz, and 

 the amount which leaves the region in the same time by the oppos 

 ite face is f Qu + % V djc\ dydz. The two faces together give a 



gain of -- -= dxdydz per unit time. Calculating in the same 



way the effect of the flow across the remaining faces, we have for 

 the total gain of Q due to the flow across the boundary the formula 



d.Qu d.Qv d.Qw 



~\~ 7 1 7 



\dxdydz (15). 



V dx dy dz 



First, let us consider the change of mass, i. e. we put Q = p, the 

 mass per unit volume. Since the quantity of matter in any region 

 can vary only in consequence of the flow across the boundary, the 

 expressions (14) and (15) must in this case be equal ; this gives 

 the equation of continuity in the form (8). 



Next, let us take the change of momentum, making Q = pit, 

 the momentum parallel to x per unit mass. The momentum con 

 tained in the space dxdydz is affected not only by the passage of 

 matter carrying its momentum with it across the boundary, but 

 also by the forces acting on the included matter, viz. the pressure 

 and the external impressed forces. The effect of these resolved 



* See Maxwell, On the Dynamical Theory of Gases, Phil. Trails. 1867, p. 71. 

 Also, Greeiihill, Solutions of Cambridge Problems for 1875, p. 178. 



