a;-axis, the amount of mass per unit time enter- 

 ing the cube through its left-hand face, per 

 unit time, is 



dipu) 



(..-^^).,. 



where pu stands for the value of this quantity 

 at the point {x, y, z). The amount which 

 leaves through the opposite face is 



\ dx 2/ 









8z 







^_.^---^\ 









1 







— 



^ly 



^ 







5a: 





^^^ 



"^ 





X 



The net increase in mass per unit time due to 

 flow in the a;-direction is the inflow minus the 

 outflow or 



dx 



Figure 3 — Illustrating the equation of 

 continuity. 



8x8ySz 



In an analogous manner the net increases in mass per unit time due to flow in the y- and s- 

 directions are, respectively, 



d{pv) 

 dy 



8x8y8z, 



d{pw) 

 dz 



8xSySz 



The net increase in mass per unit time for all three component directions must equal the 

 increase of mass per unit time within the cube, which is 



±(pSx8y8z) 



As the sides of the cube are fixed, this can also be written 



dp 



— Sx8y8z 

 dt 



Collecting the terms and dividing through by the volume or 8x8y8z, we obtain the equation of 

 continuity, 



dp d(pu) d(pv) d(pw) 



H + + 



dt 



dx 



dy 



dz 







^2a] 



This equation must be satisfied at all points throughout the fluid. 



In the subsequent chapters on two- and three-dimensional flow we are mainly concerned 

 with fluids of constant density so that p does not vary in space or in time. For this case, 

 dp/dt becomes zero and the equation of continuity takes the form, after canceling out p, 



Jm+^ + ^^0 [2b] 



dx dy 



dw 

 dz 



