Mass Balances for Multidimensional Flows

The foregoing mass balances were for a small number of flows in and out. Obviously, the same idea can be readily applied to a much larger number of flows in and out. One equation often used in theoretical fluid mechanics (see Chap. 10) is the mass balance equation for an arbitrary point in space. We find this equation by defining the coordinates and components of the local fluid velocity, as shown in Fig. 3.5. Our system is a small open-faced cube. 

The mass balance for this system in rate form is 

Now we let Lx, Ay, and Az each approach zero siihultaneously, so that the cube shrinks to a point. Taking the limit of the three ratios on the right-hand side of this equation, we find the partial derivatives  

If the density is constant or the density changes are small enough to be neglected, this simplifies to i 

By letting Ax, Ay, and Az approach zero we have shrunk the system to a single point. Thus Eq. 3.36 is the mass balance for any point in space it is often called the general continuity equation Equation 3.37 is the mass balance for any point in space which contains a constant-derisity fluid. 

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