7. Derivation of the Continuity Equation in Cartesian Coordinates

[Previous Article: How Euler Derived the Continuity Equation]
The continuity equation is an expression of a fundamental conservation principle, namely, that of mass conservation. It is a statement that fluid mass is conserved: all fluid particles that flow into any fluid region must flow out. To obtain this equation, we consider a cubical control volume inside a fluid. Mass conservation requires that the the net flow through the control volume is zero. In other words, all fluid that is accumulated inside the control volume (due to compressibility for example) + all fluid that is flowing into the control volume must be equal to the amount of fluid flowing out of the control volume.
Accumulation + Flow In = Flow Out

The mass of the control volume at some time t is

The time rate of change of mass in the control volume is

Now we can compute the net flow through the control volume faces. Starting with the x direction, the net flow is

Similarly, the net flow through the y faces is

while that through the z faces is

Upon adding up the resulting net flow and diving by the volume of the fluid element (i.e. dxdydz), we get the continuity equation in Cartesian coordinates

Voila!