14.5 Fluid dynamics  (Page 3/9)

Since liquids are essentially incompressible, the equation of continuity is valid for all liquids. However, gases are compressible, so the equation must be applied with caution to gases if they are subjected to compression or expansion.

Mass conservation

The rate of flow of a fluid can also be described by the mass flow rate or mass rate of flow. This is the rate at which a mass of the fluid moves past a point. Refer once again to [link] , but this time consider the mass in the shaded volume. The mass can be determined from the density and the volume:

The mass flow rate is then

where $\rho$ is the density, A is the cross-sectional area, and v is the magnitude of the velocity. The mass flow rate is an important quantity in fluid dynamics and can be used to solve many problems. Consider [link] . The pipe in the figure starts at the inlet with a cross sectional area of $A_1$ and constricts to an outlet with a smaller cross sectional area of $A_2$ . The mass of fluid entering the pipe has to be equal to the mass of fluid leaving the pipe. For this reason the velocity at the outlet $\left(v_2\right)$ is greater than the velocity of the inlet $\left(v_1\right)$ . Using the fact that the mass of fluid entering the pipe must be equal to the mass of fluid exiting the pipe, we can find a relationship between the velocity and the cross-sectional area by taking the rate of change of the mass in and the mass out:

[link] is also known as the continuity equation in general form. If the density of the fluid remains constant through the constriction—that is, the fluid is incompressible—then the density cancels from the continuity equation,