The Continuity equation
The mass of a moving fluid doesn’t change as it flows. This leads to an important relationship called the continuity equation. Consider a portion of a flow tube between two stationary cross sections with areas A1 and A2 (Fig. 12.21). The fluid speeds at these sections are v1 and v2, respectively. As we mentioned above, no fluid flows in or out across the side walls of such a tube. During a small time interval dt, the fluid at A1 moves a distance ds1 = v1 dt, so a cylinder of fluid with height v1 dt and volume dV1 = A1v1 dt flows into the tube across A1. During this same interval, a cylinder of volume dV2 = A2v2 dt flows out of the tube across A2.
Let’s first consider the case of an incompressible fluid so that the density r has the same value at all points. The mass dm1 flowing into the tube across A1 in time dt is dm1 = rA1v1 dt. Similarly, the mass dm2 that flows out across A2 in the same time is dm2 = rA2v2 dt. In steady flow the total mass in the tube is constant, so dm1 = dm2 and
rA1v1 dt = rA2v2 dt or
The product Av is the volume flow rate dV>dt, the rate at which volume crosses a section of the tube:
The mass flow rate is the mass flow per unit time through a cross section. This is equal to the density r times the volume flow rate dV/dt.
Equation (12.10) shows that the volume flow rate has the same value at all points along any flow tube (Fig. 12.22). When the cross section of a flow tube decreases, the speed increases, and vice versa. A broad, deep part of a river has a larger cross section and slower current than a narrow, shallow part, but the volume flow rates are the same in both. This is the essence of the familiar maxim, “Still waters run deep.” If a water pipe with 2-cm diameter is connected to a pipe with 1-cm diameter, the flow speed is four times as great in the 1-cm part as in the 2-cm part.
We can generalize Eq. (12.10) for the case in which the fluid is not incompressible. If r1 and r2 are the densities at sections 1 and 2, then
If the fluid is denser at point 2 than at point 1 1r2 7 r12, the volume flow rate at point 2 will be less than at point 1 1A2v2 6 A1v12. We leave the details to you. If the fluid is incompressible so that r1 and r2 are always equal, Eq. (12.12) reduces to Eq. (12.10)