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 A_{1} and A_{2} (Fig. 12.21). The fluid speeds at these sections are v_{1} and v_{2}, 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 A_{1} moves a distance ds_{1} = v_{1} dt, so a cylinder of fluid with height v1 dt and volume dV_{1} = A_{1}v_{1} dt flows into the tube across A_{1}. During this same interval, a cylinder of volume dV_{2} = A_{2}v_{2} dt flows out of the tube across A_{2}.
Let’s first consider the case of an incompressible fluid so that the density r has the same value at all points. The mass dm_{1} flowing into the tube across A_{1} in time dt is dm_{1} = rA_{1}v_{1} dt. Similarly, the mass dm2 that flows out across A_{2} in the same time is dm_{2} = rA_{2}v_{2} dt. In steady flow the total mass in the tube is constant, so dm_{1} = dm_{2} and
rA_{1}v_{1} dt = rA_{2}v_{2} dt or
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The product Av is the volume flow rate dV>dt, the rate at which volume crosses a section of the tube:
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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.
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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 r_{1} and r_{2} 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)