14.5.1 The Equation of Continuity

FIGURE 14.25a shows smoke being used to help engineers visualize the airflow around a car in a wind tunnel. The smoothness of the flow tells us this is laminar flow. But notice also how the individual smoke trails retain their identity. They don’t cross or get mixed together. Each smoke trail represents a streamline in the fluid. FIGURE 14.25b illustrates three important properties of streamlines.

A photograph labeled (“a”) of a sports car. Lines labeled Streamline move over the car.

The figure labeled (b) illustrates rules for drawing streamlines.

Full Alternative Text

A key observation is that the flow speed changes as the streamlines get closer together or farther apart. In FIGURE 14.26, where an ideal fluid flows through a tube, the streamlines get closer together as the tube gets narrower, so the flow speed must increase. To understand quantitatively how the speed changes, focus on the shaded volume of fluid as it moves through the tube.

The figure consists of two parts labeled Before and After that illustrate the speed changes as a fluid flows from an area of higher to lower cross-section.

FIGURE 14.26 Full Alternative Text

An ideal fluid is incompressible, so this shaded portion of the fluid moves forward without changing volume. If the flow speed on the left side of the tube is $v_{1}$ , then during a time interval $Δ t$ the shaded fluid moves forward distance $Δ x_{1} = v_{1} Δ t$ and “vacates” volume $Δ V_{1} = A_{1} Δ x_{1} = A_{1} v_{1} Δ t$ . (New fluid moves in to fill this volume, but we’re focusing on the shaded portion of the fluid.) On the right side of the tube, where the flow speed is $v_{2}$ , the shaded fluid moves forward distance $Δ x_{2} = v_{2} Δ t$ and occupies a new volume $Δ V_{2} = A_{2} Δ x_{2} = A_{2} v_{2} Δ t$ . These two volumes must be equal, leading to the conclusion that

v_{1} A_{1} = v_{2} A_{2}

(14.18)

Equation 14.18 is called the equation of continuity, and it is one of two important equations for the flow of an ideal fluid. The equation of continuity says that flow is faster in narrower parts of a tube, slower in wider parts. You’re familiar with this conclusion from many everyday observations. For example, water is shot from the narrow nozzle in FIGURE 14.27 much faster than it’s flowing through the wider hose.

A photograph of a person spraying water through a garden hose.

The quantity

Q = \frac{Δ V}{Δ t} = v A

(14.19)

is called the volume flow rate. The SI units of $Q$ are $m^{3} / s$ , although in practice $Q$ may be measured in ${cm}^{3} / s$ , liters per minute, or, in the United States, gallons per minute. Another way to express the meaning of the equation of continuity is to say that the volume flow rate is constant at all points in a tube.

EXAMPLE 14.10 River rafting

A 20-m-wide, 4.0-m-deep river with a triangular cross section flows at a lazy 1.0 m/s. As it makes its way to the sea, the river enters a gorge where rocky walls confine it to a width of 4.0 m and a depth of 6.0 m, again with a triangular cross section. What is the river’s volume flow rate? And what is its speed through the gorge?

MODEL Model the flowing river as an ideal fluid.

SOLVE A triangular river cross section of width $w$ and depth $d$ has cross-section area $A = \frac{1}{2} w d$ . In the first part of the river,

A_{1} = \frac{1}{2} (20 m)(4.0 m) = 40 m^{2}

and thus the river’s flow rate is

Q = v_{1} A_{1} = (1.0 {m/s)(40 m}^{2}) = 40 m^{3} / s

The volume flow rate is the same at all points along the river, including the gorge. The gorge has cross-section area

A_{2} = \frac{1}{2} (4.0 m)(6.0 m) = 12 m^{2}

Thus the river’s speed through the gorge is

v_{2} = \frac{Q}{A_{2}} = \frac{40 m^{3} /s}{12 m^{2}} = 3.3 m / s

REVIEW A river speed of 3.3 m/s, or about 7 mph, is typical of a Class III whitewater-rafting river. For a 6-m-deep river, some whitewater and waves on the surface don’t greatly affect our assumption of smooth flow through a perfectly triangular cross section.

14.5.1 The Equation of Continuity

FIGURE 14.25 Particles in an ideal fluid move along streamlines.

FIGURE 14.26 The flow speed changes as a tube’s cross-section area changes.

FIGURE 14.27 Narrowing the cross section increases the water’s speed.

EXAMPLE 14.10 River rafting