11.8.  The Equation of Continuity

Have you ever used your thumb to control the water flowing from the end of a hose, as in Figure 11.27? If so, you have seen that the water velocity increases when your thumb reduces the cross-sectional area of the hose opening. This kind of fluid behavior is described by the equation of continuity. This equation expresses the following simple idea: If a fluid enters one end of a pipe at a certain rate (e.g., 5 kilograms per second), then fluid must also leave at the same rate, assuming that there are no places between the entry and exit points to add or remove fluid. The mass of fluid per second (e.g., 5 kg/s) that flows through a tube is called the mass flow rate.

Figure 11.27  When the end of a hose is partially closed off, thus reducing its cross-sectional area, the fluid velocity increases.

Figure 11.28 shows a small mass of fluid or fluid element (dark blue) moving along a tube. Upstream at position 2, where the tube has a cross-sectional area A₂, the fluid has a speed v₂ and a density r₂. Downstream at location 1, the corresponding quantities are A₁, v₁, and r₁. During a small time interval Dt, the fluid at point 2 moves a distance of v₂Dt, as the drawing shows. The volume of fluid that has flowed past this point is the cross-sectional area times this distance, or A₂v₂Dt. The mass Dm₂ of this fluid element is the product of the density and volume: Dm₂=r₂A₂v₂Dt. Dividing Dm₂ by Dt gives the mass flow rate (the mass per second):

(11.7a)

Similar reasoning leads to the mass flow rate at position 1:

(11.7b)

Since no fluid can cross the sidewalls of the tube, the mass flow rates at positions 1 and 2 must be equal. But these positions were selected arbitrarily, so the mass flow rate has the same value everywhere in the tube, an important result known as the equation of continuity. The equation of continuity is an expression of the fact that mass is conserved (i.e., neither created nor destroyed) as the fluid flows.

Figure 11.28  In general, a fluid flowing in a tube that has different cross-sectional areas A1 and A2 at positions 1 and 2 also has different velocities v1 and v2 at these positions.

EQUATION OF CONTINUITY

The mass flow rate (rAv) has the same value at every position along a tube that has a single entry and a single exit point for fluid flow. For two positions along such a tube  (11.8)  where    r=fluid density (kg/m3)    A=cross-sectional area of tube (m2)    v=fluid speed (m2) SI Unit of Mass Flow Rate: kg/s

The density of an incompressible fluid does not change during flow, so that r₁=r₂, and the equation of continuity reduces to

(11.9)

The quantity Av represents the volume of fluid per second that passes through the tube and is referred to as the volume flow rate Q:

(11.10)

Equation 11.9 shows that where the tube’s cross-sectional area is large, the fluid speed is small, and, conversely, where the tube’s cross-sectional area is small, the speed is large. Example 11 explores this behavior in more detail for the hose in Figure 11.27.

Concept Simulation 11.1

The speed of an incompressible fluid, such as water, changes as it flows through a hose whose cross-sectional area changes. In this simulation the user can select different cross-sectional areas and see how the speed of the water changes as it moves from one region to another. The equation of continuity states that the volume flow rate, Q=Av, remains constant at all points along the hose, where A is the cross-sectional area and v is the speed of the water. Thus, as A increases, v decreases, and vice versa. Related Homework: Conceptual Question 19, Problems 52, 87

A garden hose has an unobstructed opening with a cross-sectional area of 2.85×10–4 m2, from which water fills a bucket in 30.0 s. The volume of the bucket is 8.00×10–3 m3 (about two gallons). Find the speed of the water that leaves the hose through (a) the unobstructed opening and (b) an obstructed opening with half as much area. Reasoning  If we can determine the volume flow rate Q, the speed of the water can be obtained from Equation 11.10 as v=Q/A, since the area A is given. The volume flow rate can be found from the volume of the bucket and its fill time. Solution (a) The volume flow rate Q is equal to the volume of the bucket divided by the fill time. Therefore, the speed of the water is  (11.10)  (a) Water can be considered incompressible, so the equation of continuity can be applied in the form A1v1=A2v2. Since A2=½A1 we find that  (11.9)

The next example applies the equation of continuity to the flow of blood.

In the condition known as atherosclerosis, a deposit or atheroma forms on the arterial wall and reduces the opening through which blood can flow. In the carotid artery in the neck, blood flows three times faster through a partially blocked region than it does through an unobstructed region. Determine the ratio of the effective radii of the artery at the two places. Reasoning  Blood, like most liquids, is incompressible, and the equation of continuity in the form of A1v1=A2v2 (Equation 11.9) can be applied. In applying this equation, we use the fact that the area of a circle is p r2. Problem solving insight The equation of continuity in the form A1v1=A2v2 applies only when the density of the fluid is constant. If the density is not constant, the equation of continuity is r1A1v1=r2A2v2. Solution From Equation 11.9, it follows that The ratio of the radii is

Water flows from left to right through the five sections (A, B, C, D, E) of the pipe shown in the drawing. In which section(s) does the water speed increase, decrease, and remain constant? Treat the water as an incompressible fluid.     Speed Increases   Speed Decreases   Speed is Constant   a.   A, B   D, E   C   b.   D   B   A, C, E   c.   D, E   A, B   C   d.   B   D   A, C, E   e.   A, B   C, D   E  Background: The equation of continuity holds the key here. The fact that water can be treated as an incompressible fluid is important. For similar questions (including calculational counterparts), consult Self-Assessment Test 11.2. This test is described at the end of Section 11.11.

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