The equation reduces to show that the volume flow rate into the pipe equals the volume flow rate out of the pipe.
Geometry for deriving the equation of continuity. The amount of liquid entering the cross-sectional (shaded) area must equal the amount of liquid leaving the cross-sectional area if the liquid is incompressible.
Summary
Flow rate
Q is defined as the volume
V flowing past a point in time
t , or
where
V is volume and
t is time. The SI unit of flow rate is
but other rates can be used, such as L/min.
Flow rate and velocity are related by
where
A is the cross-sectional area of the flow and
v is its average velocity.
The equation of continuity states that for an incompressible fluid, the mass flowing into a pipe must equal the mass flowing out of the pipe.
Conceptual questions
Many figures in the text show streamlines. Explain why fluid velocity is greatest where streamlines are closest together. (
Hint: Consider the relationship between fluid velocity and the cross-sectional area through which the fluid flows.)
Consider two different pipes connected to a single pipe of a smaller diameter, with fluid flowing from the two pipes into the smaller pipe. Since the fluid is forced through a smaller cross-sectional area, it must move faster as the flow lines become closer together. Likewise, if a pipe with a large radius feeds into a pipe with a small radius, the stream lines will become closer together and the fluid will move faster.
The Huka Falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions. On average, the river has a flow rate of about 300,000 L/s. At the gorge, the river narrows to 20-m wide and averages 20-m deep. (a) What is the average speed of the river in the gorge? (b) What is the average speed of the water in the river downstream of the falls when it widens to 60 m and its depth increases to an average of 40 m?
(a) Estimate the time it would take to fill a private swimming pool with a capacity of 80,000 L using a garden hose delivering 60 L/min. (b) How long would it take if you could divert a moderate size river, flowing at
into the pool?
What is the fluid speed in a fire hose with a 9.00-cm diameter carrying 80.0 L of water per second? (b) What is the flow rate in cubic meters per second? (c) Would your answers be different if salt water replaced the fresh water in the fire hose?
a. 12.6 m/s; b.
; c. No, the flow rate and the velocity are independent of the density of the fluid.
Water is moving at a velocity of 2.00 m/s through a hose with an internal diameter of 1.60 cm. (a) What is the flow rate in liters per second? (b) The fluid velocity in this hose’s nozzle is 15.0 m/s. What is the nozzle’s inside diameter?
Prove that the speed of an incompressible fluid through a constriction, such as in a Venturi tube, increases by a factor equal to the square of the factor by which the diameter decreases. (The converse applies for flow out of a constriction into a larger-diameter region.)
If the fluid is incompressible, the flow rate through both sides will be equal:
Water emerges straight down from a faucet with a 1.80-cm diameter at a speed of 0.500 m/s. (Because of the construction of the faucet, there is no variation in speed across the stream.) (a) What is the flow rate in
? (b) What is the diameter of the stream 0.200 m below the faucet? Neglect any effects due to surface tension.
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