Content check — College Physics, Chapter 12: Fluid Dynamics and Its Biological and Medical Applications

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 12: Fluid Dynamics and Its Biological and Medical Applications

33 problems · 1 figure

What is the average flow rate in

{cm}^{3} /s

of gasoline to the engine of a car traveling at 100 km/h if it averages 10.0 km/L?

Answer

2.78 {cm}^{3} /s

12-3 §12.1 source ↗

Blood is pumped from the heart at a rate of 5.0 L/min into the aorta (of radius 1.0 cm). Determine the speed of blood through the aorta.

Answer

27 cm/s

12-5 §12.1 source ↗

The Huka Falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions (see Figure 12.29). 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?

Water rushes over a fall. — Figure 12.29 in the book ↗

Answer

(a) 0.75 m/s

(b) 0.13 m/s

12-7 §12.1 source ↗

(a) As blood passes through the capillary bed in an organ, the capillaries join to form venules (small veins). If the blood speed increases by a factor of 4.00 and the total cross-sectional area of the venules is

10.0 {cm}^{2}

, what is the total cross-sectional area of the capillaries feeding these venules? (b) How many capillaries are involved if their average diameter is

10.0 μ m

Answer

(a) $40.0 {cm}^{2}$

(b) $5 . 09 \times 10^{7}$

12-9 §12.1 source ↗

(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 to fill if you could divert a moderate size river, flowing at

5000 m^{3} /s

, into it?

Answer

(a) 22 h

(b) 0.016 s

12-11 §12.1 source ↗

(a) 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?

Answer

(a) 12.6 m/s

(b) $0.0800 m^{3} /s$

12-13 §12.1 source ↗

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?

Answer

(a) 0.402 L/s

(b) 0.584 cm

12-15 §12.1 source ↗

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

{cm}^{3} /s

? (b) What is the diameter of the stream 0.200 m below the faucet? Neglect any effects due to surface tension.

Answer

(a) $127 {cm}^{3} /s$

(b) 0.890 cm

12-17 §12.2 source ↗

Verify that pressure has units of energy per unit volume.

Answer

\begin{array}{l} P & = & \frac{Force}{Area}, \\ (P)_{units} & = & {N/m}^{2} = N \cdot {m/m}^{3} = {J/m}^{3} \\ = & energy/volume \end{array}

12-19 §12.2 source ↗

If the pressure reading of your pitot tube is 15.0 mm Hg at a speed of 200 km/h, what will it be at 700 km/h at the same altitude?

Answer

184 mm Hg

12-21 §12.2 source ↗

Every few years, winds in Boulder, Colorado, attain sustained speeds of 45.0 m/s (about 100 mi/h) when the jet stream descends during early spring. Approximately what is the force due to the Bernoulli effect on a roof having an area of

220 m^{2}

? Typical air density in Boulder is

1.14 {kg/m}^{3}

, and the corresponding atmospheric pressure is

8 . 89 \times 10^{4} {N/m}^{2}

. (Bernoulli’s principle as stated in the text assumes laminar flow. Using the principle here produces only an approximate result, because there is significant turbulence.)

Answer

2 . 54 \times 10^{5} N

12-23 §12.2 source ↗

(a) What is the pressure drop due to the Bernoulli effect as water goes into a 3.00-cm-diameter nozzle from a 9.00-cm-diameter fire hose while carrying a flow of 40.0 L/s? (b) To what maximum height above the nozzle can this water rise? (The actual height will be significantly smaller due to air resistance.)

Answer

(a) $1 . 58 \times 10^{6} {N/m}^{2}$

(b) 163 m

12-25 §12.3 source ↗

Hoover Dam on the Colorado River is the highest dam in the United States at 221 m, with a maximum output of 1300 MW. The dam generates electricity with water taken from a depth of 150 m and an average flow rate of

650 m^{3} /s

. (a) Calculate the power in this flow. (b) What is the ratio of this power to the facility’s average of 680 MW?

Answer

(a) $9.56 \times 10^{8} W$

(b) 1.41

12-27 §12.3 source ↗

The left ventricle of a resting adult’s heart pumps blood at a flow rate of

83.0 {cm}^{3} /s

, increasing its pressure by 110 mm Hg, its speed from zero to 30.0 cm/s, and its height by 5.00 cm. (All numbers are averaged over the entire heartbeat.) Calculate the total power output of the left ventricle. Note that most of the power is used to increase blood pressure.

Answer

1.26 W

12-29 §12.4 source ↗

(a) Calculate the retarding force due to the viscosity of the air layer between a cart and a level air track given the following information—air temperature is

20º C

, the cart is moving at 0.400 m/s, its surface area is

2 . 50 \times 10^{−2} m^{2}

, and the thickness of the air layer is

6.00 \times 10^{- 5} m

. (b) What is the ratio of this force to the weight of the 0.300-kg cart?

Answer

(a) $3 . 02 \times 10^{- 3} N$

(b) $1 . 03 \times 10^{- 3}$

12-31 §12.4 source ↗

A glucose solution being administered with an IV has a flow rate of

4.00 {cm}^{3} /min

. What will the new flow rate be if the glucose is replaced by whole blood having the same density but a viscosity 2.50 times that of the glucose? All other factors remain constant.

Answer

1 . {60 cm}^{3} /min

12-33 §12.4 source ↗

A small artery has a length of

1 . 1 \times 10^{- 3} m

and a radius of

2.5 \times 10^{- 5} m

. If the pressure drop across the artery is 1.3 kPa, what is the flow rate through the artery? (Assume that the temperature is

37º C

Answer

8.7 \times 10^{- 11} m^{3} /s

12-35 §12.4 source ↗

The arterioles (small arteries) leading to an organ, constrict in order to decrease flow to the organ. To shut down an organ, blood flow is reduced naturally to 1.00% of its original value. By what factor did the radii of the arterioles constrict? Penguins do this when they stand on ice to reduce the blood flow to their feet.

Answer

0.316

12-37 §12.4 source ↗

(a) Suppose a blood vessel’s radius is decreased to 90.0% of its original value by plaque deposits and the body compensates by increasing the pressure difference along the vessel to keep the flow rate constant. By what factor must the pressure difference increase? (b) If turbulence is created by the obstruction, what additional effect would it have on the flow rate?

Answer

(a) 1.52

(b) Turbulence will decrease the flow rate of the blood, which would require an even larger increase in the pressure difference, leading to higher blood pressure.

12-39 §12.4 source ↗

Using the equation of the previous problem, find the viscosity of motor oil in which a steel ball of radius 0.8 mm falls with a terminal speed of 4.32 cm/s. The densities of the ball and the oil are 7.86 and 0.88 g/mL, respectively.

Answer

225 mPa \cdot s

12-41 §12.4 source ↗

A layer of oil 1.50 mm thick is placed between two microscope slides. Researchers find that a force of

5 . 50 \times 10^{- 4} N

is required to glide one over the other at a speed of 1.00 cm/s when their contact area is

6.00 {cm}^{2}

. What is the oil’s viscosity? What type of oil might it be?

Answer

0 . 138 Pa \cdot s,

Olive oil.

12-43 §12.4 source ↗

Example 12.8 dealt with the flow of saline solution in an IV system. (a) Verify that a pressure of

1 . 62 \times 10^{4} {N/m}^{2}

is created at a depth of 1.61 m in a saline solution, assuming its density to be that of sea water. (b) Calculate the new flow rate if the height of the saline solution is decreased to 1.50 m. (c) At what height would the direction of flow be reversed? (This reversal can be a problem when patients stand up.)

Example 12.8 What Pressure Produces This Flow Rate? in the book ↗

Answer

(a) $1 . 62 \times 10^{4} {N/m}^{2}$

(b) $0 . {111 cm}^{3} /s$

12-45 §12.4 source ↗

During a marathon race, a runner’s blood flow increases to 10.0 times her resting rate. Her blood’s viscosity has dropped to 95.0% of its normal value, and the blood pressure difference across the circulatory system has increased by 50.0%. By what factor has the average radii of her blood vessels increased?

Answer

1.59

12-47 §12.4 source ↗

An oil gusher shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. Neglecting air resistance but not the resistance of the pipe, and assuming laminar flow, calculate the gauge pressure at the entrance of the 50.0-m-long vertical pipe. Take the density of the oil to be

900 {kg/m}^{3}

and its viscosity to be

1.00 ({N/m}^{2}) \cdot s

(or

1.00 Pa \cdot s

). Note that you must take into account the pressure due to the 50.0-m column of oil in the pipe.

Answer

2 . 95 \times 10^{6} {N/m}^{2}

(gauge pressure)

12-51 §12.5 source ↗

Verify that the flow of oil is laminar (barely) for an oil gusher that shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. The vertical pipe is 50 m long. Take the density of the oil to be

900 kg {/m}^{3}

and its viscosity to be

1.00 ({N/m}^{2}) \cdot s

(or

1.00 Pa \cdot s

Answer

N_{R} =

1.99 \times 10^{3} < 2000

12-53 §12.5 source ↗

Calculate the Reynolds numbers for the flow of water through (a) a nozzle with a radius of 0.250 cm and (b) a garden hose with a radius of 0.900 cm, when the nozzle is attached to the hose. The flow rate through hose and nozzle is 0.500 L/s. Can the flow in either possibly be laminar?

Answer

(a) nozzle: $1 . 27 \times 10^{5}$ , not laminar

(b) hose: $3 . 51 \times 10^{4}$ , not laminar.

12-55 §12.5 source ↗

Concrete is pumped from a cement mixer to the place it is being laid, instead of being carried in wheelbarrows. The flow rate is 200.0 L/min through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the pump is

8 . 00 \times 10^{6} {N/m}^{2}

. Verify that the flow of concrete is laminar taking concrete’s viscosity to be

48.0 (N/ m^{2}) \cdot s

, and given its density is

2300 kg/ m^{3}

Answer

2.54 << 2000, laminar.

12-57 §12.5 source ↗

What is the greatest speed of blood flow at

37º C

in an artery of radius 2.00 mm if the flow is to remain laminar? What is the corresponding flow rate? Take the density of blood to be

1025 kg / m^{3}

Answer

1.02 m/s

$1.28 \times 10^{–2} L/s$

12-59 §12.5 source ↗

Gasoline is piped underground from refineries to major users. The flow rate is

3 . 00 \times 10^{–2} m^{3} /s

(about 500 gal/min), the viscosity of gasoline is

1.00 \times 10^{–3} ({N/m}^{2}) \cdot s

, and its density is

680 {kg/m}^{3}

. (a) What minimum diameter must the pipe have if the Reynolds number is to be less than 2000? (b) What pressure difference must be maintained along each kilometer of the pipe to maintain this flow rate?

Answer

(a) $≥ 13.0 m$

(b) $2 . 68 \times 10^{- 6} {N/m}^{2}$

12-61 §12.5 source ↗

Unreasonable Results

A fairly large garden hose has an internal radius of 0.600 cm and a length of 23.0 m. The nozzleless horizontal hose is attached to a faucet, and it delivers 50.0 L/s. (a) What water pressure is supplied by the faucet? (b) What is unreasonable about this pressure? (c) What is unreasonable about the premise? (d) What is the Reynolds number for the given flow? (Take the viscosity of water as $1.005 \times 10^{–3} (N / m^{2}) \cdot s$ .)

Answer

(a) 23.7 atm or $344 lb/ {in}^{2}$

(b) The pressure is much too high.

(d) $5.27 \times 10^{6}$ > > 3000, turbulent, contrary to the assumption of laminar flow when using this equation.

12-62 §12.7 source ↗

You can smell perfume very shortly after opening the bottle. To show that it is not reaching your nose by diffusion, calculate the average distance a perfume molecule moves in one second in air, given its diffusion constant

D

to be

1.00 \times 10^{–6} m^{2} /s

Answer

1 . 41 \times 10^{- 3} m

12-64 §12.7 source ↗

Oxygen reaches the veinless cornea of the eye by diffusing through its tear layer, which is 0.500-mm thick. How long does it take the average oxygen molecule to do this?

Answer

1 . 3 \times 10^{2} s

12-66 §12.7 source ↗

Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.

Answer

0.391 s

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Problems and answers are adapted from College Physics 2e by Paul Peter Urone, Roger Hinrichs, Kim Dirks and Manjula Sharma, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/college-physics-2e/pages/1-introduction-to-science-and-the-realm-of-physics-physical-quantities-and-units. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.