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**Flow and Resistance as Causes of Pressure Drops**

You may have noticed that water pressure in your home might be lower than normal on hot summer days when there is more use. This pressure drop occurs in the water main before it reaches your home. Let us consider flow through the water main as illustrated . We can understand why the pressure P_{1} to the home drops during times of heavy use by rearranging

to

where, in this case, P_{2} is the pressure at the water works and R is the resistance of the water main. During times of heavy use, the flow rate Q is large. This means that P_{2}-P_{1} must also be large. Thus P_{1} must decrease. It is correct to think of flow and resistance as causing the pressure to drop from P_{2} to P_{1 }. P_{2} - P_{1} = RQ is valid for both laminar and turbulent flows.

During times of heavy use, there is a significant pressure drop in a water main, and P_{1 }supplied to users is significantly less than P_{2} created at the water works. If the flow is very small, then the pressure drop is negligible, and P_{2 } P_{1}

We can use P_{2} - P_{1} = RQ to analyze pressure drops occurring in more complex systems in which the tube radius is not the same everywhere. Resistance will be much greater in narrow places, such as an obstructed coronary artery. For a given flow rate Q, the pressure drop will be greatest where the tube is most narrow. This is how water faucets control flow. Additionally, R is greatly increased by turbulence, and a constriction that creates turbulence greatly reduces the pressure downstream. Plaque in an artery reduces pressure and hence flow, both by its resistance and by the turbulence it creates.

Human circulatory system, showing average blood pressures in its major parts for an adult at rest. Pressure created by the heart’s two pumps, the right and left ventricles, is reduced by the resistance of the blood vessels as the blood flows through them. The left ventricle increases arterial blood pressure that drives the flow of blood through all parts of the body except the lungs. The right ventricle receives the lower pressure blood from two major veins and pumps it through the lungs for gas exchange with atmospheric gases – the disposal of carbon dioxide from the blood and the replenishment of oxygen. Only one major organ is shown schematically, with typical branching of arteries to ever smaller vessels, the smallest of which are the capillaries, and rejoining of small veins into larger ones. Similar branching takes place in a variety of organs in the body, and the circulatory system has considerable flexibility in flow regulation to these organs by the dilation and constriction of the arteries leading to them and the capillaries within them. The sensitivity of flow to tube radius makes this flexibility possible over a large range of flow rates.

Schematic of the circulatory system. Pressure difference is created by the two pumps in the heart and is reduced by resistance in the vessels. Branching of vessels into capillaries allows blood to reach individual cells and exchange substances, such as oxygen and waste products, with them. The system has an impressive ability to regulate flow to individual organs, accomplished largely by varying vessel diameters.

Each branching of larger vessels into smaller vessels increases the total cross-sectional area of the tubes through which the blood flows. For example, an artery with a cross section of 1 cm^{2 }may branch into 20 smaller arteries, each with cross sections of, with a total of. In that manner, the resistance of the branchings is reduced so that pressure is not entirely lost. Moreover, because and A increases through branching, the average velocity of the blood in the smaller vessels is reduced. The blood velocity in the aorta () is about 25 cm/s, while in the capillaries ( in diameter) the velocity is about 1 mm/s. This reduced velocity allows the blood to exchange substances with the cells in the capillaries and alveoli in particular.**Section Summary**

1. Laminar flow is characterized by smooth flow of the fluid in layers that do not mix.

2. Turbulence is characterized by eddies and swirls that mix layers of fluid together.

3. Fluid viscosity η is due to friction within a fluid. Representative values. Viscosity has units of (N/m^{2})s or Pa.s.

4. Flow is proportional to pressure difference and inversely proportional to resistance:

Q = P_{2 }- P_{1}/R.

5. For laminar flow in a tube, Poiseuille’s law for resistance states that

6. Poiseuille’s law for flow in a tube is

7. The pressure drop caused by flow and resistance is given by

P_{2} - P_{1} = RQ.**Conceptual Questions**

Explain why the viscosity of a liquid decreases with temperature—that is, how might increased temperature reduce the effects of cohesive forces in a liquid? Also explain why the viscosity of a gas increases with temperature—that is, how does increased gas temperature create more collisions between atoms and molecules?

When paddling a canoe upstream, it is wisest to travel as near to the shore as possible. When canoeing downstream, it may be best to stay near the middle. Explain why.

Why does flow decrease in your shower when someone flushes the toilet?

Plumbing usually includes air-filled tubes near water faucets . Explain why they are needed and how they work.

The vertical tube near the water tap remains full of air and serves a useful purpose.**Problems & Exercises**

(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 20C, the cart is moving at 0.400 m/s, its surface area is 2.5010^{-2}m^{2}, and the thickness of the air layer is 6.0010^{-5}m.

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

(a) 3.0210^{-3}N

(b) 1.0310^{-3}

What force is needed to pull one microscope slide over another at a speed of 1.00 cm/s, if there is a 0.500-mm-thick layer of 20C water between them and the contact area is 8.00 cm^{2}?

A glucose solution being administered with an IV has a flow rate of 4.00cm^{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.

1.60 cm^{3}/min

The pressure drop along a length of artery is 100 Pa, the radius is 10 mm, and the flow is laminar. The average speed of the blood is 15 mm/s.

(a) What is the net force on the blood in this section of artery?

(b) What is the power expended maintaining the flow?

A small artery has a length of 1.110^{-3}m and a radius of 2.510^{-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 37C.)

8.710^{-11}m^{3}/s

Fluid originally flows through a tube at a rate of 100cm^{3}/s. To illustrate the sensitivity of flow rate to various factors, calculate the new flow rate for the following changes with all other factors remaining the same as in the original conditions.

(a) Pressure difference increases by a factor of 1.50.

(b) A new fluid with 3.00 times greater viscosity is substituted.

(c) The tube is replaced by one having 4.00 times the length.

(d) Another tube is used with a radius 0.100 times the original.

(e) Yet another tube is substituted with a radius 0.100 times the original and half the length, and the pressure difference is increased by a factor of 1.50.

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.

0.316

Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what factor must the radius of an artery be increased in order to increase blood flow by a factor of 10?

(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?

(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.

A spherical particle falling at a terminal speed in a liquid must have the gravitational force balanced by the drag force and the buoyant force. The buoyant force is equal to the weight of the displaced fluid, while the drag force is assumed to be given by Stokes Law, . Show that the terminal speed is given by

where is the radius of the sphere, ρ_{s} is its density, and is the density of the fluid and η the coefficient of viscosity.

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.

A skydiver will reach a terminal velocity when the air drag equals their weight. For a skydiver with high speed and a large body, turbulence is a factor. The drag force then is approximately proportional to the square of the velocity. Taking the drag force to be and setting this equal to the person’s weight, find the terminal speed for a person falling “spread eagle.” Find both a formula and a number for , with assumptions as to size.

A layer of oil 1.50 mm thick is placed between two microscope slides. Researchers find that a force of is required to glide one over the other at a speed of 1.00 cm/s when their contact area is . What is the oil’s viscosity? What type of oil might it be?

or

Olive oil.

(a) Verify that a 19.0% decrease in laminar flow through a tube is caused by a 5.00% decrease in radius, assuming that all other factors remain constant, as stated in the text.

(b) What increase in flow is obtained from a 5.00% increase in radius, again assuming all other factors remain constant?

The flow of saline solution in an IV system.

(a) Verify that a pressure of 1.6210^{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.)

(a) 1.62104N/m2

(b) 0.111 cm3/s

(c) 10.6 cm

When physicians diagnose arterial blockages, they quote the reduction in flow rate. If the flow rate in an artery has been reduced to 10.0% of its normal value by a blood clot and the average pressure difference has increased by 20.0%, by what factor has the clot reduced the radius of the artery?

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?

1.59

Water supplied to a house by a water main has a pressure of early on a summer day when neighborhood use is low. This pressure produces a flow of 20.0 L/min through a garden hose. Later in the day, pressure at the exit of the water main and entrance to the house drops, and a flow of only 8.00 L/min is obtained through the same hose.

(a) What pressure is now being supplied to the house, assuming resistance is constant?

(b) By what factor did the flow rate in the water main increase in order to cause this decrease in delivered pressure? The pressure at the entrance of the water main is 5.0010^{5} N/m^{2}, and the original flow rate was 200 L/min.

(c) How many more users are there, assuming each would consume 20.0 L/min in the morning?

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 900kg/m^{3 }and its viscosity to be 1.00 (N/m^{2}).s (or ). Note that you must take into account the pressure due to the 50.0-m column of oil in the pipe.

2.9510^{6} N/m^{2} (gauge pressure)

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 8.0010^{6} N/m^{2}.

(a) Calculate the resistance of the hose.

(b) What is the viscosity of the concrete, assuming the flow is laminar?

(c) How much power is being supplied, assuming the point of use is at the same level as the pump? You may neglect the power supplied to increase the concrete’s velocity.**Construct Your Own Problem**

Consider a coronary artery constricted by arteriosclerosis. Construct a problem in which you calculate the amount by which the diameter of the artery is decreased, based on an assessment of the decrease in flow rate.

Consider a river that spreads out in a delta region on its way to the sea. Construct a problem in which you calculate the average speed at which water moves in the delta region, based on the speed at which it was moving up river. Among the things to consider are the size and flow rate of the river before it spreads out and its size once it has spread out. You can construct the problem for the river spreading out into one large river or into multiple smaller rivers.**Footnotes**

1. 1 The ratios of the viscosities of blood to water are nearly constant between 0°C and 37°C.

2. 2 See note on Whole Blood.

**Problems & Exercises**

(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 20C, the cart is moving at 0.400 m/s, its surface area is 2.5010^{-2} m^{2}, and the thickness of the air layer is 6.0010^{-5}m.

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

(a) 3.0210^{-3} N

(b) 1.0310^{-3}

What force is needed to pull one microscope slide over another at a speed of 1.00 cm/s, if there is a 0.500-mm-thick layer of 20C water between them and the contact area is 8.00cm^{2}?

A glucose solution being administered with an IV has a flow rate of 4.00cm^{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.

1.60 cm^{3}/min

The pressure drop along a length of artery is 100 Pa, the radius is 10 mm, and the flow is laminar. The average speed of the blood is 15 mm/s.

(a) What is the net force on the blood in this section of artery?

(b) What is the power expended maintaining the flow?

A small artery has a length of 1.110^{-3}m and a radius of 2.510^{-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 37C.)

8.710^{-11}m^{3}/s

Fluid originally flows through a tube at a rate of 100cm^{3}/s. To illustrate the sensitivity of flow rate to various factors, calculate the new flow rate for the following changes with all other factors remaining the same as in the original conditions.

(a) Pressure difference increases by a factor of 1.50.

(b) A new fluid with 3.00 times greater viscosity is substituted.

(c) The tube is replaced by one having 4.00 times the length.

(d) Another tube is used with a radius 0.100 times the original.

(e) Yet another tube is substituted with a radius 0.100 times the original and half the length, *and* the pressure difference is increased by a factor of 1.50.

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.

0.316

Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what factor must the radius of an artery be increased in order to increase blood flow by a factor of 10?

(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?

(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.

A spherical particle falling at a terminal speed in a liquid must have the gravitational force balanced by the drag force and the buoyant force. The buoyant force is equal to the weight of the displaced fluid, while the drag force is assumed to be given by Stokes Law, F_{s} = 6πrηv. Show that the terminal speed is given by

where R is the radius of the sphere, ρ_{s} is its density, and ρ_{1} is the density of the fluid and η the coefficient of viscosity.

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.

225mPa.sA skydiver will reach a terminal velocity when the air drag equals their weight. For a skydiver with high speed and a large body, turbulence is a factor. The drag force then is approximately proportional to the square of the velocity. Taking the drag force to be and setting this equal to the person’s weight, find the terminal speed for a person falling “spread eagle.” Find both a formula and a number for v_{t}, with assumptions as to size.

A layer of oil 1.50 mm thick is placed between two microscope slides. Researchers find that a force of 5.5010^{-4}N is required to glide one over the other at a speed of 1.00 cm/s when their contact area is 6.00cm^{2}. What is the oil’s viscosity? What type of oil might it be?

0.138 Pa -s,

or

Olive oil.

(a) Verify that a 19.0% decrease in laminar flow through a tube is caused by a 5.00% decrease in radius, assuming that all other factors remain constant, as stated in the text.

(b) What increase in flow is obtained from a 5.00% increase in radius, again assuming all other factors remain constant?

The flow of saline solution in an IV system.

(a) Verify that a pressure of 1.6210^{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.)

(a) 1.6210^{4}N/m^{2}

(b) 0.111cm^{3}/s

(c) 10.6 cm

When physicians diagnose arterial blockages, they quote the reduction in flow rate. If the flow rate in an artery has been reduced to 10.0% of its normal value by a blood clot and the average pressure difference has increased by 20.0%, by what factor has the clot reduced the radius of the artery?

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?

1.59

Water supplied to a house by a water main has a pressure of 3.0010^{5}N/m^{2} early on a summer day when neighborhood use is low. This pressure produces a flow of 20.0 L/min through a garden hose. Later in the day, pressure at the exit of the water main and entrance to the house drops, and a flow of only 8.00 L/min is obtained through the same hose.

(a) What pressure is now being supplied to the house, assuming resistance is constant?

(b) By what factor did the flow rate in the water main increase in order to cause this decrease in delivered pressure? The pressure at the entrance of the water main is 5.0010^{5} N/m^{2}, and the original flow rate was 200 L/min.

(c) How many more users are there, assuming each would consume 20.0 L/min in the morning?

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 900kg/m^{3 }and its viscosity to be 1.00(N/m^{2}).s (or 1.00 Pa.s). Note that you must take into account the pressure due to the 50.0-m column of oil in the pipe.

2.9510^{6} N/m^{2}(gauge pressure)

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 8.0010^{6}N/m^{2}.

(a) Calculate the resistance of the hose.

(b) What is the viscosity of the concrete, assuming the flow is laminar?

(c) How much power is being supplied, assuming the point of use is at the same level as the pump? You may neglect the power supplied to increase the concrete’s velocity.**Construct Your Own Problem**

Consider a coronary artery constricted by arteriosclerosis. Construct a problem in which you calculate the amount by which the diameter of the artery is decreased, based on an assessment of the decrease in flow rate.

Consider a river that spreads out in a delta region on its way to the sea. Construct a problem in which you calculate the average speed at which water moves in the delta region, based on the speed at which it was moving up river. Among the things to consider are the size and flow rate of the river before it spreads out and its size once it has spread out. You can construct the problem for the river spreading out into one large river or into multiple smaller rivers.**Footnotes**

1. 1 The ratios of the viscosities of blood to water are nearly constant between 0°C and 37°C.

2. 2 See note on Whole Blood.

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