# Garden Hose Analogy for Blood Vessel Resistance

I've been trying to reconcile the resistance component of Poisseuille's law with a mental analogy of a garden hose; specifically, I had assumed that the effects of reducing the radius of a blood vessel (of an arteriole in vasoconstriction for instance) would be the same as when you put your thumb over the opening of a hose, causing water to leave at an increased velocity. However, the law instead predicts that a decrease in radius like this would increase resistance, actually decreasing flow by a proportional amount equal to r^4. Why doesn't arteriolar flow actually end up increasing because of the pressure from the arteries, much like its "hose equivalent"? Does the hose analogy even work in this scenario? What am I missing that separates the two scenarios?

It comes down to the distinction between velocity and flow rate.

While you are right that the water would leave a an increased velocity if you put your thumb over the end of the hose, this is deceptive, because the water is exiting the closed system, so it can go absolutely anywhere. So the fact that the velocity is increased could give one the impression that the flow rate is increased, but in fact it is the inverse.

Imagine that instead of putting your thumb on the end of the hose, you pinched the hose in the middle instead. This would result in the water flowing out of the end of the hose at a reduced flow rate (lower volume of water coming out per minute) and velocity (lower speed of the individual water particles). Why?

It might help to take a snapshot of the place where you are pinching the hose. the pressure behind the pinch zone goes up, causing the water to pass through the pinch zone at a faster velocity. However, because the hole is smaller, less water is able to pass through. When it reaches the other side of the pinch zone, it slows down again because it has a very low pressure, and the smaller volume of water must fill the entire tube. However, because the flow rate was lowered because of the pinch zone, the amount of water at the end of the pinch zone is lower. Consequently, the water flows slower and the flow rate is reduced simultaneously.

Another analogy that might help would be the comparison between a garden hose and a fire hose. Let's say they have the same flow rate (same volume coming out of the hose per unit time). You can immediately realise that for the garden hose to put out the same amount of water that the fire hose is putting out in a minute, the water would have to be flowing many times faster in terms of velocity.

• Grunwald JE, Petrig BL Riva CE, Sinclair SH. Blood velocity and volumetric flow rate in human retinal vessels. Invest. Ophthalmol. Vis. Sci. 1985;26(8):1124-1132. Available online: https://iovs.arvojournals.org/article.aspx?articleid=2159754
• Baker M, Wayland H. On-line volume flow rate and velocity profile measurement for blood in microvessels. Microvasc. Res. 1974;7(1):131-143. Available online: https://doi.org/10.1016/0026-2862(74)90043-0

At a constant volumetric flow-rate, the product of velocity and vesicle cross-section area is constant. In a real-life scenario, the volumetric flow-rate decreases when you hold your finger over the opening. So while water exits 'faster', at higher velocity, you would fill less bottles of water per minute.

## Researcher discovers previously rejected function in the brain's blood vessels

IMAGE: Close-up of precapillary sphincter (the strong red mark in the middle of the greenly marked blood flow) from two-photon microscope. According to the research results, these squeezing muscle cells are. view more

Credit: Lauritzen Lab, University of Copenhagen

Allegedly, they should not exist in the brain, the so-called precapillary sphincters - a kind of squeezing 'muscle clamp' between the larger and smaller vessels of the bloodstream.

Nevertheless, Assistant Professor Søren Grubb from the Department of Neuroscience at the University of Copenhagen has indeed shown the sphincters in mice.

'In the early '10s, a Japanese review study concluded that there was no evidence that pre-capillary sphincters should exist in the heart, brain and muscular connective tissue,' he says and continues:

'Since then, scientists have focused a lot on pericytes - muscle cells that can regulate the resistance in the smallest blood vessels. At the same time, however, they have somehow missed a great resistance right between some arterioles and capillaries: The sphincters. Perhaps because the discovery of the pericytes has received more attention among all the blood vessels of the brain'.

Functions as a water faucet or a sluice system

As blood flows through the brain, it flows from arteriole to vein through the capillaries. The latter are the smallest blood vessels in the body, but incredibly important. It is here that the blood and the brain exchange oxygen and nutrients.

Søren Grubb explains that the precapillary sphincters may be compared to a kind of thermostat that distributes the pressure between the branches of the blood vessels. A bit like a faucet adjusting the pressure between a water pipe and a garden hose.

As the muscle clamp relaxes, more blood cells will flow through its passage and the pressure in the following blood vessels will increase. When the clamp contracts, a bottleneck forms, which lowers the pressure further down the blood flow.

'In this way, it also works a bit like a sluice system to irrigate fields: You may have a roaring river, but by diverting water from the river and making sluices that can regulate the amount of water for each field, you can distribute the water to many areas, says Søren Grubb.

'Conversely, if the sluice shuts down or is clogged, the field will quickly dry out', he adds.

Potential for dementia and migraine

Based on that picture, Søren Grubb assumes that the pre-capillary sphincters may play a major role for disturbances of the brain's blood supply and blood pressure.

If the assumption holds true, the discovery of the clamping muscles in the brain will potentially affect the treatment of diseases such as migraine, Alzheimer's and vascular dementia - all associated with an accumulation of waste products that may stem from blood vessel defects.

Already, the research group Lauritzen Lab, of which Søren Grubb is part, has tested a model for migraine with aura. The model confirms the hypothesis, but the Assistant Professor emphasises that further research is still needed in connection with disorders:

'We have shown that the precapillary sphincter is found in the brain. The rest is still speculative. But perhaps more researchers will start working on it, now that they know that the sphincters are there'.

Disclaimer: AAAS and EurekAlert! are not responsible for the accuracy of news releases posted to EurekAlert! by contributing institutions or for the use of any information through the EurekAlert system.

allnurses is a Nursing Career & Support site. Our mission is to Empower, Unite, and Advance every nurse, student, and educator. Our members represent more than 60 professional nursing specialties. Since 1997, allnurses is trusted by nurses around the globe.

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1-612-816-8773

allnurses is a Nursing Career & Support site. Our mission is to Empower, Unite, and Advance every nurse, student, and educator. Our members represent more than 60 professional nursing specialties. Since 1997, allnurses is trusted by nurses around the globe.

allnurses.com, INC, 7900 International Drive #300, Bloomington MN 55425
1-612-816-8773

## Laminar Flow Confined to Tubes—Poiseuille’s Law

What causes flow? The answer, not surprisingly, is pressure difference. In fact, there is a very simple relationship between horizontal flow and pressure. Flow rate is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as

where and are the pressures at two points, such as at either end of a tube, and is the resistance to flow. The resistance includes everything, except pressure, that affects flow rate. For example, is greater for a long tube than for a short one. The greater the viscosity of a fluid, the greater the value of Turbulence greatly increases whereas increasing the diameter of a tube decreases

If viscosity is zero, the fluid is frictionless and the resistance to flow is also zero. Comparing frictionless flow in a tube to viscous flow, as in Figure 4, we see that for a viscous fluid, speed is greatest at midstream because of drag at the boundaries. We can see the effect of viscosity in a Bunsen burner flame, even though the viscosity of natural gas is small.

The resistance to laminar flow of an incompressible fluid having viscosity through a horizontal tube of uniform radius and length such as the one in Figure 5, is given by

This equation is called Poiseuille’s law for resistance after the French scientist J. L. Poiseuille (1799–1869), who derived it in an attempt to understand the flow of blood, an often turbulent fluid.

Figure 4. (a) If fluid flow in a tube has negligible resistance, the speed is the same all across the tube. (b) When a viscous fluid flows through a tube, its speed at the walls is zero, increasing steadily to its maximum at the center of the tube. (c) The shape of the Bunsen burner flame is due to the velocity profile across the tube. (credit: Jason Woodhead)

Let us examine Poiseuille’s expression for to see if it makes good intuitive sense. We see that resistance is directly proportional to both fluid viscosity and the length of a tube. After all, both of these directly affect the amount of friction encountered—the greater either is, the greater the resistance and the smaller the flow. The radius of a tube affects the resistance, which again makes sense, because the greater the radius, the greater the flow (all other factors remaining the same). But it is surprising that is raised to the fourth power in Poiseuille’s law. This exponent means that any change in the radius of a tube has a very large effect on resistance. For example, doubling the radius of a tube decreases resistance by a factor of

Taken together, and give the following expression for flow rate:

This equation describes laminar flow through a tube. It is sometimes called Poiseuille’s law for laminar flow, or simply Poiseuille’s law .

### Example 1: Using Flow Rate: Plaque Deposits Reduce Blood Flow

Suppose the flow rate of blood in a coronary artery has been reduced to half its normal value by plaque deposits. By what factor has the radius of the artery been reduced, assuming no turbulence occurs?

Assuming laminar flow, Poiseuille’s law states that

We need to compare the artery radius before and after the flow rate reduction.

With a constant pressure difference assumed and the same length and viscosity, along the artery we have

So, given that we find that

Therefore, a decrease in the artery radius of 16%.

This decrease in radius is surprisingly small for this situation. To restore the blood flow in spite of this buildup would require an increase in the pressure difference of a factor of two, with subsequent strain on the heart.

Fluid Temperature (ºC) Viscosity
η(mPa·s)
Gases
Air 0 0.0171
20 0.0181
40 0.0190
100 0.0218
Ammonia 20 0.00974
Carbon dioxide 20 0.0147
Helium 20 0.0196
Hydrogen 0 0.0090
Mercury 20 0.0450
Oxygen 20 0.0203
Steam 100 0.0130
Liquids
Water 0 1.792
20 1.002
37 0.6947
40 0.653
100 0.282
Whole blood 1 20 3.015
37 2.084
Blood plasma 2 20 1.810
37 1.257
Ethyl alcohol 20 1.20
Methanol 20 0.584
Oil (heavy machine) 20 660
Oil (motor, SAE 10) 30 200
Oil (olive) 20 138
Glycerin 20 1500
Honey 20 2000–10000
Maple Syrup 20 2000–3000
Milk 20 3.0
Oil (Corn) 20 65
Table 1. Coefficients of Viscosity of Various Fluids

The circulatory system provides many examples of Poiseuille’s law in action—with blood flow regulated by changes in vessel size and blood pressure. Blood vessels are not rigid but elastic. Adjustments to blood flow are primarily made by varying the size of the vessels, since the resistance is so sensitive to the radius. During vigorous exercise, blood vessels are selectively dilated to important muscles and organs and blood pressure increases. This creates both greater overall blood flow and increased flow to specific areas. Conversely, decreases in vessel radii, perhaps from plaques in the arteries, can greatly reduce blood flow. If a vessel’s radius is reduced by only 5% (to 0.95 of its original value), the flow rate is reduced to about of its original value. A 19% decrease in flow is caused by a 5% decrease in radius. The body may compensate by increasing blood pressure by 19%, but this presents hazards to the heart and any vessel that has weakened walls. Another example comes from automobile engine oil. If you have a car with an oil pressure gauge, you may notice that oil pressure is high when the engine is cold. Motor oil has greater viscosity when cold than when warm, and so pressure must be greater to pump the same amount of cold oil.

Figure 5. Poiseuille’s law applies to laminar flow of an incompressible fluid of viscosity η through a tube of length l and radius r. The direction of flow is from greater to lower pressure. Flow rate Q is directly proportional to the pressure difference P2−P1, and inversely proportional to the length l of the tube and viscosity η of the fluid. Flow rate increases with r 4 , the fourth power of the radius.

### Example 2: What pressure Produces This Flow Rate?

An intravenous (IV) system is supplying saline solution to a patient at the rate of through a needle of radius 0.150 mm and length 2.50 cm. What pressure is needed at the entrance of the needle to cause this flow, assuming the viscosity of the saline solution to be the same as that of water? The gauge pressure of the blood in the patient’s vein is 8.00 mm Hg. (Assume that the temperature is .)

Assuming laminar flow, Poiseuille’s law applies. This is given by

where is the pressure at the entrance of the needle and is the pressure in the vein. The only unknown is

is given as 8.00 mm Hg, which converts to Substituting this and the other known values yields

This pressure could be supplied by an IV bottle with the surface of the saline solution 1.61 m above the entrance to the needle (this is left for you to solve in this chapter’s Problems and Exercises), assuming that there is negligible pressure drop in the tubing leading to the needle.

allnurses is a Nursing Career & Support site. Our mission is to Empower, Unite, and Advance every nurse, student, and educator. Our members represent more than 60 professional nursing specialties. Since 1997, allnurses is trusted by nurses around the globe.

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## Blood Vessels

The blood vessels make up the vascular system. The three main parts are: arteries, which consist of vessels taking blood away from the heart veins, which are vessels that take blood to the heart and capillaries, which are the vessels that connect the arteries and veins together to form the complete circuit.

The vascular tree allows the vascular system to create multiple pathways for the blood to flow, so that it can ultimately reach all areas of the body and deliver oxygen to the cells. These pathways appear as branches from aorta. At each branch or bifurcation, the original is split into two new pathways. At each bifurcation, the diameter of the vessel also gets smaller.

Arteries are the vessels that carry the flow of blood from the heart. The arteries can be broken down into smaller constituent components beginning with the aorta, followed by the large arteries, which branch into progressively smaller arteries, and finally the arterioles.

All arteries have similar layered structures, with the primary difference being their diminishing size. The layers are: the innermost, or tunica intima, the center section, or tunica media, and the outermost, or tunica adventitia. These vessels are made up primarily of elastin (which acts as a connective tissue that resumes its primary shape after stretching), collagen (the connective tissue between the layers), and smooth muscle (which provides active constriction and relaxation of the vessels). 3 There is also a network of very fine blood vessels traveling through the arterial walls allowing for the diffusion of oxygen to take place for the cells and tissue that make up this region.

A cross section of an artery reveals that the innermost section consists of a layer of endothelial cells, connective tissue, and basement membrane. The inner section of the blood vessels (tunica intima) has a thin layer of the endothelial cells in order to prevent blood clotting and lower turbulence created by the flow of blood. The middle layer of the arteries (tunica media) are made up primarily of a "prominent layer of elastic tissue" which helps to provide the flexibility needed for vessels to constrict and relax in order to help regulate blood flow in the system. The outermost layer (tunica adventitia) is made up "mostly of stiff collagenous fibers". 4

The elastic tissue in the arteries are concentrically distributed and attached by smooth muscle cells and connective tissue. As bifurcations occur, from the aorta down through large and small arteries, the number of elastic laminae decreases with distance from the aorta, but the amount of smooth muscles increases as well as the relative wall thickness of the vessels. This is a needed byproduct of the function of the arteries: their ability to constrict and relax helps to control the flow of blood and thus blood pressure. 5

The coronary arteries are a network of small arteries found on the surface of the heart. These vessels are extremely important to the health of an individual, as they provide the pathways for blood flow, and therefore oxygen delivery, to the tissue of the heart. Due to the small diameter of these vessels, they are susceptible to blockage.

The arterioles are small diameter blood vessels at the end of the arterial system. This is the area where the greatest amount of pressure drop takes place in the circulatory system, as this is where regional variations in blood flow are regulated. Although similar in structure to the rest of the artery system, there are some key differences that should be noted. Similar to the other parts of the arterial system these areas have the ability to contract restricting blood flow, and also relax allowing for greater flow.

Capillaries are largely ignored in middle school textbooks, simply being referred to as the region where the arteries and the veins are connected. There is little discussion about why they are so small or what there purpose is. The reality is that the capillaries are an integral part of the circulatory system for without them there would not be any transport of oxygen to the cells of the body and those cells would perish.

Capillary structure is simpler than arterial structure. The capillaries are made of endothelial cells in a single layer joined together with molecular cement. The diameter of the capillaries varies throughout the circulation, but is often the width of a red cell, meaning the cells need to line up in order to pass through these vessels. This confluent monolayer of endothelial cells does not allow blood cells to seep through the capillary wall, but it does allow for a leakage of oxygen, glucose, carbon dioxide, and even some proteins. This leakage, which is often driven by molecular diffusion, gives the capillary bed an essential role in the circulatory system. 6

The veins carry the flow of blood back to the heart. In the systemic circulation, venous blood is low in oxygen content but high in carbon dioxide. In the pulmonary veins, which are bringing blood back to the heart from the lungs, the veins contain oxygenated blood. The smallest veins, called venules, are directly connected to the capillaries. Smaller veins join together to form larger veins, with the largest vessels returning blood to the heart. The blood flowing through the veins is passive in nature as the veins do not have the smooth muscle that the arteries have and therefore do not contract and relax in order to promote higher or lower resistance to flow. Flow through veins, which occurs at low pressure, is unidirectional by the presence of valves within many veins, allowing the blood to only flow towards the heart. The walls of veins are collagenous, similar to the arterial wall, and veins have the same trilayer structure as arteries, but the layers are less distinct. There are smooth muscles associated with the tunica media of some veins, but they are not as organized or abundant as in arteries. Veins, like all vessels in the circulatory system, are covered with endothelial cells on their lumenal surface (the side containing the blood). 7

## 88 Viscosity and Laminar Flow Poiseuille’s Law

When you pour yourself a glass of juice, the liquid flows freely and quickly. But when you pour syrup on your pancakes, that liquid flows slowly and sticks to the pitcher. The difference is fluid friction, both within the fluid itself and between the fluid and its surroundings. We call this property of fluids viscosity. Juice has low viscosity, whereas syrup has high viscosity. In the previous sections we have considered ideal fluids with little or no viscosity. In this section, we will investigate what factors, including viscosity, affect the rate of fluid flow.

The precise definition of viscosity is based on laminar, or nonturbulent, flow. Before we can define viscosity, then, we need to define laminar flow and turbulent flow. (Figure) shows both types of flow. Laminar flow is characterized by the smooth flow of the fluid in layers that do not mix. Turbulent flow, or turbulence , is characterized by eddies and swirls that mix layers of fluid together.

(Figure) shows schematically how laminar and turbulent flow differ. Layers flow without mixing when flow is laminar. When there is turbulence, the layers mix, and there are significant velocities in directions other than the overall direction of flow. The lines that are shown in many illustrations are the paths followed by small volumes of fluids. These are called streamlines. Streamlines are smooth and continuous when flow is laminar, but break up and mix when flow is turbulent. Turbulence has two main causes. First, any obstruction or sharp corner, such as in a faucet, creates turbulence by imparting velocities perpendicular to the flow. Second, high speeds cause turbulence. The drag both between adjacent layers of fluid and between the fluid and its surroundings forms swirls and eddies, if the speed is great enough. We shall concentrate on laminar flow for the remainder of this section, leaving certain aspects of turbulence for later sections.

Try dropping simultaneously two sticks into a flowing river, one near the edge of the river and one near the middle. Which one travels faster? Why?

(Figure) shows how viscosity is measured for a fluid. Two parallel plates have the specific fluid between them. The bottom plate is held fixed, while the top plate is moved to the right, dragging fluid with it. The layer (or lamina) of fluid in contact with either plate does not move relative to the plate, and so the top layer moves at while the bottom layer remains at rest. Each successive layer from the top down exerts a force on the one below it, trying to drag it along, producing a continuous variation in speed from to 0 as shown. Care is taken to insure that the flow is laminar that is, the layers do not mix. The motion in (Figure) is like a continuous shearing motion. Fluids have zero shear strength, but the rate at which they are sheared is related to the same geometrical factors and as is shear deformation for solids.

The graphic shows laminar flow of fluid between two plates of area . The bottom plate is fixed. When the top plate is pushed to the right, it drags the fluid along with it.

A force is required to keep the top plate in (Figure) moving at a constant velocity , and experiments have shown that this force depends on four factors. First, is directly proportional to (until the speed is so high that turbulence occurs—then a much larger force is needed, and it has a more complicated dependence on ). Second, is proportional to the area of the plate. This relationship seems reasonable, since is directly proportional to the amount of fluid being moved. Third, is inversely proportional to the distance between the plates . This relationship is also reasonable is like a lever arm, and the greater the lever arm, the less force that is needed. Fourth, is directly proportional to the coefficient of viscosity, . The greater the viscosity, the greater the force required. These dependencies are combined into the equation

which gives us a working definition of fluid viscosity />. Solving for /> gives

which defines viscosity in terms of how it is measured. The SI unit of viscosity is . (Figure) lists the coefficients of viscosity for various fluids.

Viscosity varies from one fluid to another by several orders of magnitude. As you might expect, the viscosities of gases are much less than those of liquids, and these viscosities are often temperature dependent. The viscosity of blood can be reduced by aspirin consumption, allowing it to flow more easily around the body. (When used over the long term in low doses, aspirin can help prevent heart attacks, and reduce the risk of blood clotting.)

### Laminar Flow Confined to Tubes—Poiseuille’s Law

What causes flow? The answer, not surprisingly, is pressure difference. In fact, there is a very simple relationship between horizontal flow and pressure. Flow rate is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as

where and are the pressures at two points, such as at either end of a tube, and is the resistance to flow. The resistance includes everything, except pressure, that affects flow rate. For example, is greater for a long tube than for a short one. The greater the viscosity of a fluid, the greater the value of . Turbulence greatly increases , whereas increasing the diameter of a tube decreases .

If viscosity is zero, the fluid is frictionless and the resistance to flow is also zero. Comparing frictionless flow in a tube to viscous flow, as in (Figure), we see that for a viscous fluid, speed is greatest at midstream because of drag at the boundaries. We can see the effect of viscosity in a Bunsen burner flame, even though the viscosity of natural gas is small.

The resistance to laminar flow of an incompressible fluid having viscosity through a horizontal tube of uniform radius and length , such as the one in (Figure), is given by

This equation is called Poiseuille’s law for resistance after the French scientist J. L. Poiseuille (1799–1869), who derived it in an attempt to understand the flow of blood, an often turbulent fluid.

Let us examine Poiseuille’s expression for to see if it makes good intuitive sense. We see that resistance is directly proportional to both fluid viscosity and the length of a tube. After all, both of these directly affect the amount of friction encountered—the greater either is, the greater the resistance and the smaller the flow. The radius of a tube affects the resistance, which again makes sense, because the greater the radius, the greater the flow (all other factors remaining the same). But it is surprising that is raised to the fourth power in Poiseuille’s law. This exponent means that any change in the radius of a tube has a very large effect on resistance. For example, doubling the radius of a tube decreases resistance by a factor of .

Taken together, and give the following expression for flow rate:

This equation describes laminar flow through a tube. It is sometimes called Poiseuille’s law for laminar flow, or simply Poiseuille’s law .

Suppose the flow rate of blood in a coronary artery has been reduced to half its normal value by plaque deposits. By what factor has the radius of the artery been reduced, assuming no turbulence occurs?

Assuming laminar flow, Poiseuille’s law states that

We need to compare the artery radius before and after the flow rate reduction.

With a constant pressure difference assumed and the same length and viscosity, along the artery we have

So, given that , we find that .

Therefore, , a decrease in the artery radius of 16%.

This decrease in radius is surprisingly small for this situation. To restore the blood flow in spite of this buildup would require an increase in the pressure difference of a factor of two, with subsequent strain on the heart.

Coefficients of Viscosity of Various Fluids
Fluid Temperature (ºC) Viscosity
Gases
Air 0 0.0171
20 0.0181
40 0.0190
100 0.0218
Ammonia 20 0.00974
Carbon dioxide 20 0.0147
Helium 20 0.0196
Hydrogen 0 0.0090
Mercury 20 0.0450
Oxygen 20 0.0203
Steam 100 0.0130
Liquids
Water 0 1.792
20 1.002
37 0.6947
40 0.653
100 0.282
Whole blood 1 20 3.015
37 2.084
Blood plasma 2 20 1.810
37 1.257
Ethyl alcohol 20 1.20
Methanol 20 0.584
Oil (heavy machine) 20 660
Oil (motor, SAE 10) 30 200
Oil (olive) 20 138
Glycerin 20 1500
Honey 20 2000–10000
Maple Syrup 20 2000–3000
Milk 20 3.0
Oil (Corn) 20 65

The circulatory system provides many examples of Poiseuille’s law in action—with blood flow regulated by changes in vessel size and blood pressure. Blood vessels are not rigid but elastic. Adjustments to blood flow are primarily made by varying the size of the vessels, since the resistance is so sensitive to the radius. During vigorous exercise, blood vessels are selectively dilated to important muscles and organs and blood pressure increases. This creates both greater overall blood flow and increased flow to specific areas. Conversely, decreases in vessel radii, perhaps from plaques in the arteries, can greatly reduce blood flow. If a vessel’s radius is reduced by only 5% (to 0.95 of its original value), the flow rate is reduced to about of its original value. A 19% decrease in flow is caused by a 5% decrease in radius. The body may compensate by increasing blood pressure by 19%, but this presents hazards to the heart and any vessel that has weakened walls. Another example comes from automobile engine oil. If you have a car with an oil pressure gauge, you may notice that oil pressure is high when the engine is cold. Motor oil has greater viscosity when cold than when warm, and so pressure must be greater to pump the same amount of cold oil.

Poiseuille’s law applies to laminar flow of an incompressible fluid of viscosity through a tube of length and radius . The direction of flow is from greater to lower pressure. Flow rate is directly proportional to the pressure difference , and inversely proportional to the length of the tube and viscosity of the fluid. Flow rate increases with , the fourth power of the radius.

An intravenous (IV) system is supplying saline solution to a patient at the rate of through a needle of radius 0.150 mm and length 2.50 cm. What pressure is needed at the entrance of the needle to cause this flow, assuming the viscosity of the saline solution to be the same as that of water? The gauge pressure of the blood in the patient’s vein is 8.00 mm Hg. (Assume that the temperature is .)

Assuming laminar flow, Poiseuille’s law applies. This is given by

where is the pressure at the entrance of the needle and is the pressure in the vein. The only unknown is .

Solving for yields

is given as 8.00 mm Hg, which converts to . Substituting this and the other known values yields

This pressure could be supplied by an IV bottle with the surface of the saline solution 1.61 m above the entrance to the needle (this is left for you to solve in this chapter’s Problems and Exercises), assuming that there is negligible pressure drop in the tubing leading to the needle.

### 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 in (Figure). We can understand why the pressure to the home drops during times of heavy use by rearranging

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

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

We can use 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 , the pressure drop will be greatest where the tube is most narrow. This is how water faucets control flow. Additionally, 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.

(Figure) is a schematic of the 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.

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 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 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

• Laminar flow is characterized by smooth flow of the fluid in layers that do not mix.
• Turbulence is characterized by eddies and swirls that mix layers of fluid together.
• Fluid viscosity is due to friction within a fluid. Representative values are given in (Figure). Viscosity has units of or .
• Flow is proportional to pressure difference and inversely proportional to resistance:

### 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, as shown in (Figure). Explain why they are needed and how they work.

### 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 , the cart is moving at 0.400 m/s, its surface area is , and the thickness of the air layer is . (b) What is the ratio of this force to the weight of the 0.300-kg cart?

(a)

(b)

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 water between them and the contact area is ?

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

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

Fluid originally flows through a tube at a rate of . 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.

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?

(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, is its density, and is the density of the fluid and 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?

(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?

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

(b)

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?

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 , 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 and its viscosity to be (or ). Note that you must take into account the pressure due to the 50.0-m column of oil in the pipe.

(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 . (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.

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

The ratios of the viscosities of blood to water are nearly constant between 0°C and 37°C. See note on Whole Blood.

## Researcher discovers previously rejected function in the brain’s blood vessels

Summary: Despite previous findings, researchers have found precapillary sphincters in the brain. The study reports the precapillary sphincters may play a major role in disturbances of the brain’s blood supply and blood pressure.

Source: University of Copenhagen

Allegedly, they should not exist in the brain, the so-called precapillary sphincters – a kind of squeezing ‘muscle clamp’ between the larger and smaller vessels of the bloodstream.

Nevertheless, Assistant Professor Søren Grubb from the Department of Neuroscience at the University of Copenhagen has indeed shown the sphincters in mice.

‘In the early 󈧎s, a Japanese review study concluded that there was no evidence that pre-capillary sphincters should exist in the heart, brain and muscular connective tissue,’ he says and continues:

‘Since then, scientists have focused a lot on pericytes – muscle cells that can regulate the resistance in the smallest blood vessels. At the same time, however, they have somehow missed a great resistance right between some arterioles and capillaries: The sphincters. Perhaps because the discovery of the pericytes has received more attention among all the blood vessels of the brain’.

Functions as a water faucet or a sluice system

As blood flows through the brain, it flows from arteriole to vein through the capillaries. The latter are the smallest blood vessels in the body, but incredibly important. It is here that the blood and the brain exchange oxygen and nutrients.

Søren Grubb explains that the precapillary sphincters may be compared to a kind of thermostat that distributes the pressure between the branches of the blood vessels. A bit like a faucet adjusting the pressure between a water pipe and a garden hose.

As the muscle clamp relaxes, more blood cells will flow through its passage and the pressure in the following blood vessels will increase. When the clamp contracts, a bottleneck forms, which lowers the pressure further down the blood flow.

‘In this way, it also works a bit like a sluice system to irrigate fields: You may have a roaring river, but by diverting water from the river and making sluices that can regulate the amount of water for each field, you can distribute the water to many areas, says Søren Grubb.

‘Conversely, if the sluice shuts down or is clogged, the field will quickly dry out’, he adds.

Potential for dementia and migraine

Based on that picture, Søren Grubb assumes that the pre-capillary sphincters may play a major role for disturbances of the brain’s blood supply and blood pressure.

If the assumption holds true, the discovery of the clamping muscles in the brain will potentially affect the treatment of diseases such as migraine, Alzheimer’s and vascular dementia – all associated with an accumulation of waste products that may stem from blood vessel defects.

Close-up of precapillary sphincter (the strong red mark in the middle of the greenly marked blood flow) from two-photon microscope. According to the research results, these squeezing muscle cells are in the brain most often found at the early branches of blood vessels in the upper layers of the cerebral cortex. Image is credited to Lauritzen Lab, University of Copenhagen.

Already, the research group Lauritzen Lab, of which Søren Grubb is part, has tested a model for migraine with aura. The model confirms the hypothesis, but the Assistant Professor emphasises that further research is still needed in connection with disorders:

## Regulation of Vascular Tone

The study of vascular tone regulation has been based traditionally on layer-specific mechanisms. Three different layers form the blood vessel wall: intima, media, and adventitia. The intima is a monolayer of endothelial cells, which separates circulating blood from the medial layer underneath. The media consists of concentric layers of smooth muscle cells and elastic lamella, varying in number depending on the vessel size. The outer layer of the vessel, the tunica adventitia, is formed of collagen bundles, elastic fibers, fibroblasts, and vasa vasorum. It also harbors perivascular nerve endings.

The course of vascular function research has changed through the years. Most early functional studies characterized vasoconstrictor and vasodilator agents and their receptor types and subtypes. At the same time, a bulk of investigation focused on the neural regulation of medial function, characterized perivascular innervation in the adventitia and adventitial-medial border, and described both vasoconstrictor and vasodilator neurotransmitters. The identification in the 1980s of nitric oxide as an endothelium-derived relaxing factor reoriented vascular function studies of the next 2 decades. As a consequence, the endothelial layer is now considered as a paracrine tissue, which produces and releases a variety of contractile and relaxant factors that modulate medial function directly and indirectly through modulation of neurotransmitter release. During this time, the adventitia was regarded as a structural support for the media and its functional role was ignored. However, in recent years there is increasing evidence of a direct modulation of this layer on blood vessel function in a variety of situations. 1,2 The development of an easy method to remove the adventitia 3 will enable determination of the functional contribution of this layer in the near future and help to define the complex interactions and feedbacks between vascular layers.

### The Fat Connection

On top of these regulatory mechanisms, the precise understanding of vascular function requires the characterization and definition of the interactions between the blood vessel and its environment. In this regard, it has to be kept in mind that many blood vessels are surrounded by adipose tissue in variable amounts. In 1991, Soltis and Cassis 4 demonstrated that perivascular fat reduced vascular contractions to noradrenaline in rat aorta. The originality of this work was to analyze the role of a tissue that was considered a passive structural support for the artery and that is, still, routinely removed for isolated blood vessel studies. This finding was reexamined by Löhn et al, 5 who confirmed the inhibitory action of perivascular fat on aortic contractions to a variety of vasoconstrictors. This anticontractile action is induced by a transferable proteic factor released by adipocytes, which the authors called adipocyte-derived relaxing factor (ADRF), in analogy to endothelium-derived relaxing factor. The inhibitory action of ADRF is mediated by tyrosine kinase pathways and opening of ATP-dependent K + (KATP) channels. In a second work, 6 the same group characterized that the mechanism of ADRF release from rat aortic periadventitial tissue was dependent on calcium and cAMP.

In this context, this issue of Hypertension features an interesting study, which provides new insights and perspective into the role of periadventitial fat in the regulation of vascular tone. Verlohren et al 7 describe that perivascular fat has a vasodilatory effect on Sprague-Dawley rat mesenteric arteries, which involves the activation of vascular smooth muscle voltage-dependent K + channels (Kv). Interestingly, the channels activated by fat in mesenteric arteries (Kv) differ from the channels proposed to be activated in the aorta (KATP), 5 suggesting that there are vascular regional differences in the effects of perivascular adipose tissue or, as the authors suggest, the existence of different ADRFs. A second relevant experimental observation is that the anticontractile effect of ADRF positively correlates with the amount of perivascular fat. As shown in this work, the resting membrane potential of mesenteric vascular smooth muscle cells is more hyperpolarized in intact mesenteric rings surrounded by fat than in rings without fat. These results strongly suggest that perivascular adipose tissue contributes to the maintenance of basal mesenteric artery tone. Whether similar results can be observed in other strains or species remains to be determined.