Goldfarb D. Biophysics DeMYSTiFied

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302 Biophysics D e mystifieD

This tells us that if A

is greater than A

, then P

will be less than P

. In other

words, as an incompressible fluid flows from a larger tube into a smaller tube, the

pressure of the fluid decreases.

Laminar Flow

Where viscous friction cannot be ignored, the friction tends to slow the flow of

the fluid. The viscosity is due to molecular attractions between the fluid mol-

ecules, and between the fluid and the walls of the tube or channel in which the

fluid is flowing. The fluid closest to the vessel walls moves the slowest. The

fluid in the center of the vessel moves the fastest. This type of flow where the

velocity of the fluid changes layer by layer, or according to the distance from

the vessel walls, is called laminar flow.

The flow rate (volume per unit time past a certain point in the channel or

vessel) is given by the cross-sectional area times the average fluid velocity. This

is similar to the frictionless case except that we must use average velocity

because the velocity varies at different points within the cross section of the

tube.

Q 5 A v

avg

(12-18)

Normally friction tends to slow objects down. But if the average velocity is

kept constant, then the result of the friction will be a drop in pressure as the

fluid moves along a length of tube. The pressure drop represents a reduction in

potential energy as the potential energy (of pressure) is converted into kinetic

energy in order to overcome the friction and keep the fluid moving at a con-

stant average velocity. For a viscous (nonfrictionless) fluid, the relationship

between the flow rate, the fluid’s viscosity, the radius of a tube, and the pressure

drop along a given length of tube is given by Poiseuille’s Law.

RP P

−

()

(12-19)

where R is the radius of the tube,  (the greek letter, eta) is the viscosity of the

fluid, L is the length of the tube from point 1 to point 2, and P

2 P

is the drop

in pressure in going from point 1 to point 2. The interesting thing to note about

Poiseuille’s law is the dependence on the fourth power of the radius. This means,

for example, that doubling the radius will increase the volumetric flow rate

16 times. It also means that for a given (constant) flow rate and viscosity (as we

typically have in the case of blood in arteries), a small increase in radius will

chapter 12 physiological and anatomical Biophysics 303

make the pressure drop significantly less. This is easy to see by rearranging Eq.

(12-19) to solve for the pressure drop.

−=

(12-20)

Equation (12-20) tells us how much the pressure will drop, due to viscosity,

as a fluid flows along a tube with a given Q, , L, and R (i.e., with a given flow

rate, viscosity, length of tubing, and radius). The pressure drop is inversely pro-

portional to the fourth power of the radius. This means that increasing the

radius just a little bit will make the pressure drop much smaller (so the effect

of viscosity is greatly reduced). It also means that for a liquid such as blood that

has a relatively small viscosity to begin with it takes only a moderate radius for

the viscous pressure drop to be small enough for Bernoulli’s equation to become

a very good approximation.

still struggling

in the case of blood flowing in the body, the volumetric flow rate, Q, is constant.

yes it goes up and down slightly with the pumping of the heart but on average

it is constant along the length of a given blood vessel; that is, the rate of blood

flowing into one end of a vessel is the same as is flowing out of the other end. in

such a case where the flow rate is constant, viscous friction reduces the pressure.

the flow rate is the same at both ends of the vessel, but the pressure is different.

however, if this pressure change is very small, then the situation can be ap-

proximated by a frictionless fluid (Bernoulli’s equation). since the pressure drop

is inversely proportional to the radius to the fourth power, a small change in the

radius can make a big change to the pressure drop. A slight increase to the

radius can make the pressure drop small and insignificant so that Bernoulli’s

equation can be applied. conversely, a decrease in the radius can cause the pres-

sure drop to be large enough to cause significant negative health effects.

Turbulent Flow

As the velocity of fluid flow increases, a point is reached where the flow is no

longer smooth and laminar. Instead eddies are formed. These are small cur-

rents of fluid that flow backward, opposite the direction of the overall flow.

See Fig. 12-3.

304 BIOPHySICS D e mystifieD

The velocity at which flow changes from laminar to turbulent is called the

critical flow velocity. The critical flow velocity is proportional to the viscosity,

and to a property of the fluid known as the Reynolds number. It is inversely

proportional to the fluid density and to the radius of the tube.



(12-21)

From Eq. (12-21) we see that increased viscosity and increased Reynolds

number (



) increase the critical velocity; this makes it less likely at a given

velocity that the flow will be turbulent, whereas increasing the radius or the

fluid density increases the likelihood of turbulent flow.

Figure 12-3 • Turbulent fluid flow with

eddies (circular currents).

still struggling

A higher critical velocity means the fluid must flow faster in order for turbulent

flow to occur. Therefore, high critical velocity means more velocities at which

flow is not turbulent, and so less possibility for turbulent flow, Thus Eq. (12-21)

tells us that increasing viscosity makes it harder for turbulence to occur (due to

a higher critical velocity). Whereas increasing radius makes it easier for turbu-

lence to occur (due to a lower critical velocity). In regard to blood this means

that if turbulence occurs, it is most likely to occur in the larger blood vessels and

less likely in the smaller ones.

chapter 12 physiological and anatomical Biophysics 305

PROBLEM 12-2

The volumetric flow rate of blood in a person at rest is about 5 L per minute.

If the aorta (the central artery carrying blood from the heart) has a diam-

eter of 2 cm, what is the kinetic energy of blood flowing through the

aorta?

SOLUTION

From Eq. (12-18) we can calculate the average velocity of blood through

the aorta. A liter is 1000 cm

. The cross-sectional area of a tube with a di-

ameter of 2 cm (radius 1 cm or 10

m) is

A 5 π r

5 (3.1416) (10

5 3.1416  10

So the velocity is

avg

3633

(5000cm min) (10m/cm)(1min/6/

−

00s)

3.1416 10 m

0.2653 m/s



−

The kinetic energy is given by the last term in Bernoulli’s equation

[Eq. (12-12)].

Kinetic

5 (1/2)  v

5 (0.5) (1060 kg/m

) (0.2653 m/s)

5 140.6 J/m

PROBLEM 12-3

The volumetric flow rate of blood in a person at rest is about 5 L per minute.

The beginning of the aorta (the part closest to the heart) has a diameter of

3 cm, but the end of the aorta in the lower abdomen has a diameter of only

1.75 cm. How much faster is the blood traveling at the end of the aorta

than it is at the beginning?

SOLUTION

Five liters per minute is (5000 cm

/min) (10

/cm

) (1 min/60 s) 5

8.333  10

/s. We can use Eq. (12-18) to calculate the average velocity

at each end of the aorta. At the beginning where the diameter is 3 cm, the

radius is 0.015 m, so

avg

8 333 10 m/s

(3 1416) (0 015

55 5

πr

−

0 1179 m/s

PROBLEM

The volumetric flow rate of blood in a person at rest is about 5 L per minute.

If the aorta (the central artery carrying blood from the heart) has a diam-

PROBLEM

The volumetric flow rate of blood in a person at rest is about 5 L per minute.

SOLUTION

From Eq. (12-18) we can calculate the average velocity of blood through

✔

PROBLEM

The volumetric flow rate of blood in a person at rest is about 5 L per minute.

The beginning of the aorta (the part closest to the heart) has a diameter of

PROBLEM

The volumetric flow rate of blood in a person at rest is about 5 L per minute.

SOLUTION

Five liters per minute is (5000 cm

✔

306 Biophysics D e mystifieD

At the end of the aorta, where the diameter is 1.75 cm, the radius is

0.00875 m, so

avg

8.333 10 m/s

(3.1416) (0.00875

55 5



πr

−5

0.3464 m/s

The answer is that the blood is traveling with a velocity 2.94 times faster at

the end of the aorta compared with the beginning.

Arteriosclerosis

Arteriosclerosis is a disease characterized by a hardening on the arteries. The

most common form of arteriosclerosis, called atherosclerosis, involves a buildup

of fatty deposits on the artery walls which later becomes hardened with cal-

cium deposits. Aside from the loss of elasticity in the arterial walls, there is a

decrease in arterial radius. If we take Eq. (12-18) and put it in the form v

avg

Q/(r

), we clearly see the inverse square relationship between average veloc-

ity and radius. This inverse square relationship means a 30% decrease in arterial

radius will double the velocity of the blood. And a 50% decrease in arterial

radius will quadruple the velocity of blood. The kinetic energy of the blood

increases 16-fold (since kinetic energy is proportional to the velocity

squared).

This significant increase in kinetic energy occurs through a proportional

decrease in blood pressure. We can see from Eq. (12-20) that the pressure drop

due to viscous friction is inversely proportional to the radius to the fourth

power. The result is that a 50% decrease in radius will create a pressure drop

along the length of a blood vessel that is 16 times larger than it would be

otherwise.

All of this combined can be very detrimental to one’s health. One of the

worst effects is the possibility of turbulent blood flow in the arteries. While a

decrease in radius will increase the critical flow velocity, making turbulent flow

less likely [Eq. (12-21)], it does so only to the first power of the radius; whereas

the same decrease in radius increases the velocity by the square of the radius.

So the velocity increases much faster than the critical flow velocity, making it

much more likely that the critical flow velocity will be exceeded and smooth

laminar blood flow will be replaced by turbulent flow. This is especially danger-

ous during strenuous activity when the volumetric blood flow and average

velocity can be as much as 5 to 6 times larger than when at rest. Once turbulent

flow commences, the heart has to work a lot harder to maintain the same flow

chapter 12 physiological and anatomical Biophysics 307

rate. Turbulent flow is also dangerous because it puts more stress on the walls

of blood vessels, which can weaken the arterial walls or can dislodge some of

the deposits. These dislodged deposits then float through the bloodstream and

can cause a blockage elsewhere in the body.

Hummingbird Hovering

Many birds are able to hover, to fly in a relatively fixed position, not going up

or down, left or right, forward or backward. Many large birds, for example

seagulls, can hover for brief periods of time by adjusting their wings to ride the

breeze like a kite. This of course depends on the consistency of the wind, and

the bird’s ability to rapidly adjust to changes in wind currents. The dynamic

forces involved are rather intricate and so we will not take the time to examine

them here. However, hummingbirds can hover for extended periods of time,

with no breeze at all, by flapping their wings very rapidly.

Let’s examine what happens when a hummingbird hovers. In order to hover,

an organism must generate an upward force that is, on average, equal to its own

weight. The upward force balances out the force of gravity (the organism’s

weight) so that no upward or downward acceleration exists. For this discussion

we will ignore forces in directions other than vertical. But you should know and

consider that what follows is simplified. In practice the movements and forces

necessary for extended hovering in birds are quite complex, since the bird must

also control its horizontal motion (especially if it is trying to drink nectar from

a flower while hovering). Still, since motion in general can be mathematically

resolved into each of the three dimensions, and each dimension dealt with

independently, the discussion that follows regarding the vertical dimension is

correct for that dimension. We’re just not going to complicate matters by

including the horizontal forces and motions in our treatment.

The lifting force needed for hovering is not constant, but varies with the

movements of the wings. This is why we said that the force must be on average

equal to the weight of the bird. As long as the average force equals the weight,

then the bird can hover, provided the fluctuations in the upward force are

rapid. This is because every time the upward force is less than the weight, the

bird will accelerate downward (fall). And every time the upward force is greater

than the weight, the bird will accelerate upward (rise). If these fluctuations are

relatively small and rapid, then the rising and falling of the bird will be minimal,

or even negligible. But if the changes in lift are large or slow, then bird will end

up bobbing up and down in the air instead of hovering.

308 Biophysics D e mystifieD

Average Lift

Let’s make an approximation, or simplifying assumption, that the lifting force

is constant for the duration of each stroke of the wings. Then the only fluctua-

tion in force that we need to deal with is the difference between the force of

the downward stroke and the force of the upward stroke. With this approxima-

tion we have only two different lift forces that average over time to compensate

for the weight of the bird.

The upward lift comes from the reactive force of air pushing up on the wing

when the wing pushes down on the air (Newton’s third law). Depending on

the angle of the wing, it is possible for lift to be generated on both the down-

stroke and the upstroke. Figure 12-4 shows a diagram of a hummingbird in

horizontal flight. The beating of the wings provides both lift and forward thrust.

Compare this with Fig. 12-5 which illustrates sustained hovering flight. When

hovering, the hummingbird orients itself so that the downstroke is somewhat

of a forward stroke (relative to the ground), and the upstroke is somewhat of a

backward stroke. This orientation enables both strokes to provide lift. (We still

Figure 12-4 • Hummingbird horizontal flight.

(Courtesy of Wikimedia Commons.)

Figure 12-5 • Hummingbird sustained hovering flight. (Courtesy of Wikimedia Commons.)

chapter 12 physiological and anatomical Biophysics 309

call them downstroke and upstroke, because relative to the bird’s anatomy the

strokes are down and up.)

There are two extreme cases we can look at in terms of the lifting force

generated by the beating of the wings. At one extreme, both strokes equally

generate lift, and so the lifting force is essentially constant. At the other extreme,

lifting force is generated only on the downstroke, and no vertical force is gener-

ated by the upstroke. Let’s examine this latter case in more detail.

When lifting force is generated only on the downstroke, the force alternates

with the beating of the wings, between a positive value of upward lift and zero.

In such a case the bird will fall slightly during the upstroke (due to gravity) and

rise slightly during the downstroke (due to upward lift that is larger than the

weight of the bird). This size of the up and down motion will depend on the

amount of time for a wing stroke. On the downward stroke, the lifting force

accelerates the bird upward for the duration of the stroke. On the upward

stroke, with no lifting force, gravity accelerates the bird downward, again for the

duration of the stroke. Depending on the species, a hummingbird can beat its

wings anywhere from 20 to 100 times per second.

If we assume an equal amount of time is spent in each stroke (up or down),

then we can calculate the amount of force required on the downstroke. The

average lifting force must equal the weight of the bird. Since the upstroke

exerts zero force, the downstroke must therefore exert a force equal to 2 times

the weight of the bird in order for the average of the two numbers to equal the

weight of the bird. (If either the down- or upstroke lasted longer than the other,

then we would calculate a weighted average, weighted according to the per-

centage of time of one wing beat cycle spent in each stroke.)

PROBLEM 12-4

Assume a hummingbird beats its wings (one complete cycle of downstroke

and upstroke) 20 times per second. Assume also that the time for each

stroke is exactly half a beat cycle. Assume that the only vertical force on the

bird during the upstroke is due to gravity, and the only vertical forces dur-

ing the downstroke are gravity and a lift force equal to twice the weight of

the bird. Calculate the height of the up and down motion of the bird while

hovering.

SOLUTION

We only need to calculate the downward motion. The upward motion is

exactly the same only in the opposite direction. This is because during the

PROBLEM

Assume a hummingbird beats its wings (one complete cycle of downstroke

and upstroke) 20 times per second. Assume also that the time for each

PROBLEM

Assume a hummingbird beats its wings (one complete cycle of downstroke

SOLUTION

We only need to calculate the downward motion. The upward motion is

✔

310 Biophysics DemystifieD

downward motion the only force is the weight of the bird. During the up-

ward motion there are two forces: an upward force equal to twice the

weight of the bird, and the downward force equal to the weight of the bird.

The net force is therefore equal to the weight of the bird, but in the upward

direction.

Since F 5 ma (and the mass is constant) and since the upward and down-

ward forces are equal in magnitude (just occurring at different times in

different directions), then the accelerations are also equal in magnitude.

Therefore we can use the acceleration due to gravity to calculate how much

the bird drops during the upstroke, and assume that during the down-

stroke the bird rises by the same amount.

We know the acceleration due to gravity is 29.807 m/s

. Each stroke is

half a cycle, and there are 20 cycles (wing beats) per second. So the length

of time for one stroke is 1/2 of 1/20 of a second, or 1/40 of a second.

During the upstroke, the bird accelerates 29.807 m/s

for 1/40 of a sec-

ond. The distance the bird drops during this time is equal to the average

velocity multiplied by the time [from Eq. (12-6)].

d 5 v

avg

t (12-22)

Since the acceleration is constant, the average velocity is just the average

of the initial and final velocities [Eq. (12-5)]. We can assume the initial ve-

locity is zero (at the beginning of the upstroke). Therefore the average ve-

locity is just v

/2.

d 5 (v

/2) t (12-23)

We get the final velocity v

from Eq. (12-4) by setting v

5 0; this gives

5 a t.

Substituting this into Eq. (12-23) gives us d 5 (a t/2) t, or

d 5 (1/2) at

(12-24)

Therefore the distance the hummingbird drops during the upstroke is

d 5 (0.5) (29.807 m/s

) (0.025 s)

5 20.00306 m

At 20 beats per second the hummingbird rises and falls only 3 mm while

hovering (and it does so 20 times per second).

chapter 12 physiological and anatomical Biophysics 311

Hummingbird Wing Lift

Most birds generate lift only on their downstroke. However, most birds are not

able to hover. Many flying insects can hover, and many insects are able to rap-

idly twist or angle their wings with each stroke so that they generate lift equally

on both the upstroke and downstroke. Hummingbirds however fall somewhere

in between these two extremes, or at least the one species of hummingbird that

has been studied thoroughly in this regard.

Recent research examined hummingbird sustained hovering flight using digital

imaging particle velocimetry (DPIV). DPIV is a technique that involves suspending

small tracer particles (10 to 100 m in size) in a fluid, such as air. The particles

are illuminated with laser light, and high-speed digital video cameras record the

location of the particles as many as 500 times per second. Computer analysis can

then be used to calculate the velocity of the particles and the force required to

accelerate them to that velocity. Researchers observed rufous hummingbirds hov-

ering in a variable-speed wind tunnel, and used DPIV to analyze air currents

generated by the hummingbirds’ wings. They discovered that, while hovering,

hummingbirds generate 75% of their lift during the downstroke, and 25% of their

lift during the upstroke. In other words both strokes generate a lifting force, but

the downstroke generated 3 times as much lift as the upstroke.

We can use this information to calculate the force generated by each stroke.

Let’s call LF

downstroke

the lifting force generated by the downstroke and LF

upstroke

the lifting force generated by the upstroke. We know the average lifting force

must equal the weight of the hummingbird. Assuming that the downstroke and

upstroke take equal amounts of time, the average is

(LF

downstroke

1 LF

upstroke

)/2 5 weight (12-25)

We also know that the downstroke generates 3 times as much lift as the

upstroke.

downstroke

5 3 LF

upstroke

Substituting this into Eq. (12-25) gives

3 LF

upstroke

1 LF

upstroke

5 2 weight

upstroke

5 2 weight