Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
Chap 14B
Fluid
Dynamics
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Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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Introduction:
Once a fluid is set into motion, the expressions we have discussed thus far are inadequate to explain
or predict the properties or behavior of the fluid. The study of fluids in motion is called hydrodynamics,
and we will briefly examine a few of the guiding principles which can assist us in making sense of fluid
flow.
We will examine, in turn:
1) The continuity equation;
2) Bernoulli’s Principle
3) Viscosity and viscous flow
4) Poiseuille’s Equation;
5) Turbulence and the Reynolds' Number.
As so often is the case, we must carefully delimit any new units of measure that are pertinent to this
new area, and there are two of them that are particularly important, the volume flow rate.
Volume Flow Rate and Linear Flow Velocity
The volume flow rate is defined by the equation:
[1]
ΔV
Q=
Δt
or, in the language of calculus:
[2]
dV
Q=
dt
where Q is the volume flow rate and ΔV is the volume that flows past the point of observation in
some increment of time, Δt. A couple of things to note: V is the volume and not the velocity; the volume
flow rate is the volume of some fluid (in liters, or cubic meters, or cubic centimeters, or whatever) per unit
time, that flows past some observation point. Imagine yourself on the bank of a river, holding a stopwatch,
and with the power to count how many gallons of water are moving past you. Divide the volume by the time
elapsed on the stopwatch and you have Q. Note also, that the letter “Q” will be used in several contexts. In
thermodynamics it will represent heat, in electricity charge, in the study of oscillations, the quality factor of
an oscillator. In the context of fluid dynamics it represents volume flow rate. There are only so many letters
in the English and Greek alphabets, and there are many more concepts that need a symbol. For this reason
it is essential to cultivate the habit of being thoughtful about the context in which you are using a variable
and that, just like red, read and reed, it is the context that is all important.
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The linear flow velocity on the other hand is the actual speed of the stream of fluid. For example,
you could measure the linear flow velocity of a river by releasing a cork or some other object and
measuring how fast it was carried some known distance. A river or a pipe can have a large volume flow rate
and a very small flow velocity or a large flow velocity and a small volume flow rate. Alternatively, both
could be large or both could be small.
The river with the world’s largest flow rate, the Amazon, dumps over an average of more than
200,000 m3/s of fresh water into the Atlantic Ocean. But its linear velocity averages out to about 0.7 m/s
(about 1.5 mi/h). On the other hand, the Colorado River, near Canyonlands National Monument in SE Utah
barrels along at 6 m/s and possesses very scenic and adventurous rapids. But its volume flow rate at this
point is a mere 120 m3/s.
Finally note that volume starts with a “v” and velocity starts with a “v”. But they aren’t the same
thing and they don’t use exactly the same letter anyway: velocity uses a small “v” and Volume a capital
“V”. I know this is obvious and perhaps too obvious to mention, but experience has shown that students,
particularly beginners, get tangled up by the simplest things and so a word to the wise...so to speak.
Continuity Equation
You probably have noticed one or more of the following phenomena: A garden hose puts out a
Figure 1: Garden Hose “Experiment”
relatively slow moving stream of water until you attach one of those brass or plastic nozzles which narrow
the stream but seem to make it go faster. The same thing happens if you are conducting water from a water
faucet and pinch the hose. The water squirts out faster when the end of the hose is almost squeezed shut.
Even though you are blocking the water, it comes out faster. This is confirmed by the fact that the stream
travels further if you direct it horizontally. Figure 2 illustrates these observations quite nicely.
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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Figure 2: Pinch a hose and get more linear flow velocity
What is going on is little more than the conservation of mass. The idea is this: take any imaginary
volume in the hose: if fluid is flowing through it, the amount of matter entering that volume must equal the
amount leaving it, if the amount of matter in the volume is to stay the same. In figure 3, let’s assume that
fluid is flowing from left to right through the green pipe. Consider the mass of fluid m1 that crosses
boundary 1 and enters the center region of the pipe. A corresponding amount of fluid, m2, crosses
1
2
Figure 3: Mass flow in a tube
boundary 2. If no fluid enters or leaves the pipe in the region bounded by “1” and “2”, then the mass
flow rate out must equal the mass flow rate in:
[3]
Δm1 Δm2
=
Δt
Δt
From the definition of the density we know:
[4]
m = ρV
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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and so equation [3] becomes:
ΔV1
ΔV2
ρ1
= ρ2
Δt
Δt
[5]
If we confine ourselves to working with liquids, then, as we have seen, our fluid will be very nearly
incompressible, and so we can comfortably make the following approximation:
ρ1 = ρ2
[6]
i.e., the density is very nearly the same throughout the pipe. Equation [5] then reduces to:
ΔV1 ΔV2
=
Δt
Δt
[7]
or, if you prefer:
[8]
Q1 = Q2
In other words, for a liquid or any other incompressible fluid, the volume flow rate in must equal the
volume flow rate out. If the volume flow rate out was greater than the flow rate in (Q2>Q1), then the water
would “break up” and you would no longer have continuos flow. If Q1>Q2 the tube would rupture if the
fluid was truly incompressible. In either case, the assumption of continuous flow of an incompressible fluid
would break down.
Now consider what might happen if we no longer required the size of the tube to be uniform (see
figure 4). Once again, let’s assume that the fluid is flowing from left to right, and this time let’s examine
the flow of a small plug of fluid whose length is given by d1 at location 1, where the cross sectional area of
the flowing fluid is A1.
When this plug moves one “plug length” (i.e. d1) then another plug of exactly the same volume
must make room for it by moving at location 2. This plug of fluid has a length d2 and a cross sectional
area of A2 (again, see figure 4).
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A2
A1
d2
d1
Figure 4: Continuity in a tube of variable diameter
Equations [7] and [8] still hold (the volume flow rate is the same everywhere along the tube), but we
can restate the volume in terms of “A” and “d”:
ΔV = Ad
[9]
or, applying this to the equation of continuity for incompressible fluids [7]:
A1v1 = A2 v2
[10]
or, put another way:
[11]
v2 A1
=
v1 A2
In other words, the flow velocity is inversely proportional to the cross sectional area. Fluids speed
up when the tube in which they are flowing narrows, and they slow down when it broadens. We started out
this section by taking note of the observation that constricting a hose or a piece of plastic tubing, either
manually or using a brass nozzle, seemed to cause the fluid to exit the tube with a much higher velocity, and
now we find the reason for that to (ultimately) lie in the conservation of mass, under the alias of the
equation of continuity. There are numerous other examples of this: If you slowly push the plunger on a
syringe, the liquid inside exits the narrow end with a great deal more speed than the speed of the plunger.
Rivers flow much more rapidly when the channel is narrow than they do when it broadens.
One piece of technology that we have all used on occasion and which utilizes the physics contained
in the continuity equation and in figure 4 is shown in figure 5:
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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Figure 5: Weapon of conservation of mass destruction
A schematic diagram of one of many possible designs for a workable water pistol is shown in
figure 6:
8
3
4
2
5
1
6
7
Figure 6: Sketch of the innards of a typical water pistol
In the pistol in figure 6, the trigger (1) pushes the spring loaded piston (2) into the water chamber
using an inclined plane. It doesn’t have to use this configuration. It often moves a lever on a pivot to move
the piston--it doesn’t matter, as long as the piston moves. The pressure in the small chamber rises and
flows through the narrow tube (3) after pushing open the small check valve (4). This is where the continuity
equation applies--the narrowness of the small tube is what cause the water to speed up enough to reach
your opponent. Of course you can add a pump or even a motor, but the principle is still the same.
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The remainder of the mechanism is there to keep the water replenished., As the trigger is released,
the piston is retracted. This lowers the pressure in the small chamber and the check valve (5) opens and
water flows from the reservoir (7) through the plastic tubing (6) into the chamber, replacing the water you
just shot at your little brother. When you run out of water in the large reservoir, you remove the plug (8)
and refill.
Another important (and more serious) example of the continuity equation is that of blood flow. As
you are aware, the aorta (the main artery leading out of the left ventricle of the heart) is very wide and
branches off into narrower and narrower blood vessels which ultimately terminate in the capillaries. The
continuity equation, which predicts an increase in flow velocity as the diameter of a tube decreases, explains
why capillary cuts are so dangerous. A capillary when cut spurts out blood at a very high velocity whereas
the slow moving blood in the wide artery moves very slowly and oozes out.
Right?
If you are convinced of this, perhaps you should stay out of medicine. A cut in an artery is an
extreme emergency and is characterized by spurting blood, pushed out with each heartbeat, in considerable
volume and difficult to control (obviously in proportion to the size of the affected artery). When we get a
small cut or scrape, in contrast, we are generally injuring capillaries.
But this then still leaves us with a problem. The continuity principle seems to have broken down.
The flow velocity seems to decrease in the smaller blood vessels (the capillaries). The resolution of this
paradox lies in the fact that the aorta feeds literally billions of capillaries. The numbers and a few references
are presented in problem FLUIDS19 in your homework packet. Figure 7 gives a schematic presentation of
what is going on: put simply, the aggregate cross sectional area of the capillaries is far greater than that of
the aorta.
Figure 7: One artery feeding many, many capillaries
What this means is that the linear flow velocity of the blood decreases sharply as it enters the
smaller blood vessels. (and increases again, perhaps as it exits?) One salubrious effect of this decrease is
that the blood cells and plasma pass very slowly through the capillaries allowing ample time for nutrients
and waste products to be exchanged.
Example 14b-1--Water is flowing from left to right (region “A” to region “B” in the pipe shown in
figure 8. The diameter of the pipe is 8.00 cm in region “A” and 11.31 cm in region “B”. The flow
velocity is 3 m/s in region “A”. What is the flow velocity in region “B”.
This is a continuity equation problem and so the first thing we ought to do is write down the continuity
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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equation for incompressible fluids.
A
B
Figure 8: Section of pipe used in Example 14B-1
A1v1 = A2 v2
[10]
We can rewrite this using subscripts that relate to how we have labeled the regions in figure 8:
AAv A = AB vB
[12]
and then solve for vB :
[13]
AA
vB = v A
AB
Now before we go plugging in, I want to show you something, and this involves a bit of algebra.
The first point is, of course, a word from our sponsor--do the algebra first and punch those calculator keys
last. The second point is that, in these continuity problems, the ratio of the velocities is linear in the ratio of
the areas, but quadratic in the ratio of the radii. The third point is that, when you use the diameters in a ratio,
you get the same result as if you used the radii, or the formula for the area, because the constant factors
(such as pi) cancel out. So let’s try it out. Start by subbing for the areas:
[14]
πrA2
vB = v A 2
πrB
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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Cancel out the pi’s:
rA2
vB = v A 2
rB
[15]
Now sub for the radius:
(
2)
vB = v A d 2
( 2)
dA
[16]
2
B
Cancel out the 2’s:
d A2
vB = v A 2
dB
[17]
Now plug in numbers:
[18]
2
(0.08
m)
vB = 3.00 m s
2 = 1.50
(0.1131 m)
m
s
Again note that we don’t simply take the ratio of the diameter. We must use the ratio of the square
of the diameter because the area goes as the square of the radius and the diameter is simply 2 times the
radius. Because we are using a ratio of 2 values of the same variable (the diameter), the fact that it is related
to the radius by a factor of 2 has no bearing on our answer. The 2’s cancel out. This method is generally
applicable and worth your while to learn it and become comfortable with it.
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Bernoulli’s Principle
Try the following experiment: take a shower. No, this is not some snarky, acerbic witticism,
attempting to imply that scientists are unhygienic. It is a serious suggestion for observing a very strange
behavior in nature that crops up in the most ordinary daily activities.
If you do follow my advice and do take a shower, you should do so in a tub or stall with a hanging
curtain, and you should turn on the water at as high a velocity as possible. The lighter the curtain the better
as well. When the shower is turned on you will see the curtain pulled in at the bottom, and you may even
feel a light breeze on your ankles.
However, here is another suggestion: if you don’t want to take a shower, take a bath. This time,
observe the shape of the stream of water that comes out of the faucet--best to do this for a moderate flow
and without an aerator. The flow will narrow as the fluid falls.
Here’s a third experiment. Blow across a horizontal piece of paper. You need something that is not
very dense, not very stiff, and also not very porous. Printer paper may not work. But you can try a piece of
DYC extra scratchy hand towels and these will work fine. If they are unavailable, then use printer paper, but
hold two slices vertically, and blow between them.
Let’s talk about the water from a faucet first:
Figure 9: Thinning of a falling stream of water
Back when we studied continuity, we were studying almost always incompressible fluids, and
always fluids that were engaged in laminar or layered flow, in which the molecules were assumed to slide
past each other in layers. If the flow is indeed smooth and not turbulent then each layer keeps to itself and
the path od any element of water must follow a curve or path drawn parallel to the fluid flow.
These curves are called streamlines and the flow is called streamline flow which is synonymous
with smooth flow and laminar flow. An “artist’s conception” is in figure 10 for our falling column of
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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1
2
Figure 10: Streamlines in a falling column of water, as from a faucet
water in figure 9. There are, of course, rules associated with streamlines and here they are:
• First, as we’ve seen, streamlines never cross, and molecules stay in their own layers.
• The more the streamlines converge, the faster the fluid’s flow velocity.
• The direction of the velocity is always tangent to the streamline.
The density of the streamlines are highest at the bottom of figure 10, and this is where the linear
flow velocity is greatest. Because the stream of water is flowing fastest here (due to acceleration due to
gravity), it has to narrow down because of the same considerations that we encountered when we derived
the continuity equation. The cross sectional area must get small do maintain equal mass flow and volume
flow rate (our fluid is incompressible) along the water stream; this gives us the same result as the idea that,
as the streamlines become more concentrated, the speed increases.
All along the column of water, each little increment of volume of the fluid is doing work on the
increment below it. Pick any two positions. At position 1, the fluid above it is performing W1 on the fluid
below it and at position 2, that fluid is performing W2 on the fluid below it. The work done on the fluid
between “1” and “2” must be:
[19]
W f = W1 − W2
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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where we have considered the magnitude of W1 and W2 and not incorporated the sign inside the
variable symbol. If the fluid moves d1 at position 1 and d2 at position 2, then this turns into
W f = F1d1 − F2 d2
[20]
Now we can use an old standby, the definition of pressure:
F
P=
A
[21]
Plug this into [20]:
W f = P1 A1d1 − P2 A2 d 2
[22]
The work done on the fluid (between 1 and 2) shows up as a change in mechanical energy:
W f = ΔK + ΔU
[23]
where:
1
1
2
ΔK = m v2 − m v12
2
2
[24]
and:
[25]
ΔU = mgy2 − mgy1
Combining it all together:
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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1
1
2
2
[26] P1 A1d1 − P2 A2 d2 = m v2 − m v1 + mgy2 − mgy1
2
2
But the fluid is incompressible and this means that ΔV must be the same wherever it is located:
ΔV = A1d1 = A2 d2
[27]
So plug [27] into [26]:
[28]
1
1
2
P1ΔV − P2 ΔV = m v2 − m v12 + mgy2 − mgy1
2
2
Now divide through by ΔV:
1 m 2 1 m 2 m
m
[29] P1 − P2 =
v2 −
v1 +
gy2 −
gy1
2 ΔV
2 ΔV
ΔV
ΔV
Another old friend comes in handy:
[30]
m
ρ=
ΔV
The density is assumed to be the same everywhere:
[31]
1 2 1 2
P1 − P2 = ρv2 − ρv1 + ρgy2 − ρgy1
2
2
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Now group the 1’s and 2’s together:
[32]
1 2
1 2
P1 + ρv1 + ρgy1 = P2 + ρv2 + ρgy2
2
2
This means that the sum on the LHS must be the same everywhere and you can’t change that just
by changing subscripts. Therefore it is a constant, and [32] becomes:
[33]
1 2
P1 + ρv1 + ρgy1 = constant
2
This is Bernoulli’s equation, but, before we go into all its implications, let’s just review what we
have done. Although the derivation is a bit exhausting, all we have said is that any work done on a flowing
fluid must show up as some change in its kinetic or potential energy, which does not seem to be all that
radical a statement. We could have taken a slightly different but equivalent approach. We could have said
that the net work done on the sample was equal to the work done on it by the earth’s gravitational field
(thereby altering its potential energy) plus the work done on it by the applied pressure. By the work energy
theorem, these add up to the change in Kinetic Energy of the fluid. We would have obtained identically the
same answer.
What are the consequences of equation [33]? Well for one thing, suppose you are not changing
your altitude--your water is flowing through a pipe of some sort and so ρgy is a constant. If you increase
the velocity, “v”, somewhere along your stream tube (that’s what they call a bundle of streamlines), then
your pressure has got to go down compared to elsewhere along that tube where the flow velocity is higher.
This observation is responsible for the common verbal statement of Bernoulli’s principle:
“Where the velocity is high, the pressure is low, and where the velocity is low, the
pressure is high.”
On the other hand, if you abolish velocity: if the fluid is not flowing and you recognize this reality
by setting the velocity, “v” equal to zero, then you get [34]:
[34]
P + ρgy = constant
This is just [35], the equation for hydrostatic pressure as a function of depth, in disguise:
[35]
P − P0 = ρgh
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if we recognize the equivalence between y and h:
[36]
y = −h
On the other hand, if we use [31] and assume P1 = P2, which would be the case of a tank or bucket
with a spigot or hole at the bottom, then v2 = 0 and [31] becomes:
[37]
1 2
ρv1 + ρgy1 = ρgy2
2
[38]
v1 = 2g(y2 − y1 )
This is Torricelli’s equation, and it looks an awful lot like a kinematics equation. It was discovered
by Torricelli in 1643 nearly 100 years before Bernoulli published his more general results in 1738 in his
book, Hydrodynamica, which book his father, Johann (not his brother, Johann II) is said to have liberally
plagiarized. He was also a close friend and associate of Leonhard Euler.
Equation [38] predicts that the exit velocity of a stream of water from the bottom of a tank depends
on the height of the water in the tank above the exit hole and that’s it. It does not depend on the size of the
hole. We drilled three holes in the bottom of a plastic jar to test this (see figure 11).
Figure 11: Three holes in a jar
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When we ask students what stream will go the farthest, we get all sorts of answers. The smaller hole
has more pressure. The smaller hole has more force. The larger whole has more force (if they say “more
force is exerted by the fluid at the larger hole, we feel a little better) and Pascal’s principle is cited. Pascal’s
principal indeed can be cited here if the flow rate from the container is minimal (remember, it applies to a
confined fluid), as long as we remember that the mass of the fluid ejected from the container per unit time is
going to be proportional to the magnitude of the are of the hole. And so will the force doing the ejecting
because F=PA, but that seems to point to the exit velocity being the same for all three holes, the Torricelli
prediction. Here is what our first experiment looked like:
Figure 12: Torricelli Theorem Experiment
The large and medium sized streams seem to be exiting with the same speed, but the small stream
(hard to see but way at the rear of the curved cup) is barely landing a cm away from the bottom. We
decided that this might be a surface tension effect which impeded the flow of water at the hole. We added
some soap and got no results at all, because the soap solution was so concentrated that the surface tension
was too low for any of the three streams to form form: water needs a skin to cohere into a stream. We then
made a dilute soap solution and you can see what we found in figure 13.
Now all the streams hit the bottom of the sink at about the same horizontal distance from the jar.
This implies that their horizontal exit velocities were all nearly the same. In attempting to capture an image
of the narrow stream we may have neglected the large stream and figure 14 provides a better view of it. The
smallest stream now did, in fact, touch down at roughly the same horizontal distance as the other streams.
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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Figure 13: Dilute Soap Solution Test of Torricelli’s Theorem
Figure 14: Better view of the large stream in Torricelli’s Theorem Experiment
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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There are innumerable other examples of Bernoulli principle experiments and applications, leading
some to comment that Bernoulli’s principle is invoked whenever we don’t really understand some natural
phenomenon. In figure 15 we show what happens if you do the experiment described above in which you
blow between 2 sheets of paper:
Figure 15: Lowering the pressure between 2 sheets of paper by blowing between them
This brings us to the shower. We have taken care of the bath tub already with our discussion of the
faucet. We expect some lowering of pressure in the accelerating stream of water, due to its acceleration
under the influence of gravity (figure 9), but the major insight provided by the narrowing stream is that of
the effectiveness of the continuity equation in predicting the properties of fluid flow.
The shower is another matter, and figure 16 compares the bottom of the curtain under water-on vs.
water-off conditions:
Figure 16: The curtain is definitely pushed in towards the shower, when the water is
turned on (right-hand frame)
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There is a tremendous amount of controversy over this explanation. Some argue that the
explanation lies in the rising of hot steam. This is contradicted by reports that the phenomenon is readily
observed in cold showers. Another suggestion is that condensation of steam on the shower curtain pulls it
in and there is even an electromagnetic explanation. One recent study modeled the flow of droplets and
gases using modern hydrodynamic software and suggested that a horizontal vortex (sort of like a sideways
tornado) was formed in a shower and pulled air in from the surroundings. Unfortunately the model was
formulated without a person actually being in the shower, a shortcoming in the minds of many critics.
The Bernoulli Principle explanation is relatively simple: accelerating water from the shower head
lowers the pressure perpendicular to the streamlines of the liquids and the entrained gases in a shower. The
pressure in the stream gets lower as it falls and this pressure drop is felt in the surrounding gas
(atmosphere) and so the higher pressure air in the rest of the room pushes in on the curtain.
One question: where does the air flowing in towards the shower stream go? Is it dissolved in the
gas and washed down the drain? Does it rise and float out of the shower at the top? If so, why?
So what is the right answer? I honestly don’t know. So why talk about it? Beats me, except to ask
all of you why everything has to have a point. Scientific discussions, even the endless debates that
accompany problems like the shower curtain problem, are enjoyable and intellectually stimulating and they
deepen our understanding of physics. So there.
Since our circulatory systems involve the flow of fluids, we would expect that many of their
functions (and pathologies) evoke fluid dynamics explanations. A TIA (transient ischemic attack), for
example, whose symptoms can mimic a stroke, can occur when a constriction in the subclavian artery
Figure 17: Continuity and Bernoulli explain TIA
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causes the blood velocity to speed up (remember the continuity principal). The Bernoulli principle predicts
that this will cause a local lowering of the blood pressure, causing blood which should flow through the
basilar artery to the brain, to back flow into the shoulder, causing dizziness or weakness. The constriction
can be caused by arterial deposits or it can (rarely) happen as a result of an awkward movement during
exertion. The situation is depicted schematically in figure 17 (drawing after that appearing in Giancoli:
Physics, 5th Ed.).
Another example is associated with the formation of an air embolism during a certain class of
surgeries. An air embolism occurs when a volume of air is entrained in the circulatory system. It often
causes injury to the right heart and to the lungs by interfering with blood flow. It is particularly at risk of
occurring during surgical procedures in which the operating site is significantly above the level of the heart.
Once again our friend:
[35]
P − P0 = ρgh
provides the key insight. If "h" (the depth of the fluid) is small, the pressure may become lower than
atmospheric pressure, and air may enter the circulatory system. This is particularly significant in the veins,
where the blood pressure is small to begin with. But the blood flow in the veins also contributes to a
lowering of pressure along the streamlines and heightens the chances that air will be entrained.
The occurrence of venous air embolisms is very high (estimates run as high as 100%!!) in so-called
sitting craniotomies. It is also high in many procedures performed on the head and neck if the patient must
be positioned in a sitting position.
The morbidity of these embolisms depends on how rapidly air is entrained and on the actual
amount of air that enters the circulatory system. In risky procedures, a catheter is often inserted into the
right atrium (at the junction between the superior vena cava and the right atrium) so that air can be aspirated
out of the patient.
This effect (Bernoulli) is also what is behind airplane flight. An airplane wing is shaped so that air
which flows over it at a faster speed than under it. As a consequence (Bernoulli's Principle sez) the pressure
under the wing will exceed the pressure above it and the wing will be lifted. The background explanation
for oh-so-many years went something like this: The air has a longer path to flow above the wing than under
it. Since the airstreams must merge once more after traversing the wing, the top airstream must move farther
in the same amount of time than the one underneath. By Bernoulli’s principle, then, where the velocity is
lower, the pressure is greater and you get lift (see figure 18).
The major problem with this analysis is the (false) requirement that the airstreams “meet” in the
sense that it takes equal time for the air to travel above and below the wing. In fact, the air above the wing
can travel at twice the velocity (or mare) than the air below the wing. The key principle to invoke is the
continuity principle (see figure 19). The shape of the wing pushes the airstreams above it closer together
than those below and the continuity principle demands that the linear flow velocity of the airstreams above
the wing increase to compensate. The shape of the wing narrows the space that the airstreams must
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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Figure 18: Cross Section of Airplane Wing
flow (the streamtube) through just as surely as the narrowing in a pipe or an obstruction in an artery. At
this point, Bernoulli’s principle applies and where the velocity is greater, the pressure is less, and this is
above the wing. Thus we get lift (see figure 19).
Figure 19: Squeezing Streamlines
One factor that substantially affects the amount of lift that we get from a given airfoil is the socalled angle of attack, i.e. the angle that the wing makes with the forward velocity vector of the airplane. A
second factor is the forward velocity of the airplane. These factors are summed up in the so-called “Lift
Equation”:
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
[35]
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L = cL ρAv2
Here L is the lift, cL is the lift coefficient, ρ is the density of air, A is the surface area of the wings,
and v is the airspeed. The lift coefficient contains the dependencies on wing design and angle of attack and
must be extracted experimentally. In general, at low angles of attack, the Lift is linearly dependent on angle.
However, as the angle of attack increases beyond a threshold value (which varies depending on wing
design) the tendency to stall increases and the additional lift decreases.
At low airspeeds, such as during take off and landings the v2 term becomes less important and it is
important to increase the value of cL by increasing the angle of attack. There are methods to circumvent the
stalling associated with increasing the angle of attack too much, and still get additional lift at low speeds:
these generally involve the addition of additional hinged surfaces such as flaps on the wings. The lift
generated by deflecting air from the bottom surface of the wing is sometimes called momentum transfer lift,
and lift originating in the compression of the streamtube above the wings is identified with Bernoulli lift
and referred to as Dynamic lift. At small angles of attack, dynamic lift is more important when the velocity
is high, whereas, at slow speeds, the use of a larger angle of attack. or of flaps and other surfaces, is
required to offset the loss of dynamic lift due to the reduction in air speed.
However, it is important to note that this discussion is an approximation; that dynamic lift and
momentum transfer, as we have defined them, are both present all of the time and act in concert to provide
lift. Air flow over both the upper and lower surfaces contributes to lift at all times.
In addition to lift provided by the wings, the fuselage and all parts of the plane with horizontal
surfaces provide lift. Likewise, spoilers (on top of the wings) are often used to slow the plane down on
landing and also to “spoil” the left generated by the top surface of the wing.
The Bernoulli principle is not limited in its applicability to major airports. It also rears its head in
Figure 20: Bunsen Burner
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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our kitchens (as well as in chemistry labs). Air has to be mixed with street gas (mostly methane) for
complete combustion to occur in stoves and in Bunsen Burners, leading to the hottest flame, and
minimizing the production of carbon monoxide. To so this, a rotatable sleeve fitted with holes, lets gas flow
from atmospheric pressure to the lower pressure inside the burner tube. The lower pressure is a result of
the forward flow velocity of the street gas, lowering the pressure compared to atmospheric pressure, as
predicted by equation [33] (see figure 20).
Viscous Flow
Thus far we have confined ourselves to frictionless laminar flow. This may not be obvious, but at no
time have we invoked frictional forces, intermolecular forces or collisions between molecules. In fact we
explicitly ruled them out in our discussions of streamlines in which mass elements were explicitly
forbidden from crossing from one streamline to another. This self-discipline provided us with a set of
assumptions that permitted the derivation of two very powerful and insightful equations: the equation of
continuity and Bernoulli’s equation.
But these equations do not address the reality of fluid flow which is strongly affected by (among
other things) the intermolecular forces present in the fluid. Bernoulli’s equation, for example, makes no
mention of energy losses to heat or compressibility.
We now allow each layer to interact with its neighbor. We will still assume laminar flow, but
Figure 21: Laminar flow without viscosity
now each layer can interact with its neighbors. Figure 21 shows the velocity profile expected for laminar
flow without viscosity. The liquid flows like a plug down tube with no variation in the magnitudes of the
velocities as you examine them from the edge to the center and back again. If we allow for forces between
adjacent layers, the picture resembles more that depicted in figure 22. The layers nearest the wall of the
Figure 22: Laminar flow with viscosity
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pipe flow the slowest because the flow velocity of the wall is zero, and the flow velocity increases towards
the center of the tube or pipe. The situation is reminiscent of my star wars blowpop holder (Figure 23) in
Figure 23: Star Wars Blow Pop holder (retracted in inset)
this sense: When you shake the “sabre”, the blow-pop telescopes out. It has to move faster than the other
blue layers to accomplish this. The light sabre itself barely moves at all. The stem of the blow pop is like
the center concentric layer of the fluid, and the light sabre handle represents the pipe. The layers in between
are represented by the blue telescoping cylinders.
The reason this happens can be summed up as: Nature abhors a discontinuity. The idea that a
flowing stream of water would abruptly change to a motionless hunk of copper is unphysical. There has
got to be a gradual change with the microscopic layer of water nearest the pipe wall “traveling” at zero
velocity, with each subsequent microscopic layer traveling a little bit faster. From a macroscopic perspective,
the change in velocity may be very fast near the edges, or very slow and gradual in its increase as the
observation point is moved towards the center of the pipe. But, fast or slow, it is gradual.
This phenomenon, of a stationary boundary layer, is quite common. It is one reason why fan blades
get dusty (and, more important, stay dusty) and it is why you can’t seem to blow that last layer of dust off
of your coffee table. It is also why wind speeds at ground level are slow compared to speeds just a few
stories up. Students who live in high rise dorms often report hearing howling winds outside their upper
story window, only to find, to their surprise, relatively moderate winds at the surface. Another example,
closer to the behavior of water in a pipe, is the velocity profile of water in a river. It is often the case that a
river that is roaring with rapids near the center, or that has a strong current, seems barely to be moving near
its bank.
Once we allow for relative motion of the layers of fluid and for forces between the layers, a
necessary conclusion is that the layers of fluid interact with each other. For example, the layers near the
surface of the pipe, the ones that travel at the slowest speed, slow down the layers that are adjacent to them,
which in turn slow down the lasers adjacent to them, and so on, and so on and hence these interactions slow
down the velocity of the fluid as a whole. They act as a kind of internal friction, a brake on the flow of the
fluid and are collectively referred to as viscosity.
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The viscosity then is the resistance to flow of a fluid. It varies from fluid to fluid, and it varies as a
function of temperature. In addition, its behavior relative to gases differs in some important ways from its
behavior relative to liquids.
In liquids, viscosity varies strongly as a function of the strength of the intermolecular forces.
Hydrogen bonded water has much more viscosity than does gasoline, whose molecules interact via much
weaker dispersion forces. If we compare the first few liquid alkanes with their corresponding primary
Figure 24: Viscosities of liquid alcohols and alkanes
alcohols (figure 24), we see a gradual increase as we go to higher and higher molecular weights. This
supports the idea of greater intermolecular forces as the molecular size (and polarizibility) increases. But
the vertical distance between the 2 curves reflects the stronger intermolecular forces between alcohol
molecules than between alkanes, the reason being that these molecules exhibit hydrogen bonding due to
their terminal -OH groups. The values for these viscosities can be found in Table 1 (units are mPa•s).
One interesting and related point about viscosity is that plastics manufacturers control the viscosity
and pliability of their product by varying the polymer chain length. Longer chains give more viscosity and
stiffness. The currently accepted explanation is that the longer chains are more likely to become entangled,
further adding to the interactions between adjacent molecules.
While the intermolecular force explanation we have given for the variations seen ion Figure 27 is
plausible, perhaps it isn’t the entire story. As we add carbons to the alcohol chains, are we increasing the
polarity of the molecule? If not, how do we explain that the curve for the alcohols is running away from that
for the alkanes? The dispersion forces increase for the alcohols as we increase their chain length, but
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
# Carbons
Viscosity (mPa•s)
B.P.
-27-
Viscosity (mPa•s) B.P.
Alkanes
Alcohols
5
0.224
36.06
3.619
137.98
6
0.300
68.73
4.578
157.6
7
0.387
98.4
5.810
176.45
8
0.508
125.67
7.288
195.16
8*
0.474
110
9
0.665
150.82
9.123
213.37
10
0.838
174.15
10.9
231.1
* 2,2,4,-trimethylpentane; Densities, Viscosities, and Refractive Indices for Binary and Ternary
Mixtures of Ethanol, 2-Methylpropan-2-ol, and 2,2,4-Trimethylpentane by Chen-Chieh Wang,
Hui-Wen Chen, and Chein-Hsiun Tu, J. Chem. Eng. Data, 2005, 50 (5), pp 1687–1693, Publication Date
(Web): July 2, 2005 (Article); all others straight chain compounds with data from Handbook of Chemistry
and Physics, 88th Edition, CRC Press, (2007), Ed., David R. Lide.
Table 1: Viscosities and Boiling Points for Alkanes and Alcohols
Figure 28
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they also do so for the alkanes. We can get some further insight into the situation by looking at a plot of
boiling point vs. carbon chain length (Figure 28). Note how the BP’s for the hydrocarbons seem to be
catching up to those for the primary alcohols. In fact, by the time you get up to Eicosane (616ºC) and
Eicosanol (629ºC), the BP’s are only 13 degrees apart.
This at least suggests that there may be something else working to drive up the viscosities of the
alcohols, and the plastics data suggests that this “something” might be the increasing chain length of the
molecules. If that is the case, however, isn’t the chain length also increasing for the alkanes? So if it seems
sometimes in the sciences, that there is always a question and always some contradictory fact or idea that
destroys your idea before it is born, that is because there is. It’s the nature of science, and, for those who
get comfortable with it, science becomes enjoyable.
In fact, there are other instances in which molecular shape really is the only reasonable explanation.
molecular shape can also contribute. Although 2,2,4-trimethylpentane and n-octane are both 8-carbon
alkanes, their viscosities are not the same. In figure 29, we have photographed models of each of these
compounds. If you carefully count the number of black balls (representing carbon atoms), and white balls
(representing hydrogen atoms), you will see that each compound has 8 carbons and 18 hydrogens--what’s
Figure 29: Two isomers of octane
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the difference? It turns out that the shape of the molecules are different because of the different ways the
atoms are attached to each other. The top one in figure 29, which is stringier, is called octane. The bottom
one which is more curled up and blobbier is called 2,2,4-trimethylpentane. These molecules, because they
differ only in the arrangement of the atoms are said to be isomers of one another. The n-octane, longer and
stringier as it is, is thought to more readily attract other n-octanes, than can the bottom one attract its own
bothers and sister. Consequently the intermolecular forces are less between molecules of 2,2,4trimethylpentane because the surface area for attraction is less...less than the corresponding surface area
presented by n-octane. This is why n-octane has a higher coefficient of viscosity (0.508 mPa•s) than its
isomeric mate (0.474 mPa•s).
Let’s look at one more bit of data and try to clean up the final point we want to make about
viscosity. When we look at the temperature variation of the coefficient of viscosity, we almost always see
Figure 30: Viscosity vs. Temperature for n-octane
an inverse relationship such as that depicted in Figure 30, which displays actual experimental data for noctane, one of the components of gasoline. But it’s true for almost any fluid you can imagine: the higher
the temperature, the smaller the viscosity. Think of all the things you consume or use that reflect display
this behavior: You have to heat your hot fudge in hot water in order to have it gracefully flow over your
Cherries Garcia ice cream; You have to keep that dreadful hot cheese dip in the cafeteria warm or else it will
acquire the consistency of pudding; the motor oil in your car “thins out” as your engine warms up; honey
that pours slowly onto a hot biscuit, becomes runny as it absorbs the biscuit’s warmth; lava ejected from a
volcano can flow rapidly down the slope of the cone; that left-over turkey gravy in the refrigerator flows
like hot fudge over your potatoes on being heated; it’s simply not possible to think rationally about the
coefficient of viscosity without getting hungry. The Greek letter we use for it is “eta”, written as η, and
pronounced as in “I eta slice of apple pie, and it was so warm that the juices ran out onto the plate because
of their low value of eta”...well...you get the idea.
Gas viscosity varies with temperature in a manner that is the complete opposite of liquid viscosity.
At ordinary pressure, gas molecules are very far apart (relative to their own size) most of the time and so
they rarely interact attractively. However, because of their random thermal motion, they can wander from
layer to layer in our laminar model, and, when they collide with a molecule in an adjacent layer, they affect
(i.e. impede its motion. Since the collision frequency between molecules increases as a function of speed,
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and higher temperatures increase speed, higher temperatures increase viscosity in a gas. Why not in a
liquid? Well, in fact temperature does make the molecules move faster, but this effect is tiny because the
collision frequency is already so high due to the proximity of the molecules, and because the attractive
forces dwarf the tiny contribution to viscosity from layer crossing. This means that the decrease in viscosity
that comes from breaking away from those intermolecular forces overwhelms the tiny increase one might
observe from collisions.
Poiseuille’s Equation
Jean Leonard Marie Poiseuille (1797-1869), was a Parisian who studied the flow of blood in the
human body. He discovered the physical relations between blood flow in a tube, and the length and
diameter of that tube as well as the viscosity of the fluid. The equation that bears his name contains these
relationships. It assumes laminar viscous flow along a pipe of length l and radius r from a region of higher
pressure, P1, to one of lower pressure, P2. The fluid has a viscosity coefficient, η, and the system is
Figure 31: Fluid in tube following Poiseuille’s Equation
depicted in figure 31. Poiseuille’s equation summarizes the influence the four physical quantities that
dictate the volume flow rate of an incompressible fluid in a tube: radius, length, viscosity of the fluid, and
pressure difference. It is written down in [36], with a reminder, [37], of our use of the Greek Letter, eta.
[36]
[37]
πr 4 (P1 − P2 )
Q=
8ηl
η = Coefficient of Viscosity
We also reiterate equation [1], in which we defined the volume flow rate:
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
[1]
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ΔV
Q=
Δt
Although the equation may look complicated at first, we can work through this complexity by
looking at each variable in turn
"Q" , as noted above, is the volume flow rate, and represents the rate (usually in liters per second,
but sometimes in some other unit) at which some fluid flows past some boundary in the tube.
.
"Eta", as we have seen, is the viscosity coefficient, and it appears in the denominator of the right
hand side (RHS) of the equation. SInce it appears in the denominator, the equation predicts that, when the
viscosity increases, the flow will slow down. This is precisely what happens. For example, more viscous
honey flows slower than less viscous water.
The length of the tube, "l", is also in the denominator. Again, all other things being equal, gases or
liquids will flow more slowly through longer tubes.
The pressure difference between the two ends of the tube appears in the numerator. This supports
the common observation that, if you want to blow more air through a straw, you must blow harder. Put in
the language of the physical sciences, we would say that a higher pressure at the entrance to the straw will
result in a greater volume flow rate through the straw.
Finally, the volume flow rate is most dramatically affected by the radius of the tube. The radius
appears in the equation raised to the fourth power. This means that if we cut down the radius of the tube
(for example, by squeezing it or introducing a blockage) by a factor of 2, the flow rate will decrease by a
factor of 16. You may have observed this effect while watering the lawn. A small kink in the garden hose
can result in the reduction of the flow out of the business end of the hose from a torrent to a small drip. In
fact, we use this concept in the design of water faucets, where the rate of flow of the water is controlled by a
variable width channel, which is in turn controlled by rotating the faucet handle.
Again repeating something we discussed quite a bit at the beginning of this chapter, it is important
that you not confuse the term "flow velocity" (used in the continuity principle) with "volume flow rate". A
fluid flowing in a tube can have , for example, a small volume flow rate (only a small amount of the fluid
flows through the tube each second) , but exit the tube with a high velocity (the small volume of fluid travels
rapidly). You can get a handle on this by comparing in your mind the flow of water in two different rivers: a
small stream with lots of rapids, and a wide river that flows smoothly (e.g., the Mississippi River). In the
small stream, the water flows very fast (flow velocity) as is evidenced by how fast a kayak would travel on
it. The Mississippi, on the other hand, delivers far more water to the ocean per minute (volume flow rate)
than does the white water stream. But a kayaker on the Mississippi could eat lunch as he or she slowly
drifted down the river, due to the slow linear flow velocity of the water.
Poiseuille was interested in blood circulation during his career as a Doctor, and his equation, though
approximate, very insightfully analyzes several important circulatory pathologies. The most important term
on the equation for these purposes is the term in r4. Part of the reason is that exponent which insures that
small changes in r will induce large changes in Q . Just as in a garden hose, increasing r by a factor of 2
will result in a decrease in flow of a factor of 16. This might or might not be what you want in an artery. If,
for example, your body was constricting an artery in order to limit blood flow to the peripheral arteries in
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order to conserve body heat on a cold night. In this case, narrowing of an artery is a healthy response to an
external stimulus. You might try to restrict flow in an artery in order to take someone’s blood pressure
(more on this below). On the other hand, it might turn out that one of your coronary arteries was being
narrowed by the deposition of arterial plaque. At some point in the deterioration of the inside of this artery,
your heart muscle will begin to demand more blood. The only way this can happen is if your heart beats
more forcefully, and, by this time, you are on the downward slope of a very harmful feedback loop.
Poiseuille’s equation does not allow for elasticity in the arterial wall, and so an artery with a modest
blockage can still open to permit the bolus of blood when the left ventricle contracts. However, again
unfortunately, arterial plaque tends to stiffen and calcify, diminishing and sometimes eliminating the safety
factor that is afforded by the flexibility of the arterial walls.
In coronary bypass surgery, a tube, usually fashioned from a vein in the leg, shunts the blood
around the narrowed portion of the artery, in effect bringing "r" in Poiseuille's equation back up to normal
values. In balloon catheterization, the arterial plaque is pushed up against the artery wall by an inflatable
balloon, again increasing "r" to a livable value (one in which "Q" is large enough to properly oxygenate the
heart muscle). This can be reinforced by inserting a metal mesh stent in the artery. Medication must be
taken to guard against further deposition of tissue on the stent.
Generally η and l do not significantly vary, although there are diseases of the circulatory system
which can alter the value of η adversely. These mostly involve the overproduction of red blood cells.
Turbulence and Measuring Blood Pressure
Some rivers flow smoothly and others flow erratically, with rapids and whirlpools. This contrast
illustrates the difference between smooth laminar flow and turbulent flow in fluids. These two flow regimes
can also be observed in fluids flowing through pipes and tubes. The single most important factor in
determining whether flow will be smooth or turbulent is the linear flow velocity of the fluid. The faster the
flow velocity, the more likely there will be turbulent flow.
The quantity that governs the transition from smooth to turbulent flow is called the Reynolds
number. The Reynolds number, ℜ, for fluid flowing in a tube, is defined in terms of the coefficient of
viscosity, η, the density, ρ, the average fluid velocity, v , and the radius of the tube, r, as follows:
[38]
2v ρr
ℜ=
η
If ℜ > 2000, the flow in the tube is turbulent, but, if it is less than 2000, the flow is smooth and
laminar. These observations lead us to a practical strategy: if, for whatever reason, we want to induce
turbulent flow, all we need to do is speed up the flow velocity. We already know a way to accomplish this
for liquids: the insight is provided by the continuity principle. If we constrict a tube enough, we can get the
flow velocity high enough that turbulent flow will result.
"So what?", I hear you ask. Here's what: whenever you measure blood pressure, you are
constricting your patients arterial blood flow by pressing on an artery. In so doing, you are inducing
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turbulent flow. You can hear the turbulence if you listen carefully with a stethoscope, and the sound of the
turbulent flow is the marker you use to evaluate the value of the blood pressure. The continuity principle
and the transition from smooth to turbulent flow (Reynolds number) are the concepts that underlie our
ability to measure blood pressure in human beings.
In measuring blood pressure, an inflatable cuff, connected to a pressure gauge, is wrapped around the
patient's arm. The clinician listens with a stethoscope for the sounds of flowing blood. Air is pumped into
the cuff until flow through the brachial artery is shut down, which corresponds to a complete disappearance
of a pulse in the brachial artery. Pressure in the cuff is then gradually reduced until the pulse reappears.
The pressure at which the pulse reappears is called the systolic pressure. The clinician continues to deflate
the cuff and listen with the stethoscope. As the pressure falls, a variety of swishing and blowing sounds are
generally heard. These sounds are called Korotkov sounds, after the Russian surgeon, Nikolai Korotkov,
who first developed this technique while serving in the russian army in 1905. The pressure at which the
Korotkov sounds disappear is called the diastolic blood pressure.
Figure 32 shows the time dependent variation of arterial blood pressure, and it illustrates how it
happens that the technique used to measure blood pressure gives physiologically meaningful data about the
patient:
Figure 32: Arterial Blood Pressure as a function of time
As noted above, the first step in measuring blood pressure is to raise the pressure in the cuff above the
point where no pulse is detected in the brachial artery. The cuff pressure is then lowered, and, at the systolic
pressure, the pulse reappears with a characteristic thump. This corresponds to falling below point "A" in the
graph. If the cuff is pressurized above "A", the artery can't open; below it, the artery can. Thus the
reappearance of the brachial pulse corresponds to the highest pressure in the heart.
As the pressure in the cuff falls through the systolic value (region "B"), it sometimes lies above the
pressure in the artery and sometimes below it, depending on where in the pulse cycle you are. Thus the
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artery is opening and closing. It is clear that, in the process of opening and closing, there must be at least
two regions in the heart cycle where the artery is still open but very narrow. At these places, the continuity
principle says the flow velocity of the blood will be quite large, large enough, it turns out, to exceed the
critical value of the Reynolds number for turbulent flow to kick in. It is this turbulent flow that we hear
through the stethoscope as the so-called Korotkov sounds. These sounds will stop when the pressure in the
cuff gets so low that the artery is open all the time (point C). For this reason, the disappearance of the
Korotkov sounds is taken to be the diastolic pressure, the lowest pressure in the artery during the resting
phase of the heart.
You might already know that there is controversy about the precise method to be used to evaluate
the diastolic limit. Even though the artery is always open at pressures below "C", it can still be narrowed,
and there may still be some turbulent sounds. Near the end of the blood pressure measurement, the
korotkov sounds become muffled and the controversy centers around whether to wait for the complete
cessation of these sounds, or to measure the pressure where the muffling is first heard. In the US, most
clinicians wait for the complete disappearance of the muffled sounds, whereas, in the UK, the onset of the
muffled sounds is employed. Fortunately, the separation between the two points is less than 10 mm Hg for
almost all patients, and, in many patients, the onset and disappearance of these muffled korotkov sounds is
very nearly simultaneous. It should be noted, however, that, in some subjects, the muffled Korotkov sounds
persist all the way down to zero pressure.
One other note: that we are able to read blood pressure at all is another little physical miracle. The
pressure increase in the cuff is transmitted both to the Brachial artery and to the pressure gauge at the end
of the tube attached to the cuff. That this pressure increase (or decrease) is felt everywhere in the fluids
enclosed in this system is what is required for the whole procedure to work. Pascal’s Principle in action!
We have all heard that typical normal blood pressure is 120/80, and we chose these limits for figure
32. When the heart beats, the blood pressure in the major arteries is at its highest, 120 torr. This is called
the systolic pressure. When the heart rests between beats, the arterial pressure falls to a lower value,
roughly 80 torr. This is called the diastolic pressure. Recently the World health Organization (WHO) has
designated 120/80 as a ceiling. Blood pressure above these values is considered to be a precursor to high
blood pressure (pre-hypertensive). Hypertension, or high blood pressure is currently diagnosed at
pressures above 140/90.
The little blip in the curve in Figure 32, just after the heartbeat, is called the dicrrotic notch. This is a
result of the closing of the aortic valve at the exit of the left ventricle. This closure occurs when the BP
inside the left ventricle falls below that of the aorta, and prevents back flow of blood into the emptying
heart.
Because blood flow follows (roughly) the laws governing viscous flow (discussed earlier in this
section), the pressure varies around the circulatory system,being highest in the arteries and lowest where the
veins connect to the atrium. In addition, we need the recall that there are really two principal circulatory
systems in the body. The pulmonary system is powered by the right heart and circulates blood to the lungs.
The systemic system is "everything else" and circulates blood to the organs and extremities of the body.
Typical ranges of values found in healthy individuals are shown in tables 2 and 3 below: It should
be noted that these are values for a reclining person, when pressure is measured at the level of the heart. If
you are sitting or standing, the pressure can be greater or less than these values depending on whether the
region under investigation is higher or lower than the heart.
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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Pulmonary Blood Pressures (Systolic)
Where
Typical Pressure in torr
Superior Vena Cava
<10 torr
Right Atrium
<10 torr
Right Ventricle
15 to 30
Pulmonary Artery
15 to 30
Arterioles
10 to 20
Capillaries
10 to 15
Left Atrium
5
Table 2
Systemic Blood Pressures (Systolic)
Where
Typical Pressure in torr
Left Atrium
5
Left Ventricle
120 torr
Aorta
120 torr
Arterioles
50 to 100
Capillaries
10 to 30
Veins
5 to 15
Superior Vena Cava
< 10 torr
Table 3
It should be obvious why it is necessary to measure the blood pressure at a standardized height (the level of
the heart). The reason for this is that hydrostatic pressure increases the deeper you are in a fluid according
to the now familiar equation:
Introductory Physics © James J DeHaven, Ph.D., 2010-2011, Chap 14B
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P − P0 = ρgh
[35]
back into the picture and we don't want him there,deceptively adding (or subtracting) from the actual
physiological blood pressure. "Normal" blood pressure is 120/80 or so they say. But it is essential that it
be measured at an anatomically standard level: at the same height as the heart. As I write this, i have just
measured my blood pressure with one of those wrist cuffs you can buy at the drugstore. I have measured it
(sitting down) with my wrist in three different positions:
Hand raised
Wrist at heart level
Hand dangling between my legs
Here are the results:
My Blood Pressure (10/5/2004)
Where
BP
Hand Raised
102/58
Wrist at Heart Level
122/85
Hand Between Legs
173/127
Table 4: My Blood Pressure
Don't worry about the slight elevation of the center value. BP is very variable, and my wrist device always
measures 5-10 points above the doctor--I have checked it at his office several times. If my BP were
measured with my hand between my legs (as two medical staff have insisted in some recent office visits), I
would be diagnosed incorrectly with high blood pressure.
The variation is explained (as we noted above) by the equation for pressure as a function of depth in a fluid
[35]. The calculation is easy and the result on the mark.
In the circulatory system, the arterial pressure has two components: the pressure provided by the pumping
action of the heart and the hydrostatic pressure due to the height of the column of blood above the position
where the pressure is measured. The lower (i.e., below the heart) that the pressure is taken, the greater the
hydrostatic contribution.
Low hydration, for example,after surgery, can further lower the blood pressure at the level of the head and
cause dizziness in standing patients.