Worksheet on water flow in a pipe
1. As we start this worksheet, it’s important to remind ourselves of the
differences between liquids (like water) and gases.
a. Compressibility:
i. Water is [choose one] more/equally/less compressible than air.
ii. Do you have everyday experiences of the compressibility of gas or
water? If so, what are they?
iii. Let’s put some numbers to compressibility. Consider two “blocks” of
water with the same number of molecules, one at the top of a lake at
atmospheric pressure (about 105Pa) and one 10 meters down in the
lake (about 2 x 105Pa). The percent change in volume is given by
the following relationship:
𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑉𝑜𝑙𝑢𝑚𝑒 =
Δ𝑉 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
=
𝑉
2 𝑥 109 𝑃𝑎
(the number 2 x 109Pa tells us how hard it is to compress). What is
the fractional change in the volume of water? Is this big enough to
notice by eye?
iv. Now let’s compare that to air. Imagine a “block” of air, one with P=
105Pa and the other with P= 2.0x105Pa. If the number of molecules
and the temperature is the same, what is the ratio of V1 to V2? Is this
difference big enough to notice by eye?
Dan Young and Dawn Meredith 11/2014
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v. Given your answers to the last problem, comment on whether
each of the following statements is true or false, and how you
know.
1. PV=nRT describes water as well as gases.
2. The volume of water really does not change much under
normal conditions, so we can consider it essentially
incompressible.
3. Water is more like wood than air as far as compressibility
goes – it can exert a large force (=pressure*area) without
looking like it’s doing anything.
2. Thinking about liquid motion in a pipe that has a changing cross section.
a. What happens to the velocity of the water when you put your finger
over the end of a water hose, closing off some of the nozzle?
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Worksheet on water flow in a pipe
b. Now we want to build on these experiences and make them
quantitative.
The sketch below shows a cross section of a pipe with varying cross
section. Assume that the pipe width (coming out of the page) is the
same all along the pipe, and is one block wide. Water totally fills the
pipe, and is moving to the right. (Idea adapted from Knight’s “Physics
for Scientists and Engineers”)
A
B
C
Six blocks of water pass by plane A in one second. (Shade in the blocks)
i. How many blocks must pass by plane B in one second assuming that
water is incompressible? (Shade in the blocks)
ii. How many blocks must pass by plane C in one second? (Shade in the
blocks)
iii. Explain your reasoning.
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Worksheet on water flow in a pipe
c. If the velocity of the fluid at plane A is 2cm/s, what is the velocity at
plane B? At plane C? Hint: use velocity = x/t and the information
on the previous page.
d. In describing fluid flow, the flow rate Q (volume/s = cm3/s) moving
past a plane is an important parameter of the system. What is the
flow rate past plane A if each block is 1 cm3? Calculate Q in two
ways – both as volume/s and cross sectional area*velocity.
e. What is the value of Q at plane B (calculated in two ways)?
f. What is the value of Q at plane C (calculated in two ways)?
g. Are your previous three answers consistent with the following
equation:
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝑄 = 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 = 𝐴𝑟𝑒𝑎 ∗ 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
If not, what evidence or experience says it should not be constant?
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Worksheet on water flow in a pipe
h. Consider water flowing at 3m/s in a large pipe. This large pipe splits into two
smaller pipes, each with half of the cross
sectional area of the first pipe.
b
In using the continuity equation to find
the velocity in b, should we consider the
cross sectional area in pipe b alone, or the
a
total cross sectional area in b and c, or
something else? Explain your reasoning
c
i.
What is the velocity in each of the smaller pipes?
j.
Consider water flowing at 3m/s in a large pipe. If this large pipe splits into eight
smaller pipes, each with one quarter of the cross sectional area of the first pipe,
what is the velocity in each of the smaller pipes? Explain how you know.
k. Let’s apply these ideas to blood flow. We want blood to flow fairly quickly in the
aorta over the long distance from your heart to the individual cells, but once the
blood is in the capillaries and needs to exchange nutrients, oxygen and carbon
dioxide with the cells, the blood needs to move slowly to allow the diffusion of
these molecules. Given this, is the total cross section of ALL the capillaries
greater than, about equal to, or less than the cross section of the aorta?
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Worksheet on water flow in a pipe
3. The last question focused on why the velocity must change if the pipe cross
section changes. In this question we will think about how the velocity
changes in a pipe of varying cross section. On the next page you will see a
sketch of the cross section of the pipe filled with water and flowing right, as
well as several sets of axes. For this simple situation, we will assume there is
no friction or viscosity in the pipe.
a. Pressure will be important to this discussion of how the velocity
changes, so let’s begin with a quick review of pressure.
i. Considering yourself submerged in a swimming pool, where
was the pressure exerted on you? (e.g. just on the top of your
head? Just on your sides? All over?)
ii. What object exerts this pressure?
b. Now let’s return to velocity. Using what you learned in the last
question about velocity, qualitatively sketch the velocity along the
pipe (see the axes ahead 2 pages).. (Don’t worry about numbers, just
note if velocity increases, decreases or stays the same.)
c. Using the definition of acceleration, and your velocity plots, sketch the
acceleration along the pipe. (Don’t worry about numbers, just note if
acceleration is positive, negative or zero.)
d. Is it pressure or pressure gradients that cause acceleration? Use the
example of water moving up a straw to justify your answer.
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Worksheet on water flow in a pipe
e. Given your last few answers, fill in the following table that asks you
about pressure gradient between the points labeled on the horizontal
axis.
Pressure gradient
(subscript refers to
position label)
P1-P2
Circle correct answer for sign of
pressure gradient
P4-P5
Positive/zero/negative
P5-P6
Positive/zero/negative
P6-P7
Positive/zero/negative
P7-P8
Positive/zero/negative
P9-P10
Positive/zero/negative
P10-P11
Positive/zero/negative
P11-P12
Positive/zero/negative
P13-P14
Positive/zero/negative
Positive/zero/negative
f. Use the information about pressure gradients in your table, and the
given initial P0 on the plot (the big black dot) to sketch the pressure in
the pipe. (Don’t worry about numbers, just note if pressure increases,
decreases or stays the same.)
g. Summarize your findings: Pressure is the largest in a pipe of moving
liquid when the cross sectional area is the [choose one]
largest/smallest.
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Worksheet on water flow in a pipe
Pressure
acceleration
velocity
Pipe
0
0
0
1
2
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3
4
5
6
7
8
9
10
11
12
13
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Worksheet on water flow in a pipe
4. In this last section, we want to check if the results in the previous section are
consistent with other ideas we have seen before.
a. Bernoulli’s equation (which you will see in lecture) given below
describes changes in pressure and velocity. (The version we give here
is accurate only if the pipe is horizontal.)
1
1
𝑃1 + 𝜌1 𝑣12 = 𝑃2 + 𝜌2 𝑣22
2
2
Here P is the pressure,  (rho) is the density of the fluid and v is the
velocity of the fluid. The subscripts one and two indicate these values
at two different locations in the pipe. In this question we will verify
that this equation agrees with what you already determined
qualitatively.
Given Bernoulli’s equation above, if v1 increases, what happens to P1
in order to keep the left side of the equation constant?
Does this agree with what you have on your sketches?
Note that if we rewrite Bernoulli’s equation, it says that the work done
by the pressure gradient changes the kinetic energy of the water. This
means that Bernoulli’s equation can be looked at from the energy
point of view as well as the forces point of view.
𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑏𝑦 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝑃1 − 𝑃2 =
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1
1
𝜌2 𝑣22 − 𝜌1 𝑣12
2
2
Worksheet on water flow in a pipe
a. Another student says, “Using PV=nRT, the pressure should be lower in
regions with larger volume, assuming that temperature and number of
molecules are constant, because the water has more room to spread out.”
Is this consistent with what you found? If not, what is the flaw in their
argument?
b. A fellow student says “The pressure must be larger in regions of smaller
pipe radius because pressure is force/area. Since force is the same, and
area is smaller, the pressure must be bigger.” Is this consistent with what
you found? If not, what is the flaw in their argument? Hint: can we
accurately compare the force exerted by two different blocks of water just
by looking at the water? (This one is tough! Spend no more than five
minutes here, and at least write down a few ideas even if you don’t feel
you have a complete answer.)
5. Putting it altogether. Look back over this worksheet and summarize the big
ideas you have been exploring. Write this summary for an imaginary classmate
who missed lab this week.
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Worksheet on water flow in a pipe