P h ys ic s E ducat ion 49 (4) 436

D Ivanov et al

Testing Bernoulli’s law

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© 2014 IOP Publishing Ltd

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Phys. educ.

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10.1088/0031-9120/49/4/436

Physics education

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P a P e r s

1. IntroductionBernoulli’s law is one of the important equa-tions of fluid dynamics [1, 2]. It is often used to qualitatively explain many phenomena in hydro- and aerodynamics and also allows for some very impressive demonstrations. It is, however, rarely considered in an exact numerical way, it is mostly treated qualitatively [3–6]. We provide some new ways to test it numerically that could be used as early as high school.

Bernoulli’s law is usually tested with a tube that has two different cross-sections and a liquid (usually water) is run through it. The difference in static pressures is measured with a differen-tial U-shaped mercury manometer. This method is inappropriate for high school use for two rea-sons. Firstly, the setup is relatively complicated. Secondly, the use of mercury in high schools is unacceptable. Because of that we propose a few new methods for testing Bernoulli’s law that are much more appropriate for high schools.

2. First methodFigure 1 depicts a laminar water jet flowing out of a vertical cylindrical tube with an internal diameter d0. Let us apply Bernoulli’s law for two

cross-sections of the jet; one of them with a diam-eter d0 coincides with the end of the tube and the other with a diameter d is at a distance H from the end of the tube. We have

ρ ρ ρ+ + = +pv

gH pv

2 2,at at

02 2

where pat is atmospheric pressure and ρ is water density. From this we obtain

+ =v gH v202 2

or

= +⎛⎝⎜

⎞⎠⎟v

v

gH

v1

2.

0

2

02

(1)

According to the continuity equation for an incompressible fluid

π π=d

vd

v4 4

,02

0

2

where v0 and v are the velocities of the jet at the cross-sections d0 and d, respectively. From this we obtain:

⎜ ⎟⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠=v

v

d

d.

0

20

4

Testing Bernoulli’s lawDragia Ivanov, Stefan Nikolov and Hristina Petrova

Faculty of Physics, Plovdiv University ‘P Hilendarski’, 24 Tsar Asen str, 4000 Plovdiv, Bulgaria

E-mail: stnikolov@uni-plovdiv.bg

AbstractIn this paper we present three different methods for testing Bernoulli’s law that are different from the standard ‘tube with varying cross-section’. They are all applicable to high-school level physics education, with varying levels of theoretical and experimental complexity, depending on students’ skills, and may even be useful for university-level educators. Our experiments are well-suited for use in project-based learning, either individually or as a single large project revolving around Bernoulli’s law. We also present examples of data we have collected with our experimental setups.

S Online supplementary data available from stacks.iop.org/PED/49/436/mmedia

JB

0031-9120/14/040436+7$33.00 © 2014 IOP Publishing Ltd

Testing Bernoulli’s law

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We put this into (1) and obtain

= +d

d

gH

v1

2.0

02

4 (2)

This result is derived from Bernoulli’s law and testing it experimentally would be testing the law itself. In order to do this we need to know the diameters d and d0, the distance H and the initial velocity of the water jet v0.

To determine the initial jet velocity v0 we need to measure the volume of water V that flows out of the tube in time t. Considering that π=S d / 40 0

2 is the cross-section of the tube, we get

π= =v

V

S t

V

d t

4.0

0 02

During the experiment we need to keep the jet velocity v0 constant. There are a number of devices that can be used to achieve constant flow; we used a version of Mariotte’s bottle. The general appearance of the experimental setup is shown in figure 2. A rubber hose is attached to the constant-flow device and the other end of it is attached to the tube the water will flow out of. Its diameter d0 is pre-measured with a calliper. The tube itself is mounted on a stand and right alongside it,

possibly on the same stand, is attached a gradu-ated ruler that we will use to measure H. It is most convenient if the zero mark of the ruler is placed at the exact same level as the end of the tube. A  smooth laminar flow is obtained through the constant-flow device.

To measure the flow rate and thus v0 we put a vessel underneath the jet, simultaneously starting a stopwatch. After a certain time t (usu-ally predetermined) we stop the stopwatch and quickly withdraw the vessel. We then measure the volume V of the collected water with a meas-uring cylinder.

The hardest part of the experiment is measur-ing the diameter of the water jet at different heights under the tube. This can be done photographically. We need to take a good close-up photograph of the water jet with the ruler alongside it. From this we can actually measure the diameters. A proper scale can be established by either using the measuring scale already in the picture along-side the water jet or by using d0 which we have already measured very precisely. This, however, is not really needed since we only need the proper value of d0 to calculate v0—all other cross-section diameters are only needed in ratios to d0 and these are the same whatever the unit of measurement

Figure 2. General setup of the first method.Figure 1. Water jet flowing from a tube.

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438 P h ys ic s E ducat ion July 2014

(so long as it is the same for both diameters). We thus do not need to perform the cumbersome scal-ing calculations. The measuring of the diameter itself can be done in several ways. The picture can be printed (as large as possible) and measure-ments can be taken from it. It may be displayed on a computer screen close-up and again measured from the screen with a ruler. We actually used some image-editing software with a built-in ruler and took the positions of the leftmost and right-most points of the jet at the different positions and subtracted them to get the diameter itself.

It is possible to take the picture with some graph paper as the background to use for the meas-urements. We did encounter some problems with that—it is hard to get both the jet and the paper into proper focus; the jet will distort the graph behind it through refraction, while the jet itself in its lower end thins out to about a millimetre so the graph paper may not be sufficiently precise for measurements. It is still a viable possibility

that can be sufficient in some circumstances and it makes the measurements much simpler.

Equation (2) can be tested in several ways. We can calculate the ratio of the diameters and the fourth root separately and compare them. A proper numerical test can be achieved by calculating the ratio of the two sides of (2)—it should be close to 1. Another way to enumer-ate the comparison is to calculate the differ-ence of the two values and divide it by their average—this gives us a dimensionless relative value of the difference between them which can also be expressed as a percentage and should be close to zero. If the experiment is carried out carefully, as we show below, it is possible to achieve a comparison precision to within five per cent.

Another way to test (2) is by rearranging it as

− =⎜ ⎟⎛⎝

⎞⎠

d

d

g

vH1

2.0

4

02 (3)

Figure 3. Water jet 1, =d 8.4 mm0 , =t 90 s, = ⋅ −V 922 10 m6 3, = −v 0.185 ms0

1.Figure 4. Water jet 2, =d 14.1 mm0 , =t 40 s,

= ⋅ −V 916 10 m6 3, = −v 0.147 ms01.

Testing Bernoulli’s law

439P h ys ic s E ducat ionJuly 2014

If we now take the expression on the left and plot it on a y-axis and put H on the x-axis we should be able to obtain a nice straight line. We can even use linear regression to calculate the coefficient g v2 / 0

2 and compare it to a preliminary calcula-tion. This technique can also be used when we have not measured v0, meaning we do not even have to measure d0, given that its absolute value is only needed for calculating that velocity—for everything else we use ratios. In practice we always obtained excellent linearity (R-squared was consistently over 0.95, sometimes over 0.99). The coefficient, however, was frequently quite different from the measured one, even in cases where the first approach gave us differences of 5–6%. This can be explained from the formulae themselves. In (2) one of the major sources of experimental erro—v0

2—is under a fourth root, which greatly diminishes its influence, whereas in (3) the other major source of experimental uncertainty, the diameters, are raised to the fourth power, significantly exacerbating any measure-ment error. Thus the linearity approach is quali-tative and less precise, but it is also simpler to measure (we do not need to know the flow rate) and allows an easy visual confirmation of the straight line and thus the formula.

We present here two of the pictures we used (figures 3 and 4) with the appropriate data sets (tables 1 and 2, figures 5 and 6) and comparisons. The full-scale pictures and spreadsheets used for data processing are in the supplementary material, available at stacks.iop.org/PED/49/436/mmedia.

In classroom practice the teacher may take some pictures in advance and record the

flow-rate data. In front of the students he can just demonstrate the experimental setup quali-tatively. Getting a good quality picture is not trivial—any light sources and windows will be reflected and refracted through the water jet, making it hard to take measurements from the photograph. It is best to place a matted white screen behind the experimental setup, between it and any light sources so that only diffuse light will illuminate the water jet. The pictures may also have to be taken with a tripod, using long exposure. In case more time is available, taking the pictures may be presented to the students as an interesting and not insignificant challenge. This experimental problem, including making the setup, carrying out the experiment, taking the pictures, collecting data and making calcu-lations, is very suitable for the purpose of pro-ject-based learning.

It is important to note that in the classic experiment with a tube of varying cross-section the water-flow suffers serious friction at the tube walls. This takes us away from the ideal model and makes experimental testing of the law more difficult. In our case, this problem is not present due to negligible air friction, and thus it takes us closer to the ideal situation. Furthermore, this water jet shape is observed daily from faucets, making this a very familiar phenomenon.

3. Second methodAnother accessible option for testing Bernoulli’s law is shown in figure 7. A vessel with a volume of several litres is placed on a stand on the labora-tory table. The bottom of the vessel is drilled and a hole is made with a diameter d of several milli-metres. A metal probe of calibrated diameter may be inserted and secured as watertight with instant glue to ensure a good round hole with an accurate size.

The hole is plugged (we can just use a finger) and the vessel is filled with water. Upon unplug-ging the hole, the water starts flowing out but the vessel is kept full by constantly refilling. This ensures a constant flow rate and a constant velo-city v of the outflowing water.

In this case Bernoulli’s law is written as

ρρ ρ+ + = +p

vgH p

v

2 2,at at

02 2

Figure 5. Linear regression for the data from table 1 (first and last column).The calculated coefficient is 574.1.

D Ivanov et al

440 P h ys ic s E ducat ion July 2014

where pat is atmospheric pressure, H is the height of the water column, ρ is the water density and v0 is the velocity at which the free surface of the water is receding. Since we keep the water level constant, =v 00 . Thus we obtain Torricelli’s for-mula for the velocity of the outflowing water

=v gH2 . (4)

We can determine this velocity experimentally. By placing a graduate vessel under the water jet and determining the time t that it takes to fill up a volume V . Since

π=Vd

vt4

2

we can obtain for the velocity

π=v

V

d t

4.

2 (5)

Comparing (4) and (5) we get

π=V

d tgH

42 .

2

(6)

This equation is subject to experimental con-firmation. In order to do this we need to deter-mine d, h, v and t. The precision of the result will mostly be influenced by the precision with which the diameter d of the opening is measured. This can be measured with a calliper during the course of the experiment or can be provided in advance as a constant of the equipment. This diameter should not be more than a few millimetres, so the water jet will not become turbulent while flowing out. The water level can be kept constant with a hose from the tap or by pouring water from another vessel. This refilling must be done from

Table 1. Data related to water jet 1, as depicted on figure 3. Diameter measurements are in arbitrary ‘photo units’. The data are presented graphically in figure 5.

H m[ ] dd

d0 + gH

v1

2

02

4 ratio deviation −d

d10

4⎜ ⎟⎛⎝

⎞⎠

0.00 55 1.00 1.00 1.000 −0.0% 0.000.01 35 1.57 1.61 0.975 −2.5% 5.100.02 30 1.83 1.88 0.975 −2.5% 10.300.03 27 2.04 2.07 0.986 −1.4% 16.220.04 25 2.20 2.21 0.994 −0.6% 22.430.05 24 2.29 2.34 0.982 −1.9% 26.580.06 23 2.39 2.44 0.980 −2.0% 31.700.07 22 2.50 2.53 0.987 −1.3% 38.060.08 21 2.62 2.62 1.001 0.1% 46.050.09 20.5 2.68 2.69 0.996 −0.4% 50.810.10 20 2.75 2.77 0.995 −0.5% 56.190.11 19.5 2.82 2.83 0.997 −0.3% 62.290.12 19 2.90 2.89 1.001 0.1% 69.22

Figure 6. Linear regression for the data from table 2 (first and last column).The calculated coefficient is 912.2.

Table 2. Data related to water jet 2, as depicted on figure 4. Diameter measurements are in arbitrary ‘photo units’. The data are presented graphically in figure 6.

H m[ ] dd

d0

+ gH

v1

2

02

4 ratio deviation ⎜ ⎟⎛⎝

⎞⎠ −d

d10

4

0.00 91 1.00 1.00 1.000 0.0% 0.000.01 53 1.72 1.78 0.963 −3.8% 7.690.02 44.5 2.05 2.09 0.976 −2.4% 16.490.03 39.5 2.30 2.31 0.998 −0.2% 27.170.04 35.5 2.56 2.47 1.036 3.5% 42.180.05 34 2.68 2.61 1.024 2.4% 50.320.06 33 2.76 2.73 1.009 0.9% 56.820.07 32 2.84 2.84 1.002 0.2% 64.400.08 30.5 2.98 2.93 1.017 1.7% 78.240.09 29.5 3.09 3.02 1.022 2.1% 89.550.10 29 3.14 3.10 1.013 1.3% 95.960.11 28 3.25 3.17 1.024 2.4% 110.570.12 27 3.37 3.24 1.040 3.9% 128.04

Testing Bernoulli’s law

4 41P h ys ic s E ducat ionJuly 2014

a small height and for a more accurate experi-ment the vessel may even be allowed to gently overflow.

The data from one of our experiments is:

= ⋅ = ⋅ = =− − −H d V t24.2 10 m, 5 10 m, 10 m , 23.5 s2 3 3 3

Plugging these into (6) shows it holds with excel-lent precision.

This experiment is quite accessible for high school, both experimentally and theoretically.

4. Third methodWe have a vertical cylindrical vessel with diame-ter D and height H filled with water (figure 8) and a small hole with diameter d drilled in its bottom. Let us determine the time T it will take for all the water to flow out.

Let us assume that at a certain moment the water level is at a height h relative to the ves-sel bottom. We will now consider a thin water layer with a thickness hd . Due to the thinness of the layer we can consider that it will flow

out with constant velocity v0 in a time td . The speed of the decrease of the water level is v and we have:

= −h v td d . (7)

According to Bernoulli’s law we have

ρ ρ ρ+ + = +pv

gh pv

2 2,at at

202

where pat is atmospheric pressure and ρ is water density. This gives us

+ =v gh v2 .202 (8)

From the continuity equation for an incompress-ible fluid we have

π π= = =⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠v

Dv

dv v

D

dv v

D

d4 4, , .

2

0

2

0

2

02 2

4

Substituting this into (8) we ultimately obtain

=−( )

vgh2

1.

D

d

4 (9)

Figure 8. General setup of the third method.Figure 7. General setup of the second method.

D Ivanov et al

442 P h ys ic s E ducat ion July 2014

Putting (9) into (7) we arrive at

− =−( )

hgh

td2

1d .

D

d

4

We separate the variables and integrate

∫ ∫−−

=( )

g

h

ht

1

2

dd .

D

d

H

T4 0

0

Finally, for the total time that it will take the liq-uid to flow out we obtain

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

= −Tg

D

dH

21 .

4 (10)

If ≫D d this further simplifies to

= ⎜ ⎟⎛⎝

⎞⎠T

D

d

H

g

22

(11)

Expressions (10) and (11) were derived from Bernoulli’s law and testing them would equate to testing the law itself. Deriving those equations is not normally accessible to high-school students and they can be given to the student as final results. The experiment itself is elementary and can easily be done in class. Because of friction (viscosity) the real times will always be a little longer than calculated, but they are still very close.

The experiment can be performed with the vessel from the second method. The hole in the bottom is plugged and the vessel is filled. Upon unplugging it, the timer is started and it is stopped when the water stops flowing. This gives us the time T . After also measuring the sizes D, d and H the expressions (9) and (10) can be tested. During one experiment we meas-ured a time of =T 213 s. This was done with a vessel of sizes = ⋅ −H 237 10 m3 , = ⋅ −d 5 10 m3 ,

= ⋅ −D 154 10 m3 , which gives us a calculated time of 208.7 s according to (11) and 208.6 s according to (10), which is in excellent agree-ment with the measured time and shows the applicability of (11).

A similar experiment can be found elsewhere [7] with an algebraic derivation of (10) and with an irregularly shaped vessel (a soda bottle), which forces the authors to seek (and, to their credit, find) more complicated techniques for measuring the drain times.

5. ConclusionThe presented methods for testing Bernoulli’s law provide a wide range of opportunities, depending on teacher requirements, needs and capabilities. They can be adapted and used, all or just some of them, based on the mathematical and experimental skills of the students and the time and equipment available to the teacher. Our methods can be used as stand-alone experiments in the traditional laboratory pract-icum, but can also be used within the framework of the modern approach of project-based learning. We hope they are useful for many of our physics- education colleagues in helping them teach this interesting and widely applicable law of physics.

Received 1 October 2013, in final form 17 November 2013 accepted 20 November 2013doi:10.1088/0031-9120/49/4/436

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