The Physics Teacher • Vol. 53, March 2015 (169 - 173)

Water Jets from Bottles, Buckets, Barrels,and Vases with Holes

Vjera LopacUniversity of Zagreb, Zagreb, Croatia

Observation of the water jets flowing from three equidistant holes on the side of a vertical cylindrical bottle is an interesting and widely used didactical experiment illustrating the laws of fluids in motion. In this paper we analyze theoretically and numerically the ranges of the stationary water jets flowing from various rotationally symmetric vessels with holes, in dependence on the height of the holes above the bottom, on thickness of the block supporting the vessel and on different shapes of the vessel profile. This investigation was motivated by controversial descriptions and illustrations repeatedly found in the physics textbooks and by the fact that previously in the physics teaching literature only the cylindrical vessel was treated.

The level of the fluid is supposed to be constant and the openings extremely small. In further text we will continue referring to water, but all conclusions will be applicable to any incompressible and inviscid fluid of density in the homogenous gravitational field g.

According to the Bernoulli equation, for a cylindrical vessel filled to the level H = const. and for the hole at the height h above the bottom, the velocity of efflux is given by Torricelli’s law

. (1)

The range of the water jet at the bottom level is

. (2)

The longest range has the jet coming from the opening in the middle of the bottle, h = H/2. The ranges of jets flowing from two holes placed at distances H/4 below and above the central hole are shorter, but equal to each other. However, in many books and papers erroneous illustrations can be found, ascribing the longest range to the lowest jet and using it as an experimental proof of the fact that the hydrostatic pressure is highest near the bottom of the bottle.

The error has been noticed and was thoroughly discussed in the didactical literature. In years before 1989, a number of papers containing the critical reviews1-5 of the erroneous results were published. They were focused on the meaning and significance of the correct result1,2,4-6. The series of notes4-8 was ended with the editor's conclusion: "The correspondence on water jets is now closed." However, publications with erroneous description of experiments continued to appear. Confusing illustrations could be found even in some globally popular books9,10. Several extensive reports followed11-13, in which

authors drew attention to the crucial role of the supporting block on which in most experiments the vessel was placed1,5,7,8. It was noticed that its height B strongly influences the range of the water jet13.

In Section 1 of this paper we evaluate two general equations. The first of them describes the shape of the parabolic jet, the second determines the jet range in dependence on the distance of the hole from the bottom and on thickness of the support, for various profiles of rotationally symmetric vessels. In Section 2 we apply these equations to the cylindrical bottle. In Section 3 several other vessel shapes are analyzed. In Section 4 results are summarized and discussed with respect to their didactical potential and historical context.

1. General equations for shapes and ranges of the water jets

We consider the water jets flowing from the orifice placed at the height h above the base level of the vessel. The vessel is filled with water up to the level H which is maintained fixed. The vessel stands on the cylindrical block of height B. The reference system for describing the jet equation is chosen so that the x-axis coincides with the horizontal base level of the support, and the y-axis with the vertical wall of the support (Fig. 1). The shape of the vessel is described by the function S(h) for . Several shapes will be considered, all depending on a positive parameter L, which denotes the difference between the largest and the smallest radius of the vessel. The tangent to the profile curve closes an angle with the vertical axis. One introduces a new shape parameter

. (3)

The water jet exits perpendicularly to the wall of the vessel. The angle can be positive or negative. If , the water flows downward. If < 0 it flows upward and the parabolic jet exhibits a typical maximum (Fig. 1).

Fig. 1. Meaning of variables h, S, , x, y, D, B, H and L.

The velocity v is given by Eq. (1) and its horizontal and vertical components are, respectively,

2

and .

(4)

Using the symbol t for time, the coordinates of a point on the parabola, obtained from Eq. (4), are

and .

(5)By combining Eqs. (4) and (5), one obtains the equation of the parabolic shape of the jet:

. (6)

With , one obtains the equation for the jet range in dependence on h:

. (7)

Generally this equation should be solved numerically. However, for smooth shapes investigated in this paper the range D can be obtained by solving Eq. (7) as a quadratic equation in and taking its larger root, which gives

. (8)

2. Cylindrical bottle with holes

Shapes and ranges of the water jets

Two carefully prepared experimental arrangements, one with the supporting block and the other without it, are shown in Figs. 2 and 3. For the cylindrical bottle , ,

and . The shape parameters are listed in row A of Table I. Equation (2) for B = 0 is obtained from Eq. (8). The calculated jets from five

holes with h/H = 0.01, 0.25, 0.50, 0.75 and 0.99 on the walls of a cylindrical bottle are shown in Fig. 4(a). Horizontal lines in the picture facilitate the comparison of water jet ranges for four different thicknesses B of the support. The calculated values of the jet range in dependence on h for different B are compared in Fig. 4(b). The lowest curve (for B = 0) is symmetrical and was already depicted in Refs. 1 and 2.

The situation where the range D increases with increasing depth H-h and where the lowest jet has the largest range is possible only if the bottle is placed on a support of sufficient thickness. For the thickness B of the supporting block the range of the water jet resulting from Eq. (8) is13

. (9)

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If D is not allowed to decrease when h decreases, there must be ; hence or . As this must be valid for any possible h (thus also for

hmin= 0), one concludes that the lowest jet has the largest range only if

, (10)

i.e., if the thickness B of the supporting block is equal to or larger than the distance H from the base level of the vessel to the free surface.

Table I. Profile equations for considered vessel shapes

Vessel shape Vessel profile Slope of the vessel profile

A Cylindrical bottle

= 0

B Bucket > 0

C Conical bottle <0

D Barrel with parabolic

profileE Sinusoidal

vaseF Corrugated

vase N > 1

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Fig. 2. Water jets from three holes on a cylindrical bottle placed on the supporting block.14

Fig. 3. Water jets from a bottle with three holes without the supporting block12. The longest range has the jet flowing from the hole in the middle of the bottle. (Set up and photo by Adrian Corona Cruz.)

Dependence of the jet range on pressure and energy

Since the pressure of the fluid depends on its density and increases with the increasing depth H–h of the hole, Eq. (2) for the range of the water jet can be rewritten as

, (11)

where . This shows that for small h the range increases, but for larger h decreases with pressure. The jet range can also be expressed by means of the kinetic energy and the gravitational potential energy of a small part of the fluid of mass m placed at the orifice at the height h above the bottom. Since

and, measured from the base level of the support, , Eq. (11) for the jet range can be written as

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. (12)

3. Water jet ranges for different vessel shapes

Bucket with holes

The bucket is a truncated conical vessel wider at the top. The experiment with the leaking bucket was suggested in Ref. 10, while Fig. 5. illustrates how easily such system can be realized in everyday life. The shape profile of the vessel is a tilted straight line, and the angle is constant and positive. Their dependence on h is shown in the row B of Table I. Parabolic jets calculated from Eq. (6) for different B, a = tan and L/H = 0.2, are shown in Fig. 6(a) and their ranges in dependence on h are shown in Fig. 6(b). Similar results and graphs hold for a conical bottle with holes.

Barrel with holes

Barrels are traditional vessels often seen in real life. An example is shown in Fig. 7. We consider here the barrel with a parabolic profile, specified in row D of Table I. Shapes of the parabolic jets and the range dependence on h for a barrel are shown in Figs. 8(a) and 8(b).

Vases with holes

A vase is a vessel with sinuous, in our case also a sinusoidal, profile. The profile and its slope are specified in the row E of Table I. The range curves for the vase exhibit oscillations that are obviously caused by the periodical nature of the shape profile. If the vessel profile is folded more than once, the shape and its slope can be generalized to equations given in row F of Table I, with values N > 1. Results are the corrugated vase shapes and somewhat surprising oscillating curves depicting the dependence of the jet range on h. In Figs. 9(a) and 9(b) the calculated results are shown for the corrugated vase with N = 8. If the vessel profile is folded more than once, the shape and its slope can be generalized to equations given in the row F of Table I, with values N > 1. Results are the corrugated vase shapes and somewhat surprising oscillating curves depicting the dependence of the jet range on h. In Figs. 9(a) and 9(b) the calculated results are shown for the corrugated vase with N = 8.

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Fig. 4. (a) Water jets from holes on the cylindrical bottle.

Fig. 4. (b) Dependence of the water jet range on h for the cylindrical bottle.

Fig. 5. Jets flowing from the bucket with holes.15

Fig. 6. (a) Water jets from holes on the walls of a bucket.

Fig. 6. (b) Dependence of the water jet range on h for the bucket.

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Fig. 7. Barrel with a hole in the middle: from it the water jet exits horizontally.17

Fig. 8 (a) Water jets from holes on the walls of a barrel.

Fig. 8. (b) Ranges of the water jets from holes on the walls of a barrel.

Fig. 9. (a) Water jets from holes on the corrugated vase with N = 8

Fig. 9.(b)Dependence of the water jet range on h for the corrugated vase with N = 8

4. Discussion and conclusions

Discussion of the calculated results

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We have investigated theoretically and numerically the water jets flowing from vessels of various shapes with holes. Interesting results were obtained for different profile shapes and support thicknesses. Comparing the jet range curves for various shapes one finds the following.a) For the highest jet (h/H = 1) the range is D = 0 for all examined shapes, except for the conical bottle for which it is negative; actually, it would be negative if the jet would not be stopped by the vessel itself.b) The lowest jet (h = 0) has the range D = 0, provided that there is no supporting block below the vessel, again for all considered shapes except for the conical bottle, for which all jets, including the lowest one, flow upward.c) There are three categories of range curves: those decreasing with h, those with one maximum for a special value of h, and the complicated curves with many maxima and minima reflecting the shape oscillations of the vessel profile.d) The derivation can be extended to many other vessel shapes, including the vessels with rough or uneven walls suggested in Ref. 17, where also the effects of deterministic chaos could be expected.

Didactical possibilities of the obtained results

We have in mind several possible applications of the obtained results in the teaching process. a) The theoretical calculations of the water jet shapes and dependence of their ranges on h do not lie beyond the usual mathematical and programming skills of high school and college students. They could be used as attractive numerical tasks for the physics class. b) The shapes encountered in such investigation can be applied, in the sense of ideas proposed in Ref. 18, to connect the subjects of physics with various geometrical objects. One can consider different parabolic shapes of the jets and compare them with the paths observed in the projectile motion. Various vessel profiles can be presented as examples of interrelationship between the geometry and the motion of flows and particles. c) Another possible use is experimental: the vessels of proposed shapes, found in the environment or custom made, can be used to observe the ranges of the water jets and compare them with results obtained by calculation. Besides rotationally symmetrical vessels, which are appropriate for visualizing the shapes of the water jets, we propose a different type of vessel that would make possible the visualization of the range curves in dependence on h. It could be produced in the laboratory or ordered from specialized shops. For the vertical wall, such vessel would have the form of a rectangular block shown in Fig. 10. The jets exiting from a series of holes made along the diagonal of the longest wall would reach the bottom level at different distances. They could leave traces depicting the lowest curve shown in Fig. 4(b). The same type of vessel could be produced with different profiles S(h), and the support could be added when needed.d) Finally, the subject offers a large choice of historical issues. There is the story of Leonardo da Vinci, who first made drawings of the jets flowing from cylindrical containers with holes11,13– albeit erroneously – in the 15th century, 130 years before Torricelli. The importance of different shapes can also be emphasized by quoting Johannes Kepler, who spent a lot of his time calculating the volumes and shapes of barrels19.

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Acknowledgments

The author wishes to thank V. Volovšek for useful comments and J. Slisko for permission to use the photo from Ref. 12.

Fig. 10. The vessel shape proposed for the experimental investigation of the jet ranges.

References

1. L. Paldy, "The water can paradox", Phys. Teach. 1, 126 (Sept. 1963).2. R. Biser, "The water can explored again", Phys. Teach. 4, 304–305 (Oct. 1966).3. Göran Grimvall, "Questionable physics tricks for children", Phys. Teach. 25, 378–379 (Sept. 1987). 4. J. K. Atkin,."The great water-jet scandal", Phys. Educ. 23, 137–138 (1988).5. J. K. Atkin," ...And nothing but the truth!", Phys. Educ. 24, 67–68 (1989).6. K. Tamuli, "Liquid flow from orifices", Phys. Educ. 23, 190–191 (1988).7. J. H. Avison, "The last drop from the water jets", Phys. Educ. 24, 259 (1989).8. J. H. Avison, "The whole truth about water jets", Phys. Educ. 23, 265 (1988).9. Mae Freeman and Ira Freeman: Fun with science (Random House Inc., New York, 1943.), p. 21.10. P.G. Hewitt: Conceptual Physics, (Little, Brown and Company, Boston, 1986), p. 184.11. Donald E. Simanek, "Student misconceptions induced by teachers and textbooks", in Didaktikogenic Physics Misconceptions (2008);http://www.lhup.edu/~dsimanek/scenario/miscon.htm

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12. J.Slisko, "Repeated errors in physics textbooks: What do they say about the culture of teaching?", In Physics Community and Cooperation, Vol. 2, GIREP-EPEc&PHEC 2009 International Conference (University of Leicester, 2009), pp. 31–47;http://physics.le.ac.uk/girep2009/ConferenceProceedings/GIREP2009_ConferenceProceedings_Volume2.pdf13. Gorazd Planinšič, Christian Ucke and Laurence Viennot, "Holes in a bottle filled with water: Which water-jet has the largest range?", published by the MUSE group in the Physics Education Division (PED) of the European Physical Society (EPS) (February 2011);http://c.ymcdn.com/sites/www.eps.org/resource/collection/016775D4-8888-474D-887F-3E33AEA5E6D0/EPSPED_MUSE_bot_holes.pdf14. http://www.kids-fun-science.com/images/sa2-kids-science-activities.jpg15. http://commonscold.typepad.com/.a/6a00d8345280a669e201348053709a970c-popup.16. Photo: V. Lopac17. "Investigating the pressure of a water column“;. Nuffield Foundation, Practical

Physics, http://www.nuffieldfoundation.org/practical-physics/investigating-pressure-water-column18. Thomas B. Greenslade Jr., "The shapes of physics", Phys. Teach. 51, 524–534 (Dec. 2013).19. Roberto Cardil (MatematicasVisuales), "Kepler: The Volume of a Wine Barrel - Introduction," Loci (January 2012). http://www.matematicasvisuales.com/english/html/history/kepler/keplerbarrel.html

Vjera Lopac is Professor of Physics at the University of Zagreb, Croatia. Her research interests include nuclear physics, nonlinear physics, numerical simulation, educational physics, metrology in physics and scientific terminology. She is author of several physics textbooks and translator of books on popularization of science.vlopac@fkit.hr

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