It is a common experience that water shoots higher when we block a garden hose outlet by our thumb. But what causes this? How high does the water go? Does water from our neighbor's garden hose reach the same height? Is there an optimum outlet blockage that results in the greatest height that water can reach? Here, we show that a competition between viscous friction along the hose and the viscous dissipation at the thumb-generated constriction results in a variable water shooting height. Through systematic analysis, we demonstrate that depending on the municipal water main pressure, and length and diameter of the hose, the maximum water height may increase, decrease, or gain an optimum as the blockage ratio of the outlet varies.
REFERENCES
1.
The municipal water main is usually a large pipe running under the street that provides water to houses and units through smaller pipes that branch off of it.
2.
Note that the pressure is expressed as gauge pressure throughout this article, that is, relative to the atmospheric pressure. Therefore, zero pressure in this manuscript means that the actual pressure is equal to the atmospheric pressure.
3.
4.
More precisely, the potential energy per unit volume.
5.
J. H.
Evans
, “
Dimensional analysis and the Buckingham Pi theorem
,” Am. J. Phys.
40
, 1815–1822
(1972
).6.
In fact, Euler's equation is Newton's second law written in the Eulerian framework.
7.
for a constant acceleration:
.
8.
From a molecular perspective and specifically for a gas, the pressure on a surface is the average of momentum exchange of bouncing molecules off that surface. When we squeeze a gas container, we confine molecules into a smaller space and therefore more molecules will hit the container's surface per unit time. Therefore, the surface feels a higher force per unit area or pressure. It is as if the container wants to expand and has potential to do some work. Therefore, although pressure is purely kinetic from a molecular point of view, this increase in the pressure looks like an increase in the potential energy of the container's gas. Similar arguments can be given for fluids and solids with the exception that the nature of the molecular source of pressure is somewhat different.
9.
Note that with viscosity, it is inevitable that we have a variable velocity profile across the pipe cross section.
10.
11.
For instance, for glass
0.001 mm, and for concrete
1 mm.
12.
C. F.
Colebrook
, “
Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws
,” J. Inst. Civil Eng.
11
, 133–156
(1939
).13.
C.
Colebrook
and
C.
White
, “
Experiments with fluid friction in roughened pipes
,” Proc. R. Soc. Lond. A
161
, 367–381
(1937
).14.
G. O.
Brown
, “
The history of the Darcy-Weisbach equation for pipe flow resistance
,” in Environmental and Water Resources History Sessions at ASCE Civil Engineering Conference and Exposition, 3-7 November 2002, Washington, D.C.
(
ASCE
, 2003
), pp. 34
–43
.15.
G.
Keady
, “
Colebrook-White formula for pipe flows
,” J. Hydraulic Eng.
124
, 96–97
(1998
).16.
J. R.
Sonnad
and
C. T.
Goudar
, “
Constraints for using Lambert w function-based explicit Colebrook–White equation
,” J. Hydraulic Eng.
130
, 929–931
(2004
).17.
R. M.
Corless
,
G. H.
Gonnet
,
D. E.
Hare
,
D. J.
Jeffrey
, and
D. E.
Knuth
, “
On the Lambert w function
,” Adv. Comput. Math.
5
, 329–359
(1996
).18.
D.
Brkić
, “
Lambert w function in hydraulic problems
,” Math. Balkanica
26
, 285–292
(2012
).19.
D.
González-Mendizabal
,
C.
Olivera-Fuentes
, and
J. M.
Guzmán
, “
Hydrodynamics of laminar liquid jets. experimental study and comparison with two models
,” Chem. Eng. Commun.
56
, 117–137
(1987
).20.
R.
Brun
and
J.
Leinhard
, “
Behavior of free laminar jets leaving Poiseuille tubes
,” in Mechanical Engineering
(
ASME
,
New York
, 1968
), Vol.
90
, p. 72
.21.
J.
Duda
and
J.
Vrentas
, “
Fluid mechanics of laminar liquid jets
,” Chem. Eng. Sci.
22
, 855–869
(1967
).22.
E.
Tuck
, “
The shape of free jets of water under gravity
,” J. Fluid Mech.
76
, 625–640
(1976
).23.
J.
Eggers
and
E.
Villermaux
, “
Physics of liquid jets
,” Rep. Prog. Phys.
71
, 036601
(2008
).24.
T.
Massalha
and
R. M.
Digilov
, “
The shape function of a free-falling laminar jet: Making use of Bernoulli's equation
,” Am. J. Phys.
81
, 733–737
(2013
).25.
P.
Sharma
and
T.
Fang
, “
Breakup of liquid jets from non-circular orifices
,” Exp. Fluids
55
, 1666
(2014
).26.
J. F.
Geer
and
J. C.
Strikwerda
, “
Vertical slender jets with surface tension
,” J. Fluid Mech.
135
, 155–169
(1983
).27.
R. P.
Benedict
,
N.
Carlucci
, and
S.
Swetz
, “
Flow losses in abrupt enlargements and contractions
,” J. Eng. Power
88
, 73–81
(1966
).28.
D. C.
Rennels
and
H. M.
Hudson
, Pipe Flow: A Practical and Comprehensive Guide
(
John Wiley & Sons
,
New York
, 2012
).© 2021 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2021
Author(s)
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