Much of the drama and complexity of fluid flow occurs because its governing equations lack unique solutions. The observed behavior depends on the stability of the multitude of solutions, which can change with the experimental parameters. Instabilities cause sudden global shifts in behavior. We have developed a low-cost experiment to study a classical fluid instability. By using an electromagnetic technique, students drive Kolmogorov flow in a thin fluid layer and measure it quantitatively with a webcam. They extract positions and velocities from movies of the flow using Lagrangian particle tracking and compare their measurements to several theoretical predictions, including the effect of the drive current, the spatial structure of the flow, and the parameters at which instability occurs. The experiment can be tailored to undergraduates at any level or to graduate students by appropriate emphasis on the physical phenomena and the sophisticated mathematics that govern them.

1.
R. A.
Shaw
, “
Particle-turbulence interactions in atmospheric clouds
,”
Annu. Rev. Fluid Mech.
35
,
183
227
(
2003
).
2.
P. S.
Marcus
, “
Prediction of a global climate change on Jupiter
,”
Nature (London)
428
,
828
831
(
2004
).
3.
B. W.
Zeff
,
B.
Kleber
,
J.
Fineberg
, and
D. P.
Lathrop
, “
Singularity dynamics in curvature collapse and jet eruption on a fluid surface
,”
Nature (London)
403
,
401
404
(
2000
).
4.
E.
Bodenschatz
,
W.
Pesch
, and
G.
Ahlers
, “
Recent developments in Rayleigh-Bénard convection
,”
Annu. Rev. Fluid Mech.
32
,
709
778
(
2000
).
5.
A. M.
Obukhov
, “
Kolmogorov flow and laboratory simulation of it
,”
Russ. Math. Surveys
38
,
113
126
(
1983
).
6.
N. T.
Ouellette
, in
Experimental and Computational Techniques in Soft Condensed Matter Physics
, edited by
J.
Olafsen
(
Cambridge U. P.
,
Cambridge
,
2010
).
7.
A.
Thess
, “
Instabilities in two-dimensional spatially periodic flows. Part I: Kolmogorov flow
,”
Phys. Fluids A
4
,
1385
1395
(
1992
).
8.
F.
Feudel
and
N.
Seehafer
, “
Bifurcations and pattern formation in a two-dimensional Navier-Stokes fluid
,”
Phys. Rev. E
52
(
4
),
3506
3511
(
1995
).
9.
Y.
Couder
, “
Two-dimensional grid turbulence in a thin liquid film
,”
J. Physique Lett.
45
,
353
360
(
1984
).
10.
H.
Kellay
,
X. -L.
Wu
, and
W. I.
Goldburg
, “
Experiments with turbulent soap films
,”
Phys. Rev. Lett.
74
,
3975
3978
(
1995
).
11.
P.
Vorobieff
and
R. E.
Ecke
, “
Fluid instabilities and wakes in a soap-film tunnel
,”
Am. J. Phys.
67
,
394
399
(
1999
).
12.
J. M.
Burgess
,
C.
Bizon
,
W. D.
McCormick
,
J. B.
Swift
, and
H. L.
Swinney
, “
Instability of the Kolmogorov flow in a soap film
,”
Phys. Rev. E
60
(
1
),
715
721
(
1999
).
13.
P.
Tabeling
,
S.
Burkhart
,
O.
Cardoso
, and
H.
Willaime
, “
Experimental study of freely decaying two-dimensional turbulence
,”
Phys. Rev. Lett.
67
,
3772
3775
(
1991
).
14.
D.
Rothstein
,
E.
Henry
, and
J. P.
Gollub
, “
Persistent patterns in transient chaotic fluid mixing
,”
Nature (London)
401
,
770
772
(
1999
).
15.
T. H.
Solomon
and
I.
Mezić
, “
Uniform resonant chaotic mixing in fluid flows
,”
Nature (London)
425
,
376
380
(
2003
).
16.
H. J. H.
Clercx
,
G. J. F.
van Heijst
, and
M. L.
Zoeteweij
, “
Quasi-two-dimensional turbulence in shallow fluid layers: The role of bottom friction and fluid layer depth
,”
Phys. Rev. E
67
,
066303
(
2003
).
17.
M. K.
Rivera
and
R. E.
Ecke
, “
Pair dispersion and doubling time statistics in two-dimensional turbulence
,”
Phys. Rev. Lett.
95
,
194503
(
2005
).
18.
L.
Rossi
,
J. C.
Vassilicos
, and
Y.
Hardalupas
, “
Electromagnetically controlled multi-scale flows
,”
J. Fluid Mech.
558
,
207
242
(
2006
).
19.
N. F.
Bondarenko
,
M. Z.
Gak
, and
F. V.
Dolzhanskii
, “
Laboratory and theoretical models of plane periodic flow
,”
Akademiia Nauk SSSR Fizika Atmosfery i Okeana
15
(
1
),
1017
1026
(
1979
).
20.
D.
Vella
and
L.
Mahadevan
, “
The ‘Cheerios effect’
,”
Am. J. Phys.
73
,
817
825
(
2005
).
21.
Kalliroscope Corporation
, ⟨www.kalliroscope.com/⟩.
22.
N. T.
Ouellette
,
H.
Xu
, and
E.
Bodenschatz
, “
A quantitative study of three-dimensional Lagrangian particle tracking algorithms
,”
Exp. Fluids
40
,
301
313
(
2006
).
23.
F.
Toschi
and
E.
Bodenschatz
, “
Lagrangian properties of particles in turbulence
,”
Annu. Rev. Fluid Mech.
41
,
375
404
(
2009
).
24.
N.
Mordant
,
E.
Lévêque
, and
J. F.
Pinton
, “
Experimental and numerical study of the Lagrangian dynamics of high Reynolds turbulence
,”
New J. Phys.
6
,
116
(
2004
).
25.
See supplementary material at http://dx.doi.org/10.1119/1.3536647 for the post-processing software. This document can also be downloaded at ⟨leviathan.eng.yale.edu⟩.
26.
S. T.
Merrifield
,
D. H.
Kelley
, and
N. T.
Ouellette
, “
Scale-dependent statistical geometry in two-dimensional flow
,”
Phys. Rev. Lett.
104
,
254501
(
2010
).
27.
Matlab is published by
MathWorks
www.mathworks.com/⟩.
28.
M. C.
Cross
and
P. C.
Hohenberg
, “
Pattern formation outside of equilibrium
,”
Rev. Mod. Phys.
65
(
3
),
851
1112
(
1993
).
29.
S.
Chandrasekhar
,
Hydrodynamic and Hydromagnetic Stability
(
Dover
,
New York
,
1961
).
30.
P. G.
Drazin
and
W. H.
Reid
,
Hydrodynamic Stability
, 2nd ed. (
Cambridge U. P.
,
Cambridge
,
2004
).

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