This article describes an updated version of the famous Taylor-Couette flow reversibility demonstration. The viscous fluid confined between two concentric cylinders is forced to move by the rotating inner cylinder and visualized through the transparent outer cylinder. After a few rotations, a colored blob of fluid appears well mixed. Yet, after reversing the motion for the same number of turns, the blob reappears in the original location as if the fluid has just been unmixed. The use of household supplies makes the device inexpensive and easy to build without specific technical skills. The device can be used for demonstrations in fluid dynamics courses and outreach activities to discuss the concepts of viscosity, creeping flows, the absence of inertia, and time-reversibility.

Flows are guaranteed to be smooth and laminar if inertial effects are overwhelmed by viscous effects. Our daily lives do not expose us to laminar flows in a sustained manner, so their properties may seem unusual and counterintuitive. Probably the most striking feature is the time-reversibility of Stokes (or “creeping”) flows, which typically occur when the Reynolds number—the ratio of the inertial to viscous forces—is much less than unity. Time reversibility implies that fluid elements retrace their motion when the boundary motion is reversed, giving the impression that the effects of mixing can be undone.

A classic demonstration of this phenomenon was introduced by Taylor in an instructional video1,2 about low Reynolds number flows, part of the series3 produced by the National Committee for Fluid Mechanics Films in the 1960s. In this video, which sparks interest even among lay persons,4 Taylor uses what is now called a Taylor-Couette apparatus (see Fig. 1(a)) filled with glycerol possessing kinematic viscosity about 1100 times that of water at 20 °C. The apparatus—named in honor of Taylor, who used it to visualize the famous toroidal vortices in 1923,5 and Maurice Couette, who used it as a viscometer in the thesis6 he defended in 1890—confines the fluid between two concentric cylinders with the inner cylinder rotating. The simplicity of this flow, its rich dynamics, and its easy experimental realization made it a popular tool for fluid dynamicists to study both laminar and turbulent flows (for a historical overview, see Ref. 7). The demonstration starts with Taylor introducing a blob of dyed glycerol in the fluid. After the inner cylinder is spun clockwise four times and the colored blob seems to be mixed with the rest of the fluid, a surprise awaits. By spinning the cylinder the same number of times in the counterclockwise direction, one recovers the original blob (except for a small amount of blur caused by diffusion). This observation seems to go against a common sense understanding of how fluid flow behaves and usually surprises both expert and non-expert audiences alike.

Fig. 1.

Taylor-Couette apparatus; (a) original demonstration of Taylor; (b) version by J. DeMoss and K. Cahill from the University of New Mexico that went viral on YouTube, exceeding 1.6 million views; (c) proposed version of the demonstration. The image in panel (c) shows the moment at which the photochromic dye is activated by the laser pointer to create colored spots.

Fig. 1.

Taylor-Couette apparatus; (a) original demonstration of Taylor; (b) version by J. DeMoss and K. Cahill from the University of New Mexico that went viral on YouTube, exceeding 1.6 million views; (c) proposed version of the demonstration. The image in panel (c) shows the moment at which the photochromic dye is activated by the laser pointer to create colored spots.

Close modal

A version of this demonstration was proposed by Heller8 in 1960, and a recent version (see Fig. 1(b)) became popular after a video9 posted by DeMoss and Cahill from the University of New Mexico went viral on YouTube (having more than 1.6 million views). Similar demonstrations have been featured in popular science news outlets10 as well as on TV shows.11 The large number of YouTube comments that question the authenticity of the video demonstrates the non-intuitive and curious nature of the demonstration.

We propose here a new version of this demonstration (see Fig. 1(c)), which is inexpensive to build, and can be shown during lectures and outreach events without any previous preparation. The use of a photochromic dye activated with a laser makes it particularly entertaining to use. This version of the Taylor-Couette apparatus can be built for less than $30 and the full demonstration, including liquid and supplies, for less than $150.

The equations that describe the conservation of mass and momentum of incompressible Newtonian fluid in the Taylor-Couette system are the Navier-Stokes equations
· v = 0 ,
(1)
v t + ( v · ) v + 1 ρ p ν 2 v = 0.
(2)
Here, t is the time, v is the fluid velocity, p is the fluid pressure, and ν = μ / ρ is the kinematic viscosity, μ being the dynamic viscosity and ρ the density (we have neglected the body forces). By considering a characteristic length scale L and a characteristic velocity U, we can make Eq. (2) dimensionless. We define the following dimensionless variables: velocity v = v / U , length r = r / L , time t = t U / L , and pressure p = p L / ( U ρ ν ) , which allows us to write Eq. (2) as
Re [ v t + ( v · ) v ] + p 2 v = 0 ,
(3)
where
Re = U L ν
(4)
is the (dimensionless) Reynolds number.
If the flow velocity and the spatial scale are small and the fluid viscosity is large, the Reynolds number becomes small and the viscous forces become dominant compared to the inertial terms. The limiting case for which Re 1 is usually referred to as Stokes, or creeping, flow. The resulting equations describing these flows are the Stokes equations
· v = 0 and p 2 v = 0.
(5)
Note that if we make the pressure dimensionless as p = p / ( ρ U 2 ) , the pressure term would be proportional to the Reynolds number and drop out, leaving us with a trivial equation. In contrast to the full Navier-Stokes equations, the Stokes equations are linear and do not depend explicitly on time. Hence the time reversal of boundary motion reverses the fluid motion, explaining why the colored blob, after being mixed, can return to its original configuration. In this particular demonstration, the fluid flows in parallel layers, or laminae, which slide past each other without mixing, hence the name “laminar flow.” We note that the flow is reversible, up to molecular diffusion effects. For a detailed treatment of Stokes flows see, e.g., Ref. 12.

The reversibility demonstration introduced by Taylor is clear and striking. Unfortunately, despite its apparent simplicity, it is not easy for a novice to establish the demonstration. No commercial scientific instrument suppliers offer a Taylor-Couette apparatus for purchase; however, a replica of the apparatus from the University of New Mexico is available13 for $550. The few low-cost alternatives proposed online (which, for example, use two drinking glasses) leave much to be desired. A relatively simple Taylor-Couette apparatus to demonstrate the generation of Taylor vortices was described in Ref. 14. In Sec. III A, we describe how to assemble such an apparatus for less than $30. In practice, however, this standard demonstration requires significant time and effort to prepare and clean up. Before the demonstration, the colored dyes need to be mixed with some of the viscous fluid and then carefully injected in the apparatus. After the demonstration, all the tools must be cleaned of fluids such as glycerol, corn syrup, or silicone oil, which are not very pleasant to handle. Moreover, unless all of the fluid is disposed of every time, the colored blobs need to be removed with a syringe or a spoon, leaving residues that make the fluid progressively dirtier.

An alternative apparatus that is easier to use would thus encourage instructors to show this demonstration routinely in class. To this end, we propose here a demonstration device that is sealed, does not require any setup or cleaning, and can be performed in less than a minute. The full apparatus, including the photochromic fluid and supplies, can be easily built by students in just a few hours for less than $150. The parts used for the demonstration and their costs are listed in Table I.

Table I.

List of materials and equipment for the assembly of the demonstration device shown in Fig. 1(c); the costs exclude shipping. The actual amount of dye and toluene used in the 16 oz (0.47 l) of silicone oil are 0.1 g and 10 ml, respectively. Note also the need to have access to a 3D printer, a fume hood, a precision scale, a spatula, a pipette, and a vial.

Description Quantity/Size Vendor/Source Cost
Spiropyran photochromic dye  1 g  TCI America  $50 
Toluene  32 oz  Consolidated Chemical/Amazon  $28 
Silicone oil, 1000 cSt  16 oz  Consolidated Chemical/Amazon  $18 
Norpro coffee & Tea pressa  25 oz  Hardware Store  $35 
Laser, 301–405 nm laser pointer  Ebay  $9 
3D printed cylinder  Afinia3D H480  — 
3D printed cap  Afinia3D H480  — 
Rubber bands, plastic straw, steel spheres  —  Found objects  — 
Description Quantity/Size Vendor/Source Cost
Spiropyran photochromic dye  1 g  TCI America  $50 
Toluene  32 oz  Consolidated Chemical/Amazon  $28 
Silicone oil, 1000 cSt  16 oz  Consolidated Chemical/Amazon  $18 
Norpro coffee & Tea pressa  25 oz  Hardware Store  $35 
Laser, 301–405 nm laser pointer  Ebay  $9 
3D printed cylinder  Afinia3D H480  — 
3D printed cap  Afinia3D H480  — 
Rubber bands, plastic straw, steel spheres  —  Found objects  — 
a

We also used a 34-oz French press, which we bought at Starbucks for $20. Note that the shaft diameter and thread size are the same with the Norpro as well as with a smaller Bodum press.

To assemble our Taylor-Couette apparatus, we used a commercial French press (for the preparation of coffee and tea) and two 3D printed parts.15 The borosilicate glass (Pyrex) beaker of the French press acts as the transparent external cylinder of the Taylor-Couette device, while the plunger acts as the base for the rotating inner cylinder—the central metal rod is used as the rotating shaft, and the bottom metal part provides the pivot around which the shaft spins. The assembly (see Fig. 2) starts by sliding the 3D printed cap and the 3D printed cylinder onto the metal rod. The diameter of the holes in the center of the cap and cylinder are slightly smaller than the diameter of the threaded shaft16 to ensure that the cylinder locks onto the shaft (the friction between the metal thread and the plastic hole should be larger than the friction between the external surface of the cylinder and the viscous fluid). To avoid buoyancy forces that lift the empty cylinder when immersed in the fluid, we added some metal spheres (about 1 cm in diameter) to the inside of the cylinder. To prevent the movement of the spheres inside the cylinder we wrapped them with paper. (Any objects with a density higher than that of the silicone oil could be used in place of the metal spheres.) The French press comes with the central rod and the bottom metal part screwed together. To let one spin around the other, we screwed the threaded cap just a few millimeters onto the threaded shaft, such that the threaded shaft would not be able to screw into the threaded metal cylinder in the center of the bottom metal part.

Fig. 2.

Schematic assembly of the Taylor-Couette setup, whose main components are a French press and a plastic cylinder made of two 3D printed parts.

Fig. 2.

Schematic assembly of the Taylor-Couette setup, whose main components are a French press and a plastic cylinder made of two 3D printed parts.

Close modal

A section of a plastic drinking straw around this metal cylinder helps the plastic cylinder spin smoothly, preventing any wobbly motion in the lower end. A rubber band around the cap of the French press ensures a snug fit in the glass beaker, and preventing any wobbly motion on the upper end. The resulting setup is a Taylor-Couette apparatus that spins effortlessly even without bearings.

For our first prototype, we did not use a 3D printed cylinder but instead used a plastic pipe filled with Styrofoam disks. In order to prevent the spinning pipe from sliding when immersed in the viscous fluid, we secured it to the shaft with a cap that was molded by hand using some polycaprolactone plastic or with glue. Centering the cylinder as well as securing it to the shaft poses some challenges, so we recommend the 3D printed alternative. If not available on site, we recommend looking for a 3D printer in a nearby makerspace.

This Taylor-Couette apparatus can be used to replicate the traditional demonstration by Taylor; it can be filled with glycerol or corn syrup, and then have some colored blobs introduced using a syringe and a flexible plastic tubing through a hole in its cap. However, this configuration requires a laborious effort for preparation and cleanup as previously described. By using a photochromic viscous fluid, it is possible to create a version that obviates all these problems.

A photochromic substance is one that changes color once illuminated by a light source. Dyes of such materials have been successfully used for flow visualization.17 However, despite being non-intrusive and effective close to solid boundaries, photochromic dyes have not replaced standard tracer particles because dyes generally do not work well in water or in large-scale apparatuses, greatly limiting their flexibility.

For our purpose, we created a highly viscous photochromic fluid using a spiropyran photochromic dye (1,3,3-Trimethylindolino-6 -nitrobenzopyrylospiran, C 19 H 18 N 2 O 3 , Tokyo Chemical Industry Co., Ltd.) dissolved in toluene and then mixed in silicone oil, following the procedure in Ref. 18. No particular tools are necessary to create the photochromic fluid, except for a fume hood to safely mix the dye with the toluene (or other non-polar organic solvents) and precision digital scales, if one wishes to characterize the exact dye concentration. We tested silicone oils with viscosities of 1,000 cSt and 12,500 cSt and a few different dye concentrations. For the demonstrations presented here, we selected the 1,000 cSt silicone oil because it looks more like fluids with which we have daily experience. The silicone oil has a concentration of dye on the order of 200 ppm diluted in 2% toluene by weight (we dissolved 0.1 g of dye in 10 ml of toluene, then mixed in 0.47 l of silicone oil). We note that toluene is toxic and should be handled with care,19 but in the small quantity and dilution used here, it does not create any serious hazard, nor does it affect the integrity of any plastic part.

The silicone oil solution is transparent, but it quickly turns purple when illuminated by UV light. To change the color of the dye, we use a 405-nm violet laser pointer (Laser 301, 165 ± 5 mW) (Ref. 20) with an adjustable focus to create thinner or thicker lines. A 5-mW laser has also been tested successfully with the only disadvantage of taking longer to activate the dye.

The procedure for the demonstration with the photochromic fluid is similar to the original demonstration of Taylor, except that instead of creating the colored dots by injecting the dye, we create them by shining the laser (see Fig. 1(c)). The steps of the demonstrations are summarized in Fig. 3. Of course, the laser makes a colored line across the full length of the gap between the two cylinders and not just a spherical blob, and allows us to create relatively complicated shapes. The time after which the colored dye becomes transparent again depends on how long the fluid is illuminated, the power of the laser, and the dye concentration. With our setup, after an illumination of 10 s, the colored spot was visible for several minutes. It is important to note that demonstrating the experiment outdoors requires special precautions because the entire photochromic liquid turns blue when exposed to sunlight.

Fig. 3.

The sequence of images illustrates the various steps of the demonstrations. First, we create three localized spots by illuminating the photochromic fluid with the laser (a). As soon as we start spinning the inner cylinder (b), the spots start smearing. After four full turns, the spots are not recognizable and appear mixed with the fluid (c). Then, after spinning the cylinder in the opposite direction (d), the fluid returns to its original configuration in which we can recognize the three blobs in their initial locations (e).

Fig. 3.

The sequence of images illustrates the various steps of the demonstrations. First, we create three localized spots by illuminating the photochromic fluid with the laser (a). As soon as we start spinning the inner cylinder (b), the spots start smearing. After four full turns, the spots are not recognizable and appear mixed with the fluid (c). Then, after spinning the cylinder in the opposite direction (d), the fluid returns to its original configuration in which we can recognize the three blobs in their initial locations (e).

Close modal

The demonstration provides an excellent insight into the concept of viscosity and the role of the Reynolds number and how the absence of the nonlinear advective term ( v · ) v fundamentally changes the flow dynamics. While everyone has an intuitive understanding of the concept of viscosity and how it differs in water or honey, few can appreciate the profound effects this quantity has on the flow. Such concepts are of fundamental importance for any basic course in fluid dynamics and can be effectively conveyed through these demonstrations. The demonstration can also be used beyond the scope of fluid mechanics to show how the world would be different without the effect of inertia. Such a demonstration was also used by Bohm as an analogy21 to illustrate the concept of “implicate order” in quantum mechanics.22 

While there is little doubt that demonstrations are among students' favorite elements in the classroom and can stimulate scientific interest in an entertaining way,23 their educational value depends on whether the students observe the demonstrations correctly, predict their outcome, and have a basic conceptual understanding of the phenomenon.24,25 The flow reversibility demonstration has the advantage of being very simple to grasp and observe, and the students' predictions are almost always overturned by the unexpected outcome.

A possible problem for students, to get them familiar with the Reynolds number, is to have them calculate this dimensionless parameter after selecting the characteristic velocity and length scales. One definition of the Reynolds number in this case is Re = Ω r ( R r ) / ν , where Ω is the angular velocity and r and R are the inner and outer cylinder radii, respectively. Here, we have taken the characteristic length to be the gap width between the cylinders, L = R r , and the characteristic velocity as U = Ω r . Assuming one revolution every eight seconds we find Ω = π / 4 s− 1, and using r = 0.024 m, R = 0.048 m, and ν = 0.001 m 2 / s, we calculate a Reynolds number Re = 0.45 . Although the condition Re 1 is not strictly satisfied, the viscous effects are larger than the inertial effects, and in this case that is enough to yield reversible effects. Using a 12,500 cSt fluid, The Reynolds number would be Re = 0.036 , and this might be a more appropriate value for emphasizing the conditions for Stokes flow.

Another problem for students is to use dimensional analysis to show that the ratio of the inertial ( v · ) v and viscous ν 2 v terms is equal to the Reynolds number. Taking U and L as the characteristic velocity and length in this problem, we have
[ inertial term ] [ viscous term ] = [ ( v · ) v ] ν [ 2 v ] = U · U / L ν U / L 2 = U L ν = Re .
(6)

Students could also be asked to predict the outcome of the demonstration knowing that it is Stokes flow, which has the property described in Sec. II. After showing the demonstration, one could then discuss the practical relevance of Stokes flow in problems of lubrication, micro-fluidics, and swimming of microorganisms. An important consequence for the swimming of microorganisms is the “Scallop theorem.”26 The name of the theorem comes from the implication that a scallop would not be able to move in a Stokes flow situation. The motion of its shell is reciprocal, meaning that the shell moves to a certain position and then goes back to the original configuration by going through the sequence in reverse. The reciprocal motion leads the scallop to return periodically to its starting position. Microorganisms can move in a Stokes flow by breaking the time-reversal symmetry in various ways, for example, with wavy or twisting motions that are not reciprocal.

Note that by adding a motor to the central shaft and using a rheoscopic fluid (e.g., Kalliroscope, water with mica powder, Viniq liqueur) the demonstration could be used in principle to visualize Taylor vortices, extending its educational value to more flows and flow transitions. Moreover, to study the flow configuration with counter-rotating cylinders, one could set the apparatus on a rotating platform such as a turntable.

We have described the design of a Taylor-Couette device that is inexpensive and easy to assemble using available commercial parts and/or 3D printed components. This apparatus is suitable for a demonstration of the reversibility of Stokes flow. In particular, we propose a version that uses a photochromic fluid, requires minimal setup time, and can be performed in less than a minute in a repeatable and particularly engaging way.

The authors acknowledge Maurizio Porfiri for the lab space, Riccardo Panciroli, Filippo Cellini, and Randy Sofia for assistance in building the experiment and conducting the tests, and Avi Ulman for useful suggestions. The authors thank Michael Gilheany from Townsend Harris High School, and Stephen Johnston and Devesh Ranjan from the Georgia Institute of Technology for helping during the tests of the photochromic solution and testing a few experimental designs. The authors also thank the anonymous reviewers for helpful comments.

1.
Geoffrey I.
Taylor
,
Film Notes for Low Reynolds-Number Flows
(
National Committee for Fluid Mechanics Film
,
Cambridge, MA
,
1967
); <http://web.mit.edu/hml/ncfmf/07LRNF.pdf>.
2.
The video
, “
Low Reynolds number flows
,” can be found at <https://www.youtube.com/watch?v=51-6QCJTAjU>. The portion dealing with kinematic reversibility begins at 13 m 14 s.
3.
National Committee for Fluid Mechanics Film
, <http://web.mit.edu/hml/ncfmf.html>
4.
Etienne Guyon and Marie Yvonne Guyon
, “
Taking fluid mechanics to the general public
,”
Annu. Rev. Fluid Mech.
46
,
1
22
(
2014
).
5.
Geoffrey I.
Taylor
, “
Stability of a viscous liquid contained between two rotating cylinders
,”
Phil. Trans. R. Soc. A
223
,
289
343
(
1923
).
6.
Maurice
Couette
, “
Etudes sur le frottement des liquides
,”
Ann. Chim. Phys.
21
,
433
510
(
1890
).
7.
Russell J.
Donnelly
, “
Taylor-Couette flow: The early days
,”
Phys. Today
44
(
11
),
32
39
(
1991
).
8.
John P.
Heller
, “
An unmixing demonstration
,”
Am. J. Phys.
28
,
348
353
(
1960
).
9.
Laminar Flow, University of New Mexico Physics and Astronomy video; <https://www.youtube.com/watch?v=p08_KlTKP50>
10.
New Scientist TV: Born to be Viral: How to unmix a mixed fluid
; <http://www.newscientist.com/blogs/nstv/2011/08/born-to-be-viral-how-to-unmix-a-mixed-fluid.html>
11.
Twist in Time – Laminar Flow; <http://www.youtube.com/watch?v=W3YZ5veN_Bg>
12.
Stephen
Childress
,
An Introduction to Theoretical Fluid Mechanics
(
American Mathematical Society, Courant Institute of Mathematical Sciences at New York
,
2009
); <http://www.ams.org/bookstore-getitem/item=cln-19>.
13.
Couette Cell for Demonstrating Laminar Flow—Apparatus or Plans Available for Purchase; <http://www.flintbox.com/public/project/6027/>
14.
Stephen J.
Van Hook
and
Michael F.
Schatz
, “
Simple demonstrations of pattern formation
,”
Phys. Teach.
35
,
391
395
(
1997
).
15.
See supplementary material at http://dx.doi.org/10.1119/1.4996901 E-AJPIAS-85-007709 for the 3D printer (.stl) files.
16.
The thread in the three French presses considered is a standard #10-32 UNC thread, with an outer diameter of 0.19" (4.8 mm) and a tap drill of #21.
17.
S.
Kurada
,
G. W.
Rankin
, and
K.
Sridhar
, “
Flow visualization using photochromic dyes: A review
,”
Opt. Laser Eng.
20
,
177
192
(
1994
).
18.
Toshiyuki
Sanada
,
Minori
Shirota
, and
Masao
Watanabe
, “
Bubble wake visualization by using photochromic dye
,”
Chem. Eng. Sci.
62
,
7264
7273
(
2007
).
19.
Toluene Safety in the Workplace; <https://www.osha.gov/Publications/OSHA3646.html>.
20.
Note that while operating powerful lasers it is necessary to use proper protective eyewear. We also recommend caution against possible laser mislabeling.
21.
Interview with Dr. David Bohm on “The Nature of Things,” CBC Canadian Radio; <https://www.youtube.com/watch?v=r-jI0zzYgIE>. The portion dealing with flow reversibility begins at 13 m 25 s.
22.
David J.
Bohm
,
Wholeness and the Implicate Order
(
Routledge
,
London and New York
,
1980
).
23.
R.
Di Stefano
, “
Preliminary IUPP results: Student reactions to in-class demonstrations and to the presentation of coherent themes
,”
Am. J. Phys.
64
,
58
68
(
1996
).
24.
Catherine
Crouch
,
Adam P.
Fagen
,
J.
Paul Callan
, and
Eric
Mazur
Classroom demonstrations: Learning tools or entertainment?
,”
Am. J. Phys.
72
,
835
838
(
2004
).
25.
Kelly
Miller
,
Nathaniel
Lasry
,
Kelvin
Chu
, and
Eric
Mazur
, “
Role of physics lecture demonstrations in conceptual learning
,”
Phys. Rev. Phys. Educ. Res.
9
,
020113-1
5
(
2013
).
26.
E. M.
Purcell
, “
Life at low Reynolds number
,”
Am. J. Phys.
45
,
3
11
(
1977
).

Supplementary Material