A classic magic trick involves filling a bucket with liquid and then turning it upside down so that the liquid stays in the bucket. This does not work under normal circumstances because of the Rayleigh–Taylor instability. Although the atmospheric pressure pushing at the free surface from below is theoretically enough to support the surface, in reality any very small perturbation or disturbance of the surface causes it to ripple and form irregularities, which grow so that there is no longer a balance between the weight of the water and the pressure. In this article, I will help students understand the physics behind these instabilities by introducing a variation on a common type of geometry known as a Hele-Shaw cell and studying the influence of various factors that are involved in the physics of the magic trick, including the viscosity and surface tension of the liquid.

All students should be familiar with ripples that can form on the surface of a liquid. A simple intuitive description of the Rayleigh–Taylor instability is that it is the growth of small ripples into larger ripples at the horizontal interface between two different fluids of different densities that occurs when the less dense fluid pushes against the denser fluid. The two fluids could both be liquids (water and oil, for example), or one could be a liquid and the other a gas. Although the evolution of this interface can be very complicated, for simplicity when working in two dimensions, one can assume that the shape of the instability begins as shown in Fig. 1 (*ρ*_{G} and *ρ*_{L} refer to the density of the gas and liquid, respectively).

The amplitudes of both the lower and the upper part of the ripple then increase over time. This instability grows almost without limit, because the ripple causes an imbalance between the pushing force of the liquid and the pushing force from the gas at the interface; this imbalance becomes greater as the ripple grows larger, which causes the instability to grow faster.

Before moving on to the Hele-Shaw cell, I will say more about the magic trick that was discussed in the introductory paragraph. I clarify that the trick I describe is slightly different from a similar experiment where a glass of water is covered with a piece of card and turned upside down. The typical elementary explanation for this is that atmospheric pressure then prevents the water from spilling out of the glass, since the force exerted from below on the card due to air pressure is greater than the force from above due to the weight of the water.^{2}

The trick I describe is a similar one where the glass is turned upside down, but this time the water somehow stays in the glass when the cover is slowly pulled away. A video of the trick can be found at Ref. 3, but the explanation in this case is rather prosaic: one has a plastic disk attached to the piece of card and simply replaces the card with the disk when the card is pulled away. A more interesting variation occurs when the card is pulled away and replaced with porous gauze, fine mesh, or even wet tissue paper. In this case, water is clearly allowed to pass through the material, but it remains in the glass (some liquid may drip out in this case, but the surface of the liquid then stabilizes). How is this possible? In fact, both these experiments can be explained qualitatively using the concept of a Rayleigh–Taylor instability.^{4} To aid in a more quantitative understanding of the physics involved, I will try to illustrate the key processes using a Hele-Shaw cell as mentioned previously.

A Hele-Shaw cell is a pair of very closely spaced plane parallel plates. We will consider a slight variation on the usual Hele-Shaw cell, where two transparent glass slides meet at the bottom at an angle so that the plates are only extremely close to parallel. The gap thickness between the tops of the slides is measured to be 1 mm, and the thickness decreases linearly as one goes to the bottom of the slides. The spacing between the top of the slides is fixed with Blu Tack, and the slides are glued in place. The slides are sealed with filler at the bottom and sides to prevent liquid from draining out as the gap thickness is filled. At the bottom of the plates, the thickness decreases so far below 1 mm that any fluid in this part is effectively trapped and not able to move. In case the construction is unclear, in Fig. 2 I provide a photograph from the side of the two glass slides stood upright and placed together (prior to use of the filler).

In terms of using the experiment in a teaching environment, one might also ask if the cell involves any expensive components in the construction. Glass slides for a microscope are cheaply available from a laboratory glassware manufacturer. The filling material I used is also inexpensive (I used ready-mixed all-purpose filler, which is available at a hardware store). The only component of our experiment that would not be easily available outside of a fluid dynamics laboratory is the low-viscosity silicone oil. There are, however, other fluids that could be used for the demonstration, and it is not essential that silicone oil be used. These fluids could be borrowed from a high school or university chemistry department (ethanol or methanol, for example). Similarly, the dye I use shortly to visualize the flow with silicone oil is not easily obtainable and would likely have to be ordered from a scientific supplier. Again, the dye is not strictly necessary and is only used to aid visualization, whereas water dye is readily available.

Once the cell is constructed, we then cover the top of the cell, turn it upside down, and carefully remove the cover so that the liquid surface is not disturbed. In Fig. 3, this principle is demonstrated first by filling the cell with water and then turning it upside down in atmospheric air. We will repeat the same experiment shortly with a different fluid to illustrate all the key physical principles involved. One can see that the fluid has shifted back and forth, driving motion and formation of bubbles in the fluid. The study of viscous fluid flow in a Hele-Shaw channel and the motion of bubbles driven by the liquid is an interesting topic that has been studied both on the experimental and numerical side, although this is not our focus here.^{5} However, no fluid has left the cell, and one can see that is visibly being held at the orifice. A small amount of blue dye has also been added at the bottom of the cell to show that this portion of the fluid has not shifted around after turning the cell upside down.

^{6}A key example here is the surface tension

*σ*of the liquid, which acts to stabilize “ripple” wavelengths smaller than a critical wavelength

*λ*given by

*g*is the acceleration due to gravity,

*ρ*

_{L}is the density of the liquid, and

*ρ*

_{G}is the density of the surrounding air. For clarity on the physics involved in this stabilizing effect, note that the definition of the critical wavelength is the same as the formula for the capillary length, a length scaling factor that relates the forces due to surface tension and gravity at an interface of two fluids. It makes intuitive sense that the longest stable wavelength

*λ*increases when

*σ*increases, since with a higher surface tension, a wave can maintain its shape over a longer distance. Conceptually, it also makes sense that the wavelength decreases when g increases, since gravity acts to flatten out droplets and reduce their curvature. Finally, it decreases when there is a larger difference between the densities of the two fluid phases, since this increases the tendency of one fluid to push into the other, which makes it harder for the perturbation to retain its initial shape.

^{3}, the surface tension is 0.072 N/m, and the dynamic viscosity is 0.8891 mPa·s. The acceleration due to gravity is assumed to be 9.8 m/s

^{2}. Neglecting the density of the gas and substituting in the above values, we obtain

*F*

_{σ}is proportional to the amount of surface the liquid is in contact with at the air–liquid interface. One might naively expect that this force should instead depend on the surface area of the interface, but we will use the perimeter of the air–liquid interface in our calculation. The cell has height 70 mm, width 50 mm, and thickness 1 mm, so the perimeter of the orifice

*P*is 102 mm. This gives us

*h*is the height of the liquid above the free surface. We have as a consequence

*h*, so we will modify this formula by multiplying by ½, which gives us

This confirms, as expected, that the force due to surface tension is smaller than the force due to the weight of the liquid, and that the main role of the surface tension is very likely the effect it has on suppressing instabilities, thereby allowing the force from below to act in a way that is effectively uniform. Atmospheric pressure is equal to 101,325 N/m^{2}, which results in a large force pushing the liquid from below (around 5 N).

*σ*= 0.015 N/m and dynamic viscosity 0.0014 Pa·s. Note, however, that this liquid is twice as viscous as water and that viscosity can also affect formation of Rayleigh–Taylor instabilities, since viscosity regularizes the flow and reduces growth rate of irregularities. When I say that the flow is regularized, I mean that it is smoothed out and made uniform, without local “dips” or “spikes” that deviate away from flatness. The exact role that viscosity plays in influencing evolution of the Rayleigh–Taylor process is quite complicated and subtle. Research has shown that surface tension is by far the dominant mode of stabilization compared to viscosity and that up to a high level of accuracy, the growth rate of the instabilities observed in an experiment only depends on the surface tension, the density-weighted average viscosity, and the effective acceleration, with relatively little direct influence from the viscosities and densities of the fluids.

^{7}In terms of a quantitative formula, the reduction of growth rate from viscosity has a relative size given by

*ν*is the kinematic viscosity of the fluid,

*t*is the time, and

*λ*is the critical wavelength discussed previously. Note that although the size of this effect is proportional to the viscosity, it will decrease very quickly when the surface tension is increased because

*λ*

^{2}, which is directly proportional to

*σ*, is in the denominator.

In Fig. 4, the cell is filled with oil, which has been dyed blue to improve visibility. As before, the cell is covered and turned upside down before removing the cover. Although this time some of the fluid drains out as the cover is removed, this quickly stabilizes, and the remaining fluid stays inside the cell against the force of gravity just as the water did. One final point to mention is that the cell that I have described has a vertically varying gap width, which introduces capillarity effects. It has been found that in a Hele-Shaw cell similar to the one I have described, there is actually an additional new destabilizing process for small wavenumbers at the liquid–gas interface inside the cell due to a combination of surface tension and vertically varying thickness. However, this additional instability is suppressed when the non-wetting fluid (i.e., the air) is in the thinner part of the cell. This occurs in our configuration, so we assume that capillary effects from variation in cell thickness do not introduce any additional instabilities that would significantly change the conclusions we arrived at above.^{8} Besides this, these new instabilities technically only come into play when both fluids are flowing in the *x-*direction, whereas in our case, there is no such flow.

It is anticipated that this simple experiment will hopefully motivate students to think not just about balancing of forces, but also the concept of Rayleigh–Taylor instabilities, which otherwise might seem rather esoteric with little relation to the real world. However, perturbations and instabilities are absolutely fundamental to physics, and it is certainly desirable that more students should learn about them. In this direction, although I have not yet used the demonstration in an introductory physics course, I have tested it on undergraduate engineering students who visited the fluid dynamics laboratory, and the students involved agreed that it was an interesting demonstration. The students also admitted that they had not thought much about the issues of instabilities of fluids before or the role that instabilities might play in nature. The students were also unsure whether the trick of holding the cell upside down would work with lower-viscosity oil even though it worked with water and were surprised to see that it did.

In terms of carrying out the experiment, I suggest that the fluid be introduced slowly with a plastic syringe and that the instructor not try to pour fluid into the cell from a beaker, which would be inefficient. The introduction of liquid can be done with a standard plastic syringe and does not require a precision syringe. Fluid that then leaks out and goes down the outside of the cell should ideally be cleaned so that the interior of the cell can be clearly seen during the demonstration. Once the cell has been filled, the cell has to be turned upside down quite quickly, but not so violently that all or almost all of the liquid is lost. The top of the cell can be covered with a finger that is then moved away from the cell. When the cell is upside down, the cell should be held steady and not shaken from side to side, which also causes leakage. In Figs. 2 and 3, I have shown that the cell is held pinched by one corner with a thumb and finger so that it is stable but the interior of the cell is not obscured from view. I recommend that the instructor practice this a few times before giving the demonstration in front of a class.

## REFERENCES

*Phys. Teach.*

*Phys. Teach*.

*J. Fluid Mech*.

*Physica D*

*Phys. Fluids*

*J. Fluid Mech*.

**Hollis Williams** *is an engineering PhD student at the University of Warwick. He is interested in various aspects of physics education and theoretical physics and has published articles on fluid dynamics, quantum mechanics, and particle physics.*