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Flutter and Tumble: The Physics of Falling Paper

A thin steel strip exhibits flutter (side-to-side motion) as it falls through a narrow water-filled tank. Produced in recent experiments, such movies have advanced physicists' understanding of what determines the path of a sheet of paper as it falls to the ground.

Despite gravity's undeniable attraction, not all falling objects travel straight downwards: the piece of paper you drop in the hall or the tree leaves fluttering to the ground in the autumn are familiar experiences. Yet it may surprise you to know that these common occurences still present a scientific challenge. The complex downward motion of paper can provide insights into such things as how to prevent aircrafts from stalling and how insects can fly with a minimum of effort.

How paper falls to the ground is a process impossible to describe exactly with the laws of physics. If you drop an 8 1/2" by 11" sheet from the top of your desk, it is nearly impossible to predict with certainty exactly where the paper will land. This is because paper falls through air. Air is a fluid, the term in physics to describe any gas or liquid. And the behavior of any fluid can be incredibly complex. The equations describing the motion of a fluid can be written out on a single line, yet their properties are a topic for entire books, a subject for specialists, and an open question in physics. It is often simply impossible to find the relevant solutions to these equations; all physicists can hope for is to find approximate solutions to them on a case-by-case basis. Understanding fluid dynamics is one of the last great classical problems in physics.

The equations for the motion of a fluid, known as the Navier-Stokes equations, relate the velocity and pressure at each point in the fluid for each instant in time. The equations also depend on the given values of density and viscosity of the fluid. The effects of the velocity on itself lead to a sort of feedback effect that makes the problem infinitely complex after a short amount of time. Put somewhat more concretely, this feedback effect gives the equations a "nonlinear " property: Small changes in the velocity at one point in the fluid will affect the overall velocity distribution of the fluid in a way that quickly becomes unpredictable over time. Furthermore, tiny, inevitable uncertainties in knowledge about the starting conditions of the fluid quickly add to the unpredictability. This is why mathematics cannot exactly predict weather patterns, and it's also why they cannot exactly describe the relatively simpler question of the path of a sheet of paper as it falls to the ground. So all physicists can hope for is to approximate the behavior of the air around the sheet of paper.

Nonetheless, scientists beginning with James Clerk Maxwell in the mid-1800s have attempted to come up with mathematically based theories that attempt to describe, at least approximately, how paper falls to the ground. Exploring this question would bring great deal of insights into the behavior of air and other fluids. Yet even modern simulations stick to the simplified case of paper falling in two dimensions. Experiments have studied falling paper in three dimensions, but the results are so complicated that they tend to yield descriptive insights rather than hard numbers.

New experiments at the Weizmann Institute in Israel have tested these 2-D theories and have, for what is believed to be the first time, yielded quantitative results that can be compared to the predictions of the theories. Their experiments provide information on what factors of the air and paper are important in determining its fall.

In the experiments, researchers dropped thin strips of metal or plastic into a thin fluid-filled tank, which forced the strips to fall in a two-dimensional plane. Thus the researchers did not drop paper itself, but stiffer counterparts such as plastic, brass, and steel strips, about 30 different sizes and weights in all. They also didn't drop them through air, but instead used one of three fluids: water, petroleum ether, and a glycerol/water mixture. Water is 1000 times more dense than the air, but pound for pound, it offers about 10 times less resistance to things which try to move through it. Physicists would say that water had 10 times less "kinematic viscosity." Similarly, the strips were quite a bit heavier than a sheet of paper. Nonetheless, by observing the motion of these falling strips, the results could be extrapolated to a much lighter piece of paper that falls through a less dense volume of air.

What happened when the researchers dropped the strips? They exhibit two types of motion. In the first type, the strip moved back and forth from side-to-side. The researchers called this "flutter." In the second type, the strip rotated end over end as it fell to the ground. The researchers termed this "tumble."

What determined whether the falling strips predominantly oscillated from side-to-side (flutter) or rotated end-over-end (tumble) was the Froude number, in this case the ratio of the time it takes for the strip to fall its own length to the time it takes for the strip to move from side to side. Longer or lighter strips, which have a low Froude number (like an 8.5 x 11" page) flutter while smaller or heavier strips, which have a higher Froude number (e.g., a business card) tend to tumble. (Try it yourself!)

(Interestingly, the Froude number is also used in the field of biomechanics, to predict the maximum speed one can walk before one must start running. In that case, the Froude number describes the interplay between the swinging energy of the legs and the pull of the earth's gravity on the legs. The Froude number was originally defined to describe the performance of sea-going ships!)

So what useful information did the Froude number provide in the falling strips experiment? The researchers discovered that Froude numbers below a certain critical value (0.7, to be exact) meant the strip would flutter; above a certain value it would tumble. For values just above the critical number, the strip would sometimes reverse its rotation direction randomly as it tumbled, but this was not observed enough times for the experimentalists to confirm that the culprit was chaos, in which small variations in the starting conditions lead in a well-defined way to unpredictable outcomes. The experiment was inconclusive on this point, but recent numerical simulations by Tanabe and Kaneko (1994) and also Aref and Jones (1993) predict chaos in the fall of paper.

In turn, the researchers found that the flutter or tumble was greatly affected by the density, and not the viscosity, of the fluid in the tank. Indeed, the "drag force" that the fluid exerts to oppose the motion of the falling strip does not originate from viscosity, as some existing theories suggest, but rather from the density of the fluid; this "Eiffel drag" is what exerts a force on the top of the Eiffel Tower as air flows past it.

This image shows swirls in the fluid ("vortices") forming behind the falling strip. These vortices form when the strip changes the direction of its side-to-side motion. Such vortices also form when an ascending airplane begins to stall. In contrast, insects may use these vortices to their advantage, to help them stay aloft during flight.

Viscosity, however, plays a major role by causing the falling strips to generate closed whirls of fluids, known as "vortices," which appear when the strip switches the direction of its side to side oscillation. Such vortices, whose role is currently being investigated, are believed to stall airplanes during takeoff but may be exploited by insects to enable them to fly with great efficiency. Birds and insects fly by flapping their wings, a much different process from the flight of jet planes, and researchers still do not understand exactly how and why flapping allows these creatures to stay aloft with such great efficiency.

This question of how insects employ air vortices will be a future area of investigation for at least one of the researchers in the experiment. So, perhaps the physics of how paper falls to the ground will help biologists to figure out the secret of insect flight!

Thanks to Andrew Belmonte, Hagai Eisenberg, and Elisha Moses for the figures and much of the text.

This research will be reported by Belmonte, Eisenberg, and Moses in an upcoming issue of Physical Review Letters.