# Cornell Physicists Solve 'Falling Paper Problem'

When you are walking across the Arts quad and your papers fall, why do they all fly in different, unpredictable directions? This is the essence of the “falling paper problem” and research by Prof. Z. Jane Wang, theoretical and applied mechanics.

Wang and Umberto Pesavento, grad, started researching two and a half years ago and were later joined by Anders Andersen, a postdoctoral associate; their findings were published in “Falling Paper: Navier-Stokes Solutions, Models of Fluid Forces and Center of Mass Elevation.”

In their quest to discover the motions of free-falling objects, the researchers used rigid metal strips of equal length, but varying thicknesses. They released the strips from a small clamp in a tank of water and tracked the freefall of the metal strips. The three motions observed were moving side to side, also known as fluttering, rolling over, also known as tumbling, and a combination of the two. Thicker strips tumbled and thinner ones fluttered when released in the tank. Their paths could be predicted after many trials. However, there were strips of moderate thickness that displayed a combination of fluttering and tumbling, resulting in very different pathways in each trial.

“Just by looking at how leaves fall you can see that there are forces produced by air acting on it — unsteady aerodynamic forces. However, the forces on falling paper are rather different than the forces on airplanes,” Wang said.

Andersen explained that regarding an airplane, air flows around the wing independently of time. The motion in this study was dependent on time. Also, regarding a falling object, one cannot assume that the fluid circulation around the object is constant; in fact, the researchers found that the fluid circulation changed with time since vortices — spiral motions of fluid within a limited area — were formed and shed, thereby making the motion time-dependent. This experiment aimed to control the outside variables, such as air and flexibility.

Using rigid metal strips in a tank allowed the researchers to investigate the fluid force acting on the objects.

This research on the motion and forces acting on falling objects is relevant in nature.

“This could help us understand basic mechanisms of how an insect flies, how leaves fall, or how fish swim,” Pesavento said. Andersen said that trees could have strategies for dispersing their seeds. One example is helicopter seeds which tumble in a similar way to the metal strips in a tank, but on an axis. He said that if a tree is in an area with little wind, the optimal free-falling motion would be tumbling, while in a place with wind, it would be most beneficial for seeds to stay in the air as long as possible. The researchers also found in the study that tumbling is slower than just parachuting down.

Archived article by Vanessa Hoffman
Sun Staff Writer