From Falling Paper Strips, Tossed Coins To Settling Snowflakes

By Majid Hassan Khan

Did you notice money raining down in part three of the Spanish TV series “Money Heist” (Spanish: La casa de Papel, “The House of Paper”) on Netflix? A blimp flew over Madrid and showered money. These falling paper bills fluttered, tumbled and followed random trajectories while descending to the streets. The behaviour has a similarity with the falling of leaves from a tree, descent of snowflakes and path of tossed coins in fountains and wishing wells.

It is a wonder why Newton never addressed the behaviour of falling leaves from the apple tree in his garden at Woolsthorpe Manor. He did perform experiments by dropping glass spheres and inflated hog bladders from a cathedral in London. A look at the influence of air around falling leaves or freely falling paper strips or the behaviour of water around a flipped coin in a pool was to come later. It took another two hundred years before a 22-year-old James Clerk Maxwell paid heed to the physics around falling paper slips. He observed the flutter and explained the changes in pressure distribution around the strip and even referred to the resistance offered to the paper by the air around it. Similar is the nature of fall when coins are tossed in fountains, pools and rivers. Falling coins rotate, flutter, tumble and descend on different trajectories.

A common term used to emphasize the similarity in physics between air and water is ‘fluid’. The story of fluid dynamics can be woven using ‘inertia’ and ‘friction’ of the flowing medium as the central characters. The idea of friction in fluids was proposed by many scientists who contributed to the evolution of the physics of fluids. Newton being the first, followed by Poiseuille and Hagen. Later Stokes further emphasised the notion of friction in fluids. The physics of fluids was evolving, and newer ideas were included to aid a better understanding. When Reynolds demonstrated the onset of turbulence in flow and Prandtl explained the effect of viscosity close to the surfaces of an object immersed in the flow, newer insights in the understanding of external flows started taking shape. Now it is widely known that the storyline is influenced by the geometry of the object around which the flow happens. Inertia and viscosity swap the roles of being the main protagonist based on the size and shape of the object around which the flow is observed.

With recent advances in fluid dynamics, we can explain that the flow physics of a freely falling paper strip is influenced by its initial state, the size and symmetry of the falling paper strip, the density difference of falling paper and air, and the viscosity of the fluid. The flows separating at the edges of the strip roll in vortices leaving an unsteady trail behind the falling paper. This unsteadiness causes an imbalance in the forces acting on the paper strip. The extent of the force imbalance on the paper strip due to the flowing fluid around it causes it to fall steadily, flutter, rotate, tumble, drift, oscillate or stably fall. All these fall behaviours are direct outcomes of the interaction of the air and the paper strip. Similar interactions happen when a coin is tossed in a wishing well or a pool of water. Flow separates at the edge of the coin, vortices are formed and shed in the wake. The coin falls steadily, flutters, rotates or shows chaotic behaviour. The physics of these simple freely falling objects when extended to ice particles and snowflakes helps understand the fall behaviour of snowflakes and the nature of flow around them which aids in explaining their growth rates.

To appreciate the fall behaviour of a freely falling paper Maxwell used a paper strip with sides two inches long and one inch wide. You can try dropping paper strips of varied sizes and observing their fall. I suggest you cut a paper strip 1 cm wide and about 5 to 10 cm long and drop it from a height with the larger side slightly tilted away from the horizontal (and the flat face of the strip looking at you). You will be amazed by the fall behaviour. After the initial settling the paper will orient itself in one direction of fall and will start spinning around an axis parallel to the longest side (an example of autorotation).  If you get the feel of the fall, you can use different shapes and enjoy the diverse fall behaviour of simple, harmless paper pieces and marvel at the complexity of the fluid dynamics of freely falling paper strip.

The understanding of fluid object interaction is of immense use in environmental science, atmospheric physics, insect flights and industrial needs. In meteorology, clarity about the fluid dynamics of falling objects helps establish the fate of ice particles and snowflakes in the atmosphere and in clouds.

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