Americans drink an average of 3.1 cups of coffee per day; for many people, the popular beverage is a morning necessity. When carrying a liquid, common sense says to walk slowly and refrain from overfilling the container. But when commuters rush out the door with coffee in hand, chances are their hastiness causes some of the hot liquid to slosh out of the cup. The resulting spills, messes, and mild burns undoubtedly counteract coffee's savory benefits.
The authors derive their inspiration from an Ig Nobel prize-winning paper describing a basic mechanical model that investigates the results of walking backwards while carrying a cup of coffee. They use both Newton's laws of physics and the basic properties of hydrodynamics to employ a so-called "paradigm" configuration, which explains how a cradle introduces an extra degree of freedom that in turn modifies the liquid's response. "The paradigm model contains the same mechanics as the pendulum but is simpler to write down," Ockendon said. "We found some experimental results on the paradigm model, which meant we could make some direct comparisons."
Variables in the initial model represent (i) a hand moving around a fixed position, (ii) the frequency of walking, typically between 1-2 Hertz, and (iii) a spring connecting the shaking hand to the mug, which slides on the table's smooth surface. Ockendon and Ockendon are most interested in the spring's effect on the motion of the liquid.
The authors discover that including a string or a pendulum between the container and the carrying hand (the forcing mechanism) lessens the rigidity and dramatically decreases the lowest resonant frequency, thus diminishing sloshing for almost all frequencies. "Our model shows that, compared to an unpivoted tank, the amplitude of the lowest resonant response will be significantly reduced, provided the length of the pendulum is greater than the length of the tank," Ockendon said.
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Ockendon, H., & Ockendon, J.R. (2017). How to Mitigate Sloshing. SIAM Review , 59(4), 905-911.
SIAM Review