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# The Faraday-Instability: Pattern formation in vibrating thin liquid films

- If a horizontal vibration is combined with a normal one, the motion of a droplet on a horizontal plane can be controled.
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Abstract:

Since the first experimental observations of Michael Faraday in 1831 it is known that a vertically vibrating liquid may show an instability of its flat free surface with respect to oscillating regular surface patterns, normally squares. These squares oscillate with half of the driver’s frequency and are in resonance with gravity waves of the unforced liquid.

In this lecture, thin liquid films on a horizontal substrate are studied in the longwave approximation. The films are parametrically excited by mechanical vertical and/or horizontal oscillations. Inertia effects are taken into account and the standard thin film formulation is extended by a second equation for the horizontal mass flow rate. Linear results based on a damped Mathieu equation as well as fully nonlinear results found numerically will be presented. We obtain standard Faraday patterns such as oscillating squares or hexagons. For lateral vibrations, long wave instabilities in form of coarsening drops are detected. These drops can move in a certain direction if an additional vertical excitation is switched on (see figure).