Lay Language Papers 2008

The Fluid Trampoline 

Dip a circular frame into a bottle of soapy water, so that the frame is spanned by a thin soap film. Now, instead of blowing a bubble, turn the frame into the horizontal, and place it on a loud speaker so that it can be vibrated up and down. If the film is stationary, a drop placed on its surface will merge with the film. However, if the film is vibrated, the droplet may bounce up and down as if on a trampoline (Figure 1).

As the frequency and amplitude of the vibrating film are varied, a number of distinct bouncing states are observed. Simple bouncing states (m, 1) are characterized by a bouncing period that is m times the forcing period of the frame. Complex bouncing states are characterized by two integers (m, n) such that one period of the trajectory corresponds to m forcing periods and n bounces of the droplet. Increasing the vibration amplitude while keeping the frequency fixed prompts the evolution from simple to complex modes via period doubling transitions that ultimately culminate in chaotic bouncing states. Time sequences of a number of bouncing states are presented in Figure 2.

The film shape and resulting force on the droplet may be uniquely expressed in terms of the drop’s vertical position. When the droplet is in contact with the film, the film responds as a linear spring with spring constant proportional to the surface tension. This insight allows us to describe the droplet dynamics in terms of a single simple equation. Numerical solution of this equation allowed us to reproduce all observed bouncing behaviors, and to rationalize all of the complex behavior observed, including the period-doubling transitions to chaos.

The theoretical description of the fluid trampoline is considerably simpler than that of the dripping faucet, and similar to that of an elastic bouncing ball, the chaotic nature of both systems having been thoroughly characterized by previous investigators. As such, in terms of both experiment and analysis, the fluid trampoline represents the simplest fluid mechanical chaotic oscillator yet explored.

Figure 1: A droplet bouncing on a horizontal soap film.

Figure 2: A number of bouncing states observed experimentally. The images are deduced by extracting a thin vertical slice along the drop centerline from each frame of a video sequence, then placing those slices side by side. The result is the vertical trajectory of the drop as a function of time. Simple bouncing states a) (1,1), b) (2,1), c) (3,1) were all observed at the same forcing. d) The complex bouncing state (3,3). e) The chaotic trajectory of a bouncing droplet.