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(a) Schematic of experimental setup: Soap solution at constant pressure head flows through a nozzle between two parallel nylon wires that are a width W apart and over a length 1 m to form a soap film. An annular ring of external radius R = 1.1 × 10−2 m hangs from a flexible string that is pivoted outside the soap film. (b) For a given flow rate, the pendulum exhibits oscillatory motion above a critical length L c . Multiple images of oscillations over several periods record the pendulum amplitude A. (c) A longer pendulum exhibits more complex oscillations, such as one with a node as marked with the blue circle. (d) Velocity field superposed on raw images shows when it is vertical and (e) when it is at its extremal position (enhanced online). [URL: http://dx.doi.org/10.1063/1.4800057.1] [URL: http://dx.doi.org/10.1063/1.4800057.2]doi: 10.1063/1.4800057.1.
Dimensionless critical length L c /L 0 vs. dimensionless ring radius R/L 0, (where L 0 is the smallest length of a fiber without an attached ring that spontaneously oscillates in a flow) at constant mass M R and flow rate, obtained experimentally (solid squares). The solid line corresponds to the best fit obtained from the first four points, with L c /L 0 = 1 + 1.2R/L 0.
(a) Dimensionless frequency ω/ω0 vs. dimensionless length L/R (bottom horizontal axis) when M R > M A (M R = 1.48 × 10−4 kg). Experimental data are shown as solid circles and a fit based on Eq. (2) is shown as a solid line; ω0 = 9.44 s−1 is the natural frequency for a gravity pendulum of length L = 10R. Top horizontal axis shows the regime when M R < M A , with the ring radius held constant at R = 1.1 × 10−2 m while increasing mass M R by adding plasticine, with M ρ = πR 2ρh being the mass of a disc of liquid of radius R. Experiment (solid squares) and theory fit (dashed line) from Eq. (4) . (b) Dimensionless frequency ω/ω r vs. rescaled velocity based on Eq. (4) in the added mass regime M A > M R & M s . Here ω r = 19.08 s−1 is the offset frequency corresponding to the mode where the ring oscillates around its pivot.
Comparison between experimentally obtained angular dynamics shown with open circles (red) and the simple theory based on a driven oscillator model (solid black curve) given by Eq. (5) with the following parameters: M R = 0.12 × 10−3 kg, h = 10 μm, V = 1 m/s, L = 5.5 × 10−2 m, γ = 80 s−1, R = 1.1 × 10−2 m. The numerical curve is intentionally offset by Δθ = 0.1 for clarity.
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