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^{1,a)}, A. Concha

^{1,b)}, R. Wood

^{1}and L. Mahadevan

^{1,2}

### Abstract

We consider the dynamics of a pendulum made of a rigid ring attached to an elastic filament immersed in a flowing soap film. The system shows an oscillatory instability whose onset is a function of the flow speed, length of the supporting string, the ring mass, and ring radius. We characterize this system and show that there are different regimes where the frequency is dependent or independent of the pendulum length depending on the relative magnitude of the added-mass. Although the system is an infinite-dimensional, we can explain many of our results in terms of a one degree-of-freedom system corresponding to a forced pendulum. Indeed, using the vorticity measured via particle imaging velocimetry allows us to make the model quantitative, and a comparison with our experimental results shows we can capture the basic phenomenology of this system.

This work was partially funded by the National Science Foundation (NSF) award No. CCF-0926148. We thank R. Zarinshenas for his help with preliminary experiments.

### Key Topics

- Torque
- 9.0
- Rotating flows
- 8.0
- Rheology and fluid dynamics
- 7.0
- Torque measurement
- 5.0
- Velocimetry
- 4.0

## Figures

(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.

doi: 10.1063/1.4800057.2.

(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.

doi: 10.1063/1.4800057.2.

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.

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.

(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.

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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