Science Advances

Supplementary Materials

This PDF file includes:

  • section S1. Analytical model for the bubble ring
  • section S2. Estimation of the different contributions to drag and study of the
    stability of air rings and DR in the presence of surface grooves
  • section S3. On the flow regimes encountered in the experiments
  • section S4. Laminar flow dynamics and DR induced by the annular bubble theory
  • section S5. Torque contribution of the top and bottom cylinder flat surfaces
  • section S6. The dynamic stability of the air rings
  • fig. S1. Schematic representation of the bubble.
  • fig. S2. Results of the simulation of the bubble growth induced by air injection,
    for w = 4.0 mm, ri = 12.5 mm, and zero rotational speed.
  • fig. S3. Results of the simulation: Dimensionless bubble pressure p (black lines) and the dimensionless bubble axial length wb/w (red lines) as a function of the hydrophobic strip radial size tb.
  • fig. S4. Results of the simulation: Dimensionless bubble pressure (black line) and bubble volume (red line) as a function of the dimensionless angular velocity.
  • fig. S5. Experimental results for the drag torque measurements with embedded circular bubbles.
  • fig. S6. Normalized cylinder torque T/(ρυ2L) as a function of the Couette Reynolds number ReC, in a log-log plot.
  • fig. S7. Normalized cylinder torque T/(ρυ2L) as a function of the Couette Reynolds number ReC, in a log-log plot.
  • fig. S8. Results of the simulation: Normalized cylinder torque C as a function of the slippage area density α, for η = 0.1 (dotted line), η = 0.5 (dashed line), and η = 0.735 (solid line, referring to the experiments).
  • fig. S9. Results of the simulation.
  • fig. S10. Torque contribution of the top and bottom cylinder flat surfaces.
  • fig. S11. Air rings’ dissolving process when the inner rotor is rotating in a 10 cm × 10 cm × 120 cm water tank.
  • Reference (54)

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Other Supplementary Material for this manuscript includes the following:

  • movie S1 (.mov format). Visualization of the flow regimes of smooth and patterned rotors at ReC = 1320.
  • movie S2 (.mov format). Visualization of the stable air rings with different thickness in all the experiment conditions with Reynolds numbers up to 1320.
  • movie S3 (.mov format). Visualization of the air ring stability in the high turbulent regime, adopting a newly developed Taylor-Couette apparatus.

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