Research ArticleMATERIALS SCIENCE

Significant and stable drag reduction with air rings confined by alternated superhydrophobic and hydrophilic strips

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Science Advances  01 Sep 2017:
Vol. 3, no. 9, e1603288
DOI: 10.1126/sciadv.1603288
  • Fig. 1 Schematics and pictures describing the physical principles governing the interactions between air bubbles and textured surfaces in the Taylor-Couette apparatus.

    (A) Schematic of the test apparatus. (B) Schematic cross-sectional profile of an air bubble on a vertical plate. (C) SEM of the jointed superhydrophobic (pho) and hydrophilic (phi) surfaces. (D) Cross-sectional view of air bubbles sticking on superhydrophobic circular spots at the center of the hydrophilic plate—shown as a reference to introduce the concept of air bubble pinning (note that this is not the test configuration discussed in this investigation). The upper air contact angle (defined as the angle measured through the air, where an air-water interface meets a solid surface), θu, of the air bubble ranges from ~27° to ~140° (from top to bottom) because of the surface energy barrier needed for the bubble to unpin from the substrate and due to the discontinuity in the surface energy. (E) Inner rotor of the TC cell showing the actual air rings generated in our experimental apparatus.

  • Fig. 2 Experimental visualization and comparison with theoretical predictions of the air ring shape in static configurations.

    (A) Bubble geometry in the meridian plane as predicted by simulations (red curves, see section S1) and measured through optical acquisitions (dark region, see Methods) for a superhydrophobic strip of 4 mm and for the internal cylinder radius ri = 12.5 mm. (B) Simulated bubble shape as a function of the hydrophobic strip size, wpho, for constant bubble thickness, TH; the graph inset (right) shows the outcomes of the simulations in terms of predicted bubble axial length, wb, and configuration as a function of the hydrophobic strip size, wpho, for different values of the bubble thickness, TH.

  • Fig. 3 Theoretical investigation of the effect of Bond number (wb/lc)2, where Embedded Image, on the bubble formation and shape evolution obtained numerically (see section S1).

    (Left) Small Bond numbers, with wpho = 2 mm. (Right) Large Bond numbers, with wpho = 6 mm. The curves in the graphs show, for a fixed wpho, the normalized bubble growth along the axial direction, wb/wpho (red curve), and the air contact angles (θu and θl) as a function of the air injected volume; the latter is expressed in terms of the bubble equivalent thickness TH. The larger the Bond number, the larger the air contact angle hysteresis of the air ring, which corresponds to a decreasing value of receding air contact angle; thus, for large Bond numbers, a larger pinning strength can be provided because of the increasing value of cosθl − cosθu.

  • Fig. 4 Experimental investigation of the effect of the Couette Reynolds number ReC, air ring radial (TH), and axial (wpho/wphi) width on measured torque and DR obtained experimentally.

    (Left) Variation of the dimensionless torque Embedded Image against ReC (red dots) for the uncoated inner rotor and for different bubble radial thickness (including the case of no air injection). The solid red lines represent the fit to the scaling Embedded Image, and n is the scaling exponent. The solid and dashed black lines represent the empirical power law describing the dimensionless torque for the transition and low-turbulent regime, respectively (the calculation methods are presented in section S3). Inset: Variation of the DR against ReC. Results are measured for a constant value of wpho = 4 mm. (Right) DR as a function of the hydrophobic strip size for different equivalent radial widths (TH) of the air ring but at constant ReC = 1320.

  • Fig. 5 Theoretical investigation of the effect of flow strips, air ring behavior, and geometrical configuration on torque obtained numerically.

    (Top) Dimensionless torque Embedded Image (see section S4) as a function of the bubble thickness TH for different texture area density α and Bond numbers. In the calculations, we have considered the case wpho = wb (bubble covering the whole hydrophobic strip), whereas the superhydrophobic/hydrophilic reference texture is characterized by Embedded Image and Embedded Image. Increasing the Bond number (green path) has no effect on the drag torque. Instead, increasing the texture area density α (red path) determines a net reduction of the water velocity on the bubble surface, resulting in a stronger reduction in the generated drag torque. The bubble thickness TH (blue path) has an influence similar to α but weaker in terms of reduction of the water speed on the bubble wall. (Bottom) Dimensionless drag torque reduction (and effective azimuthal wall velocity Embedded Image) as a function of the hydrophobic axial size wpho (here, wphowb) for TH = 0.16 mm (corresponding to the red curve in Fig. 2B). The wetting condition in the portion of the hydrophobic strip, which is uncovered by the air ring (width wphowb), is governed by the slip parameter β (0 ≤ β ≤ 1). We have analyzed two conditions, namely, no slip (β = 0) and partial slip (β = 0.043, see section S4). The DR does not increase linearly with wpho due to the occurrence of the partial bubble contact on the hydrophobic strip. Instead, a maximum in the DR exists, supporting the experimental results of Fig. 4 (right).

  • Fig. 6 Experimental results showing the velocity profiles and vorticity fields for different texture configurations.

    (A) Radial profiles of the mean azimuthal velocity. The radial position and the mean azimuthal velocity are normalized by the gap width and the rotation velocity of the inner rotor, which are (rri)/d and uθri, respectively. The mean azimuthal velocity profiles near the inner cylinder wall cannot be plotted because the measurements are impeded by the presence of the air rings. The horizontal solid line in (B) represents the axial position of the radial profile of the azimuthal velocity above the air ring. The dashed line in (B) represents the axial position of the radial profile of the azimuthal velocity above the hydrophilic strip. (B) Radial-axial vorticity fields of the gap flow measured at ReC = 660. The axial position is normalized by the annular gap width, which is z/d. First column: smooth inner cylinder; second and third columns: TH = 0.31 mm and TH = 0.62 mm of the air rings with width of 4 mm; fourth column: TH = 0.31 mm of the air ring with width of 2 mm. Note that the bubble-wall water speed reduction (proportional to DR) obtained when increasing TH from 0.31 to 0.62 mm (red to pink points) is much smaller than the reduction encountered from TH = 0 to 0.31 mm (black to red points), in agreement with the theoretical predictions.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/9/e1603288/DC1

    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 Formula 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.

    movie S1. Visualization of the flow regimes of smooth and patterned rotors at ReC = 1320.

    movie S2. Visualization of the stable air rings with different thickness in all the experiment conditions with Reynolds numbers up to 1320.

    movie S3. Visualization of the air ring stability in the high turbulent regime, adopting a newly developed Taylor-Couette apparatus.

    Reference (54)

  • 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.

    Files in this Data Supplement:

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