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Geometric constraints and optimization in externally driven propulsion

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Science Robotics  18 Apr 2018:
Vol. 3, Issue 17, eaas8713
DOI: 10.1126/scirobotics.aas8713
  • Fig. 1 Illustration of random clusters.

    Random aggregates generated using PC aggregation algorithm from 100 spherical beads with fractal dimension (A) Df = 1.5, (B) Df = 1.75, (C) Df = 2.0, and (D) Df = 2.25 and the fractal prefactor kf = 2.3.

  • Fig. 2 Results for random clusters.

    Random aggregates composed of 100 identical spherical beads with kf = 1.2 (red squares) and 2.3 (gray circles) generated with PC aggregation algorithm. Empty symbols stand for a mean value averaged over 300 configurations, bars correspond to SD from the mean, and full symbols denote the maximum values. (A) Longitudinal rotation anisotropy parameter, p. (B) Transverse anisotropy parameter, ε. (C) Maximum propulsion efficiency, δmax. (D) Three individual aggregates labeled in (B) and (C): ε ≈ 0.24 (cluster #1), δmax = 0.214 (cluster #2), and δmax = 0.177 (cluster #3).

  • Fig. 3 δ-Optimal propeller.

    Three projections of the optimal shape with an aspect ratio of 1:12 having δ = 0.246 obtained by unconstrained optimization. The shape is practically skew-symmetric and deviates considerably from an ideal helix resembling an arc with twisted ends. The leftmost image shows the “shish-kebab” optimal shape as obtained using a bead-based hydrodynamic model together with the approximate smooth envelope.

  • Fig. 4 δ*-Optimal propellers.

    Optimized shapes with an aspect ratio of 1:6 for twofold reflection symmetry (A) and skew symmetry (B) shown together with the principal axes of rotation (e1, e2, e3) and orientation of the optimal magnetization m (red arrow).

  • Fig. 5 Effect of slenderness on optimal shapes.

    δ*-Optimal structures (from Table 1) with twofold reflection symmetry (top) and skew symmetry (bottom). Propellers’ aspect ratio varies from left to right from 1:3 to 1:7 (top) and from 1:4 to 1:8 (bottom). The (red) arrow in the bottom panel corresponds to the direction of the magnetic moment m maximizing δ*.

  • Fig. 6 Velocity-frequency relationship.

    Scaled velocity-frequency curves for δ*-optimal skew-symmetric propeller with an aspect ratio of 1:4. Thick solid (red) curve stands for the optimal magnetization. Dashed (long blue dashes) curve stands for magnetization that optimizes δ at the step-out. Dotted (black) line corresponds to transverse (to the easy rotation axis e3) magnetization. Three colored solid and dashed (short dashes) curves stand for dual solutions that maximize δ at three intermediate frequencies. Thin solid line with the slope λmax is the upper bound on UZ0.

  • Table 1 δ-Optimal propellers.

    Comparison of optimal (geometric) propulsion efficiency, δmax, obtained using GA search for propellers with two mutually perpendicular symmetry planes, skew-symmetric propellers, and unconstrained shapes upon varying the filament aspect ratio/slenderness (width to length).

    Aspect ratioOne symmetry
    plane*
    Skew-symmetryUnconstrained
    1:60.1290.1530.154
    1:70.1440.1760.177
    1:80.1560.1950.196
    1:90.1670.2120.212
    1:100.1750.2250.224
    1:110.1830.2380.238
    1:120.1890.2460.246

    *The resulting shapes have twofold reflection symmetry.

    †This shape is shown in Fig. 3.

    • Table 2 δ*-Optimal propellers.

      Comparison of optimal propulsion efficiency, Embedded Image (×103), obtained using GA search for propellers with the same symmetries as in Table 1.

      Aspect ratioTwo symmetry
      planes
      Skew-symmetryUnconstrained*
      1:30.360
      1:40.3500.4140.421
      1:50.2900.3790.384
      1:60.2330.3390.341
      1:70.1920.2900.287
      1:80.1660.2500.241
      1:90.1400.2310.206
      1:100.1320.2070.186
      1:110.1110.1910.180

      *Cylindrical approximation, ε = 0.

      †This shape is used in Fig. 6.

      Supplementary Materials

      • robotics.sciencemag.org/cgi/content/full/3/17/eaas8713/DC1

        Supplementary Text

        Fig. S1. δ-Optimal propeller (Fig. 3).

        Fig. S2. δ*-Optimal propeller (Fig. 4A).

        Fig. S3. δ*-Optimal propeller (Fig. 4B).

        Fig. S4. Typical random aggregate.

        Fig. S5. Effect of magnetization on propulsion of a random aggregate.

        Table S1. Tabulated shape of the δ-optimal propeller (Fig. 3).

        Table S2. Tabulated shape of the δ*-optimal propeller (Fig. 4A).

        Table S3. Tabulated shape of the δ*-optimal propeller (Fig. 4B).

        Table S4. Tabulated shape of the δ*-optimal propeller (Fig. 6).

      • Supplementary Materials

        Supplementary Material for:

        Geometric constraints and optimization in externally driven propulsion

        Yoni Mirzae, Oles Dubrovski, Oded Kenneth, Konstantin I. Morozov, Alexander M. Leshansky*

        *Corresponding author. Email: isha{at}technion.ac.il

        Published 18 April 2018, Sci. Robot. 3, eaas8713 (2018)
        DOI: 10.1126/scirobotics.aas8713

        This PDF file includes:

        • Supplementary Text
        • Fig. S1. δ-Optimal propeller (Fig. 3).
        • Fig. S2. δ*-Optimal propeller (Fig. 4A).
        • Fig. S3. δ*-Optimal propeller (Fig. 4B).
        • Fig. S4. Typical random aggregate.
        • Fig. S5. Effect of magnetization on propulsion of a random aggregate.
        • Table S1. Tabulated shape of the δ-optimal propeller (Fig. 3).
        • Table S2. Tabulated shape of the δ*-optimal propeller (Fig. 4A).
        • Table S3. Tabulated shape of the δ*-optimal propeller (Fig. 4B).
        • Table S4. Tabulated shape of the δ*-optimal propeller (Fig. 6).

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