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Helical swimming in Stokes flow using a novel boundary-element method
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10.1063/1.4812246
/content/aip/journal/pof2/25/6/10.1063/1.4812246
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/6/10.1063/1.4812246
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

The value () of a vector field at an arbitrary location on a surface with helical symmetry can be determined from the vector ( ) on a given circumference through a rotation about the axis . The rotation angle Δφ is the angle between the projections of the surface normals at positions and in the plane. The dashed curve shows a contour along which vector fields () have the same magnitude but vary in orientation. The inset shows a view of the helical filament along its axis of symmetry.

Image of FIG. 2.
FIG. 2.

Body-centerline coordinate system. (a) A coordinate system that follows the body-centerline of a filament { } is given by two Euler angles θ, φ. Here is tangential to the body-centerline, and the circumference is a circle and normal to . (b) A view of the circumference in the body-centerline frame. Radius ρ and angle α are the associated polar coordinates in the body-centerline frame. The dots show the grid points on a single circumference.

Image of FIG. 3.
FIG. 3.

Force density on a rotating tethered helix with /Γ = 0.026, θ = π/4, κ = 40, and = 128. (a) The force density along a circumference approaches the exact value with a second-order convergence with respect to . (b) A three dimensional distribution of the stress tensor σ is computed from using the helical symmetry. The inset illustrates the direction of three components of σ: σ, σ, σ are normal, tangential, binormal to the circumference , respectively.

Image of FIG. 4.
FIG. 4.

Helix geometry. The helix rotates with rate Ω and advances at constant speed along its axis of symmetry.

Image of FIG. 5.
FIG. 5.

Comparison of the boundary-integral results with experimental measurements and theoretical predictions for the normalized swimming speed of helices of various pitch angles θ and two different aspect ratios, (a) /Γ = 0.013 and (b) /Γ = 0.026. The graphs show experimental measurements (circles and squares) and the predictions of resistive-force theory (R.F.T.) in the limit of /Γ → 0, Gray and Hancock's theory, Lighthill's approximations for finite , Lighthill and Johnson's slender-body theories, and the boundary-element method with reduced dimension (B.E.). The inset shows a zoom-in view of the rectangular window. Adapted from Ref. .

Image of FIG. 6.
FIG. 6.

(a) Normalized free-swimming speed of a helical filament in a tube of radius due to rigid rotation ‘○' and transverse wave ‘•'. Here, the geometry of helical filament is given by θ = 0.16π and /Γ = 0.013. (b) Effect of finite length on helical swimming using linear interpolation [Eq. (37) ] with = 9 and = 32, and comparison with the experiment. The result obtained by enforcing full helical-symmetry ( = 1) is also shown as a comparison.

Image of FIG. 7.
FIG. 7.

Singularity reduction. Boundary integral on a meshed surface near the singular origin (shown as the cropped section) is valued analytically. The integral on the rest of the surface is performed numerically using the trapezoidal rule.

Image of FIG. 8.
FIG. 8.

Numerical convergence of the hydrodynamic force per unit arc-length . The helical geometry is given by the ratio of the radius of the cross-section to its arc length per pitch /Γ = 0.01, pitch angle θ = π/4, and number of pitches κ = 40. (a) Relative error of the force components Δ/ as a function of the number of grids with fixed . (b) Re-plot of data in (a) against Δ , the ratio between the physical length scales regarding grids Δα and Δφ. The error starts to increase when is sufficiently large or when Δ becomes less than Δ , as shadowed in gray. (c) Relative error of the force components as a function of the number of grids . (d) Re-plot of data in (c) against Δ . Similar to (a) and (b), the solution diverges when is sufficiently large or when Δ < Δ , as shadowed in gray. The dashed lines in each figure indicate second-order convergence.

Image of FIG. 9.
FIG. 9.

Motility of a force-free helix and its numerical convergence. Helical filament with the same geometry as shown in Fig. 8 are used. (a) Relative error of the free-swimming speed Δ / as a function of the number of grids . (b) Re-plot of data in (a) against Δ . (c) Relative error as a function of the number of grids . (d) Re-plot of data in (c) against Δ . Similar dependencies as shown in convergence of force components (Fig. 8 ) are observed: the simulation result becomes divergent if grids along φ (or α) are too dense, as characterized by the ratio Δ . The dashed lines in each figure indicate second-order convergence.

Image of FIG. 10.
FIG. 10.

(a) Numerical convergence of the free-swimming of a helix (with the same geometry as shown in Figs. 8 and 9 ) with fixed ratio Δ . The dashed line shows a first order convergence. (b) Convergence of the free-swimming with first-order error subtracted using Eq. (C2) . The dashed line indicates a third order convergence.

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/content/aip/journal/pof2/25/6/10.1063/1.4812246
2013-06-28
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Helical swimming in Stokes flow using a novel boundary-element method
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/6/10.1063/1.4812246
10.1063/1.4812246
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