^{1}, Kenneth S. Breuer

^{1}and Thomas R. Powers

^{1,2}

### Abstract

We apply the boundary-element method to Stokes flows with helical symmetry, such as the flow driven by an immersed rotating helical flagellum. We show that the two-dimensional boundary integral method can be reduced to one dimension using the helical symmetry. The computational cost is thus much reduced while spatial resolution is maintained. We review the robustness of this method by comparing the simulation results with the experimental measurement of the motility of model helical flagella of various ratios of pitch to radius, along with predictions from resistive-force theory and slender-body theory. We also show that the modified boundary integral method provides reliable convergence if the singularities in the kernel of the integral are treated appropriately.

This work was supported by National Science Foundation Grant No. CBET-0854108.

I. INTRODUCTION

II. HELICAL SYMMETRY AND THE BOUNDARY INTEGRAL METHOD

III. BODY-CENTERLINE COORDINATES

IV. NUMERICAL SIMULATIONS OF A HELICAL SWIMMER

V. COMPARISON WITH EXPERIMENTS AND THEORIES

A. Experimental system

B. Resistive force theory

C. Slender-body theory

1. Lighthill's slender-body theory

2. Johnson's slender-body theory

D. Comparison

VI. OTHER APPLICATIONS

A. Confined geometry and non-rigid body

B. Non-uniform geometry

VII. DISCUSSION

### Key Topics

- Boundary integral methods
- 14.0
- Boundary element methods
- 9.0
- Linear equations
- 6.0
- Stokes flows
- 6.0
- Singularity theory
- 5.0

## Figures

The value f(x) of a vector field at an arbitrary location on a surface D with helical symmetry can be determined from the vector f(x C ) on a given circumference C 0 through a rotation about the axis x 3. The rotation angle Δφ is the angle between the projections of the surface normals at positions x and x C in the x 1–x 2 plane. The dashed curve shows a contour along which vector fields f(x) have the same magnitude but vary in orientation. The inset shows a view of the helical filament along its axis of symmetry.

The value f(x) of a vector field at an arbitrary location on a surface D with helical symmetry can be determined from the vector f(x C ) on a given circumference C 0 through a rotation about the axis x 3. The rotation angle Δφ is the angle between the projections of the surface normals at positions x and x C in the x 1–x 2 plane. The dashed curve shows a contour along which vector fields f(x) have the same magnitude but vary in orientation. The inset shows a view of the helical filament along its axis of symmetry.

Body-centerline coordinate system. (a) A coordinate system that follows the body-centerline of a filament {q i } is given by two Euler angles θ, φ. Here q 3 is tangential to the body-centerline, and the circumference C 0 is a circle and normal to q 3. (b) A view of the circumference C 0 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.

Body-centerline coordinate system. (a) A coordinate system that follows the body-centerline of a filament {q i } is given by two Euler angles θ, φ. Here q 3 is tangential to the body-centerline, and the circumference C 0 is a circle and normal to q 3. (b) A view of the circumference C 0 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.

Force density on a rotating tethered helix with a/Γ = 0.026, θ = π/4, κ = 40, and N α = 128. (a) The force density along a circumference approaches the exact value with a second-order convergence with respect to N φ. (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 nn , σ nt , σ nb are normal, tangential, binormal to the circumference C 0, respectively.

Force density on a rotating tethered helix with a/Γ = 0.026, θ = π/4, κ = 40, and N α = 128. (a) The force density along a circumference approaches the exact value with a second-order convergence with respect to N φ. (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 nn , σ nt , σ nb are normal, tangential, binormal to the circumference C 0, respectively.

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

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

Comparison of the boundary-integral results with experimental measurements and theoretical predictions for the normalized swimming speed V 0/ΩR of helices of various pitch angles θ and two different aspect ratios, (a) a/Γ = 0.013 and (b) a/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 a/Γ → 0, Gray and Hancock's theory, Lighthill's approximations for finite a, 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. 24 .

Comparison of the boundary-integral results with experimental measurements and theoretical predictions for the normalized swimming speed V 0/ΩR of helices of various pitch angles θ and two different aspect ratios, (a) a/Γ = 0.013 and (b) a/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 a/Γ → 0, Gray and Hancock's theory, Lighthill's approximations for finite a, 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. 24 .

(a) Normalized free-swimming speed V 0/ΩR of a helical filament in a tube of radius A due to rigid rotation ‘○' and transverse wave ‘•'. Here, the geometry of helical filament is given by θ = 0.16π and a/Γ = 0.013. (b) Effect of finite length on helical swimming using linear interpolation [Eq. (37) ] with N C = 9 and N φ = 32, and comparison with the experiment. The result obtained by enforcing full helical-symmetry (N C = 1) is also shown as a comparison.

(a) Normalized free-swimming speed V 0/ΩR of a helical filament in a tube of radius A due to rigid rotation ‘○' and transverse wave ‘•'. Here, the geometry of helical filament is given by θ = 0.16π and a/Γ = 0.013. (b) Effect of finite length on helical swimming using linear interpolation [Eq. (37) ] with N C = 9 and N φ = 32, and comparison with the experiment. The result obtained by enforcing full helical-symmetry (N C = 1) is also shown as a comparison.

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

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

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 a/Γ = 0.01, pitch angle θ = π/4, and number of pitches κ = 40. (a) Relative error of the force components ΔF/F as a function of the number of grids N φ with fixed N α. (b) Re-plot of data in (a) against Δl α/Δl φ, the ratio between the physical length scales regarding grids Δα and Δφ. The error starts to increase when N φ is sufficiently large or when Δl φ becomes less than Δl α, as shadowed in gray. (c) Relative error of the force components as a function of the number of grids N α. (d) Re-plot of data in (c) against Δl φ/Δl α. Similar to (a) and (b), the solution diverges when N α is sufficiently large or when Δl α < Δl φ, as shadowed in gray. The dashed lines in each figure indicate second-order convergence.

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 a/Γ = 0.01, pitch angle θ = π/4, and number of pitches κ = 40. (a) Relative error of the force components ΔF/F as a function of the number of grids N φ with fixed N α. (b) Re-plot of data in (a) against Δl α/Δl φ, the ratio between the physical length scales regarding grids Δα and Δφ. The error starts to increase when N φ is sufficiently large or when Δl φ becomes less than Δl α, as shadowed in gray. (c) Relative error of the force components as a function of the number of grids N α. (d) Re-plot of data in (c) against Δl φ/Δl α. Similar to (a) and (b), the solution diverges when N α is sufficiently large or when Δl α < Δl φ, as shadowed in gray. The dashed lines in each figure indicate second-order convergence.

Motility of a force-free helix V 0 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 ΔV 0/V 0 as a function of the number of grids N φ. (b) Re-plot of data in (a) against Δl α/Δl φ. (c) Relative error as a function of the number of grids N α. (d) Re-plot of data in (c) against Δl φ/Δl α. 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 Δl α/Δl φ. The dashed lines in each figure indicate second-order convergence.

Motility of a force-free helix V 0 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 ΔV 0/V 0 as a function of the number of grids N φ. (b) Re-plot of data in (a) against Δl α/Δl φ. (c) Relative error as a function of the number of grids N α. (d) Re-plot of data in (c) against Δl φ/Δl α. 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 Δl α/Δl φ. The dashed lines in each figure indicate second-order convergence.

(a) Numerical convergence of the free-swimming V 0 of a helix (with the same geometry as shown in Figs. 8 and 9 ) with fixed ratio Δl φ/Δl α. 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.

(a) Numerical convergence of the free-swimming V 0 of a helix (with the same geometry as shown in Figs. 8 and 9 ) with fixed ratio Δl φ/Δl α. 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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