^{1}, John F. Brady

^{1}, Rachel S. Moore

^{1}and ChE 174

^{1,a)}

### Abstract

We develop a general framework for modeling the hydrodynamic self-propulsion (i.e., swimming) of bodies (e.g., microorganisms) at low Reynolds number via Stokesian Dynamics simulations. The swimming body is composed of many spherical particles constrained to form an assembly that deforms via relative motion of its constituent particles. The resistance tensor describing the hydrodynamic interactions among the individual particles maps directly onto that for the assembly. Specifying a particular swimming gait and imposing the condition that the swimming body is force- and torque-free determine the propulsive speed. The body’s translational and rotational velocities computed via this methodology are identical in form to that from the classical theory for the swimming of arbitrary bodies at low Reynolds number. We illustrate the generality of the method through simulations of a wide array of swimming bodies: pushers and pullers, spinners, the Taylor/Purcell swimming toroid, Taylor’s helical swimmer, Purcell’s three-link swimmer, and an amoeba-like body undergoing large-scale deformation. An open source code is a part of the supplementary material and can be used to simulate the swimming of a body with arbitrary geometry and swimming gait.

The motivation for this work arose from a special topics course, ChE 174, on self-propulsion at low Reynolds number given in the spring of 2010 at the California Institute of Technology—hence the title “*Teaching Stokesian Dynamics to swim*.” Participants were Lawrence Dooling (pushers and pullers), Nicholas Hoh (pushers and pullers), Jonathan Choi (spinners) and Roseanna Zia (toroids). We thank them for their contributions to the course and this work. This work was supported in part by NSF Grant Nos. CBET 0506701 and CBET 074967.

I. INTRODUCTION

II. THE HYDRODYNAMICS OF SELF-PROPELLED MICROORGANISMS

III. THE HYDRODYNAMICS OF PARTICULATE DISPERSIONS

IV. STOKESIAN DYNAMICS

V. THE MECHANICS OF RIGID ASSEMBLIES

VI. SWIMMING VIA STOKESIAN DYNAMICS

VII. EXAMPLES

A. Toy models

1. Pushers and pullers—The implicit swimming gait

2. Spinners—The explicit swimming gait

B. Purcell’s three-link swimmer

C. Taylor’s helical swimmer

D. Amoeba-like swimming

VIII. EXTENSIONS

### Key Topics

- Hydrodynamics
- 53.0
- Tensor methods
- 40.0
- Reynolds stress modeling
- 18.0
- Kinematics
- 17.0
- Hydrological modeling
- 16.0

## Figures

(Color online) A pusher and a puller (dipoles) and a push-puller (quadrupole) are illustrated. Because the fluid cannot flow freely in the gap between the particles in response to the implicit swimming gait, a pressure gradient forms and drives the swimmer through the fluid.

(Color online) A pusher and a puller (dipoles) and a push-puller (quadrupole) are illustrated. Because the fluid cannot flow freely in the gap between the particles in response to the implicit swimming gait, a pressure gradient forms and drives the swimmer through the fluid.

The swimming speed and rate of energy dissipation of pushers/pullers are plotted. These swimmers are constructed as chains of *N* particles, the first *M* of which have the implicit swimming gait **E** _{1}. The rate of dissipation is so large because the inter-particle spacing is 2.01 *a*. As such, the non-affine deformation of neighboring particles induces large stresslets and thus a large rate of dissipation.

The swimming speed and rate of energy dissipation of pushers/pullers are plotted. These swimmers are constructed as chains of *N* particles, the first *M* of which have the implicit swimming gait **E** _{1}. The rate of dissipation is so large because the inter-particle spacing is 2.01 *a*. As such, the non-affine deformation of neighboring particles induces large stresslets and thus a large rate of dissipation.

(Color online) The most primitive spinner is built from two spherical particles rolling at fixed separation as though calendaring the fluid between them. In the limit that , the rate of energy dissipation diverges logarithmically with respect to the separation. This is a consequence of the strong lubrication forces experienced when the gap between the spheres is narrow.

(Color online) The most primitive spinner is built from two spherical particles rolling at fixed separation as though calendaring the fluid between them. In the limit that , the rate of energy dissipation diverges logarithmically with respect to the separation. This is a consequence of the strong lubrication forces experienced when the gap between the spheres is narrow.

(Color online) The toroidal swimmer moves in the same direction as the fluid flowing through its center. When the mean, minor radius of the toroid is used to rescale the swimming speed and aspect ratio, the swimming speed of the rigid assembly and a solid toroid collapse on top of one another. The rate at which fluid is pumped through its center is the most important factor in setting the swimming speed of the toroid. Therefore, the mean, minor radius sets the appropriate velocity scale , the average surface velocity of the rigid assembly.

(Color online) The toroidal swimmer moves in the same direction as the fluid flowing through its center. When the mean, minor radius of the toroid is used to rescale the swimming speed and aspect ratio, the swimming speed of the rigid assembly and a solid toroid collapse on top of one another. The rate at which fluid is pumped through its center is the most important factor in setting the swimming speed of the toroid. Therefore, the mean, minor radius sets the appropriate velocity scale , the average surface velocity of the rigid assembly.

(Color online) Purcell’s three-link swimmer is an illustration of the “simplest animal that can swim that way.” At least two degrees of freedom in the parameter space characterizing the configuration of any swimmer are needed in order to achieve net propulsion. Three possible paths through phase space are shown. The reciprocal path gives rise to no net motion while the standard path in which one rudder is held fixed while the other flaps in alternating fashion is the path studied via Stokesian Dynamics. Note, if the three links are all of the same length, then as these configurations prevent the rudders from colliding. The excluded regions of phase space are marked explicitly on the diagram.

(Color online) Purcell’s three-link swimmer is an illustration of the “simplest animal that can swim that way.” At least two degrees of freedom in the parameter space characterizing the configuration of any swimmer are needed in order to achieve net propulsion. Three possible paths through phase space are shown. The reciprocal path gives rise to no net motion while the standard path in which one rudder is held fixed while the other flaps in alternating fashion is the path studied via Stokesian Dynamics. Note, if the three links are all of the same length, then as these configurations prevent the rudders from colliding. The excluded regions of phase space are marked explicitly on the diagram.

(Color online) The swimming speed of Purcell’s three-link swimmer is plotted as a function of the maximum stroke angle, , and the number of particles composing each link, *N*. The speed is normalized by the tip speed of a rudder as it carries out its stroke. An additional normalization by a factor depending on the logarithm of the length of the rudder, denoted , which is the couple between rotation and torque for slender bodies. The curve is the swimming speed predicted by Becker *et al.* ^{26} using slender body theory for rudders moving with a constant torque difference relative to the body.

(Color online) The swimming speed of Purcell’s three-link swimmer is plotted as a function of the maximum stroke angle, , and the number of particles composing each link, *N*. The speed is normalized by the tip speed of a rudder as it carries out its stroke. An additional normalization by a factor depending on the logarithm of the length of the rudder, denoted , which is the couple between rotation and torque for slender bodies. The curve is the swimming speed predicted by Becker *et al.* ^{26} using slender body theory for rudders moving with a constant torque difference relative to the body.

Taylor’s helical swimmer as depicted in the film *Low Reynolds Number Flow*. As shown in the film, the fish-like swimmer (middle of the figure) cannot swim in high viscosity fluids because of time-reversal symmetry.

Taylor’s helical swimmer as depicted in the film *Low Reynolds Number Flow*. As shown in the film, the fish-like swimmer (middle of the figure) cannot swim in high viscosity fluids because of time-reversal symmetry.

(Color online) Taylor’s helical swimmer is constructed from spherical particles of radius *a*. The top helix has a clockwise orientation (right-handed as depicted) while the bottom helix has a counter-clockwise orientation. The spherical particles are colored in quadrants to aid visualization of the relative orientations.

(Color online) Taylor’s helical swimmer is constructed from spherical particles of radius *a*. The top helix has a clockwise orientation (right-handed as depicted) while the bottom helix has a counter-clockwise orientation. The spherical particles are colored in quadrants to aid visualization of the relative orientations.

(Color online) The velocities of helical swimmers satisfying the slender-body and long wavelength conditions () are normalized by the velocity of Taylor’s swimming sheet and plotted as a function of the ratio of swimmer length to helical arc length, denoted *s*. This results in a collapse of the data onto a universal curve that the slender-body theory matches in functional form over a wide range of *s*.

(Color online) The velocities of helical swimmers satisfying the slender-body and long wavelength conditions () are normalized by the velocity of Taylor’s swimming sheet and plotted as a function of the ratio of swimmer length to helical arc length, denoted *s*. This results in a collapse of the data onto a universal curve that the slender-body theory matches in functional form over a wide range of *s*.

(Color online) The shape of the amoeboid is constrained by conditions of constant volume and no self-interaction. The valid phase space in variables *Y*(*t*) and *Z*(*t*) is depicted as well as the circular phase space orbits studied herein. The shape of a particular amoeba is shown evolving in time (steps 1–8) with increased thickness and decreased opacity both corresponding to longer times. This amoeba arises from a circular phase space orbit with *r* = 0.5.

(Color online) The shape of the amoeboid is constrained by conditions of constant volume and no self-interaction. The valid phase space in variables *Y*(*t*) and *Z*(*t*) is depicted as well as the circular phase space orbits studied herein. The shape of a particular amoeba is shown evolving in time (steps 1–8) with increased thickness and decreased opacity both corresponding to longer times. This amoeba arises from a circular phase space orbit with *r* = 0.5.

The swimming speed of amoebas subject to a circular phase space orbit with radius *r* is compared for swimmers composed of 32, 52, 100, and 152 particles. The number of particles, *N*, was chosen as a multiple of four for symmetry purposes. The speed of two-dimensional amoebas is also depicted. It is smaller than that of the monolayer swimmers because fluid may flow freely above, below, and through the swimmers composed of spherical particles.

The swimming speed of amoebas subject to a circular phase space orbit with radius *r* is compared for swimmers composed of 32, 52, 100, and 152 particles. The number of particles, *N*, was chosen as a multiple of four for symmetry purposes. The speed of two-dimensional amoebas is also depicted. It is smaller than that of the monolayer swimmers because fluid may flow freely above, below, and through the swimmers composed of spherical particles.

(Color online) A flexible oar is constructed by connecting many rigid assemblies (dumb-bells) with linear springs. The swimming gait emerges through articulation of the springs by changing their rest length.

(Color online) A flexible oar is constructed by connecting many rigid assemblies (dumb-bells) with linear springs. The swimming gait emerges through articulation of the springs by changing their rest length.

(Color online) The swimming speed of a flexible oar is decays with increasing frequency of articulation. The swimming results from a traveling wave-type articulation of the spring rest lengths forming the connection among the swimmer’s constituent dumb-bells. The spring constant is denoted *k* and serves as a factor for normalization.

(Color online) The swimming speed of a flexible oar is decays with increasing frequency of articulation. The swimming results from a traveling wave-type articulation of the spring rest lengths forming the connection among the swimmer’s constituent dumb-bells. The spring constant is denoted *k* and serves as a factor for normalization.

## Tables

The number of particles, helical wave number, and helical wavelength studied for Taylor’s swimming helices. All combinations of these variables were employed.

The number of particles, helical wave number, and helical wavelength studied for Taylor’s swimming helices. All combinations of these variables were employed.

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