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Modeling hydrodynamic self-propulsion with Stokesian Dynamics. Or teaching Stokesian Dynamics to swim
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Image of FIG. 1.
FIG. 1.

(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.

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

(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.

Image of FIG. 4.
FIG. 4.

(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.

Image of FIG. 5.
FIG. 5.

(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.

Image of FIG. 6.
FIG. 6.

(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.

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

(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.

Image of FIG. 9.
FIG. 9.

(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.

Image of FIG. 10.
FIG. 10.

(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.

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

(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.

Image of FIG. 13.
FIG. 13.

(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.


Generic image for table
Table I.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Modeling hydrodynamic self-propulsion with Stokesian Dynamics. Or teaching Stokesian Dynamics to swim