Micro-swimmers with hydrodynamic interactions

Efficient sliding locomotion of three-link bodies

We study the efficiency of sliding locomotion for three-link bodies with prescribed joint angle motions. The bodies move with no inertia, under dry (Coulomb) friction that is anisotropic (different in the directions normal and tangent to the links) and directional (different in the forward and backward tangent directions). Friction coefficient space can be partitioned into several regions, each with distinct types of efficient kinematics. These include kinematics resembling lateral undulation with very anisotropic friction, small-amplitude reciprocal kinematics, very large-amplitude kinematics near isotropic friction, and kinematics that are very asymmetric about the flat state. In the two-parameter shape space, zero net rotation for elliptical trajectories occurs mainly with bilateral or antipodal symmetry. These symmetric subspaces have about the same peak efficiency as the full space but with much smaller dimension. Adding the second or third harmonics greatly increases the numbers of local optimal for efficiency, but only modestly increases the peak efficiency. Random ensembles with higher harmonics have efficiency distributions that peak near a certain nonzero value and decay rapidly up to the maximum efficiency. A stochastic optimization algorithm is developed to compute optima with higher harmonics. These are simple closed curves, sharpened versions of the elliptical optima in most cases, and achieve much higher efficiencies mainly for small normal friction. With a linear (viscous) resistance law, the optimal trajectories are similar in much of the friction coefficient space, and relative efficiencies are much lower except with very large normal friction.

Dynamical density functional theory with hydrodynamic interactions in confined geometries

We study the dynamics of colloidal fluids in both unconfined geometries and when confined by a hard wall. Under minimal assumptions, we derive a dynamical density functional theory (DDFT) which includes hydrodynamic interactions (HI; bath-mediated forces). By using an efficient numerical scheme based on pseudospectral methods for integro-differential equations, we demonstrate its excellent agreement with the full underlying Langevin equations for systems of hard disks in partial confinement. We further use the derived DDFT formalism to elucidate the crucial effects of HI in confined systems.

A stochastic micro-machine inspired by bacterial chemotaxis

Inspired by bacterial chemotaxis, we propose and study the dynamics of a 3D hydrodynamical search-machine at micrometer scale. Chemotactic memory of the proposed system that is borrowed from bacteria, allows it to search the fluid medium and find the required target. As the motion in micron size length scale is dominated by random forces, we analyze the statistical properties of the model. Mean square displacements, orientational correlation functions and also the chemotactic index (CI) of the system are investigated in detail. Because of the chemotactic memory, the system shows superdiffusing displacements in all directions and the diffusion exponents are anisotropic for the directions along or perpendicular to a preferred direction given by the gradient of attractant molecules transmitted by the target.

Optimization and small-amplitude analysis of Purcell's three-link microswimmer model

This work studies the motion of Purcell's three-link microswimmer in viscous flow, by using perturbation expansion of its dynamics under small-amplitude strokes. Explicit leading-order expressions and next-order correction terms for the displacement of the swimmer are obtained for the cases of a square or circular gait in the plane of joint angles. The correction terms demonstrate the reversal in movement direction for large stroke amplitudes, which has previously only been shown numerically. In addition, asymptotic expressions for Lighthill's energetic efficiency are obtained for both gaits. These approximations enable calculating optimal stroke amplitudes and swimmer's geometry (i.e. ratio of links' lengths) for maximizing either net displacement or Lighthill's efficiency.

Swarming in viscous fluids: Three-dimensional patterns in swimmer- and force-induced flows

We derive a three-dimensional theory of self-propelled particle swarming in a viscous fluid environment. Our model predicts emergent collective behavior that depends critically on fluid opacity, mechanism of self-propulsion, and type of particle-particle interaction. In "clear fluids" swimmers have full knowledge of their surroundings and can adjust their velocities with respect to the lab frame, while in "opaque fluids" they control their velocities only in relation to the local fluid flow. We also show that "social" interactions that affect only a particle's propensity to swim towards or away from neighbors induces a flow field that is qualitatively different from the long-ranged flow fields generated by direct "physical" interactions. The latter can be short-ranged but lead to much longer-ranged fluid-mediated hydrodynamic forces, effectively amplifying the range over which particles interact. These different fluid flows conspire to profoundly affect swarm morphology, kinetically stabilizing or destabilizing swarm configurations that would arise in the absence of fluid. Depending upon the overall interaction potential, the mechanism of swimming ( e.g., pushers or pullers), and the degree of fluid opaqueness, we discover a number of new collective three-dimensional patterns including flocks with prolate or oblate shapes, recirculating pelotonlike structures, and jetlike fluid flows that entrain particles mediating their escape from the center of mill-like structures. Our results reveal how the interplay among general physical elements influence fluid-mediated interactions and the self-organization, mobility, and stability of new three-dimensional swarms and suggest how they might be used to kinetically control their collective behavior.