Nonlinear Dynamics of a Microswimmer in Poiseuille Flow

Abstract

We study the three-dimensional dynamics of a spherical microswimmer in cylindrical Poiseuille flow which can be mapped onto a Hamiltonian system. Swinging and tumbling trajectories are identified. In 2D they are equivalent to oscillating and circling solutions of a mathematical pendulum. Hydrodynamic interactions between the swimmer and confining channel walls lead to dissipative dynamics and result in stable trajectories, different for pullers and pushers. We demonstrate this behavior in the dipole approximation of the swimmer and with simulations using the method of multi-particle collision dynamics.PACS numbers: 47.63. Gd, 47.63.mf, Microswimmers often have to respond to fluid flow and confining boundaries, like sperm cells in the Fallopian tubes [1] or pathogens in blood vessels [2]. Artificial microswimmers constructed with the vision to act as drugdeliverers in the human body [3] would have to swim in narrow channels like arteries. Two properties influence the swimming in microchannels. On the one hand, vortices in flow reorient the swimming direction of microorganisms. In simple shear flow, for example, microswimmers tumble due to a constant flow vorticity [4]. Vortices in Poiseuille flow in combination with bottom-heaviness due to gravitation lead to stable orientations of swimming algae cells [5]. On the other hand, microorganisms swimming near surfaces are trapped by hydrodynamic interactions [6] and ultimately escape with the help of rotational diffusion [10]. Finally, bacteria in Poiseuille flow show a net-upstream flux at the walls due to the interplay of confinement and flow vorticity [7,8]. All these examples show there is genuine interest in understanding generic features of microorganisms and artificial swimmers in Poiseuille flow.In this letter we demonstrate that the dynamics of a simple spherical microswimmer in a cylindrical Poiseuille flow can be mapped onto a conservative dynamical system with the Hamiltonian as a constant of motion. In analogy to the oscillating and circling solutions of a mathematical pendulum, we discuss in detail the swinging and tumbling motion of the microswimmer in 2D and generalize them to three dimensions. Hydrodynamic interactions with the channel wall treated in the dipole approximation introduce dissipation and the microswimmer assumes specific stable swimming trajectories depending on its type as puller or pusher.We first introduce the geometry. We consider a pointlike microswimmer that moves with a constant intrinsic swimming speed v 0 in a cylindrical microchannel where a Poiseuille flow is imposed. Using a cylindrical coordinate system (ρ, ϕ, z) with the coordinate basis (ρ,φ,ẑ), the flow is given by v f = v f (1 − ρ 2 /R 2 Ch )ẑ, where v f is the maximum flow speed in the center of the channel [ Fig. 1(a)]. In the absence of noise the equations of mo- tion for the swimmer position r and orientationê are given bywhereChφ is the flow vorticity. The swimmer orientationê = e ρρ + e ϕφ + e zẑ has the components e ρ = − cos Θ sin Ψ, e ϕ = sin Θ, e z = − cos Θ co...

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“…These theories predict a negative value for the coefficient K for pushers at low shear rates, meaning the suspension can exhibit a lower viscosity than the suspending fluid. The theoretical assessment of shear viscosity relies on an assumed statistical representation of the orientations of the bacteria, captured by a Fokker-Plank equation and a kinematic model for the swimming trajectories [25,26].Despite the large number of theoretical studies, few experiments have been conducted. With bacillus subtilis (pushers) trapped in a soap film Sokolov et al[12] have shown that a vorticity decay rate could be associated with a strong decrease of shear viscosity in the presence of bacteria.…”

“…These theories predict a negative value for the coefficient K for pushers at low shear rates, meaning the suspension can exhibit a lower viscosity than the suspending fluid. The theoretical assessment of shear viscosity relies on an assumed statistical representation of the orientations of the bacteria, captured by a Fokker-Plank equation and a kinematic model for the swimming trajectories [25,26].…”

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