Session CQ: Biolocomotion I: Micro-swimming I

Diffusion vs. locomotion

In this talk we consider small organisms self-propelling in viscous fluids. We address theoretically and numerically the interplay between fluid-based locomotion and Brownian motion. Interesting dynamics occurs on time scales close to, or larger than, the inverse rotational diffusion constant for the organism, where the cells transition from swimming to diffusing. We derive results valid for all types of swimmers, including a new diffusion constant.   

Stirring by squirmers

We analyse a simple ``Stokesian squirmer'' model for the enhanced mixing due to swimming micro-organisms [1]. The model is based on a calculation of Thiffeault \& Childress [2], where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that for the viscous case the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate nonzero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly-swimming squirmers exhibit PDFs with exponential tails and a short-time superdiffusive regime, as found previously [3]. In our case, the exponential tails are due to ``sticking'' near the stagnation points on the squirmer's surface. \\[0pt] [1] Lin, Z., Thiffeault, J.-L. \& Childress, S., arxiv.org/abs/1007.1740\\[0pt] [2] Thiffeault, J.-L. \& Childress, S., \emph{Phys. Lett. A\/} {\bf 374}, 3487 (2010). \\[0pt] [3] Leptos, K. C., Guasto, J. S., Gollub, J. P., Pesci, A. I. \& Goldstein, R. E. {\em Phys. Rev. Lett.\/} {\bf 103}, 198103 (2009).   

Hydrodynamic interaction of two unsteady squirmers

The study of pair-wise interactions between swimming microorganisms is fundamental to the understanding of the rheological and transport properties of semi-dilute suspensions. In this study, the hydrodynamic interaction of two ciliated microorganisms is investigated numerically using a boundary-element method. The microorganisms are modeled as spherical squirmers that swim by time-dependent surface deformations. The results show that the inclusion of the unsteady terms in the ciliary propulsion model has a large impact on the trajectories of the interacting cells, and causes a significant change in scattering angles with potential important consequences on the diffusion properties of semi- dilute suspensions. Furthermore, the analysis of the shear stress acting on the surface of the microorganisms revealed that the duration and the intensity of the near-field interaction are significantly modified by the presence of unsteadiness. This observation may account for the hydrodynamic nature of randomness in some biological reactions, and supersedes the distinction between intrinsic randomness and hydrodynamic interactions, adding a further element to the understanding and modeling of interacting microorganisms.   

Breaking Symmetry with Gravity: Two-Link Swimming Using Buoyant Orientation

Swimming at low Reynolds number requires the swimmer's motion to be non-reciprocal in order to break the time-reversal symmetry of the equations of motion. We demonstrate that a neutrally buoyant swimmer can achieve net motion simply by introducing a static separation between the centers of mass and buoyancy. In the presence of gravity, the swimmer passively reorients toward its natural equilibrium without changing shape. We derive the governing equations for the system and explain how to control swimming direction with parameter and stroke selection and discuss swimming efficiency for various strokes.   

Low-Reynolds-number swimming in confined geometries

We present results of a theoretical investigation into the locomotion in confined geometries (e.g. near no-slip walls) of simple circular swimmers, in two dimensions, actuated by some imposed velocity profile in their surface. It is shown how use of the reciprocal theorem of Stokes flows together with knowledge of exact solutions for certain ``dragging problems'' can lead to the derivation of explicit dynamical systems for the swimmer's linear and angular velocities.   

Analytical model of a butterfly micro-swimmer

We propose a simple mechanical model consisting of two spheroids (wings) connected by a single hinge. Unlike micro-swimmers proposed so far, this model has just one hinge, but its motion allows two degree of freedom, corresponding to open-close and twisting. Its non-reciprocal operation cycles resembles conformational motions characteristic for real protein machines and similar to the propulsion pattern of a butterfly. The net velocity and the net stall force are calculated analytically and their dependence on the model parameters is discussed.   

Acceleration of swimming bacteria at ``zero" Reynolds number

Self-propelled objects can accelerate at ``zero'' Reynolds number Re. An incompressible fluid responds ``instantaneously'' and globally to the motion of bounding surfaces. When the propulsion mechanism of a bacterial body, a helical bundle of flagella, forms and rotates, the body accelerates according to $F = ma$, where $F$ is the sum of forces exerted on the body: drag plus the thrust of the flagella. The flagellar rotation instantly moves all the surrounding fluid, as does the body`s motion on, acting on its surround. The acceleration of bacteria stopped by a collision and beginning reverse swimming is important in the analysis of the jammed phase in the onset of Zooming BioNematic (ZBN). An instantaneous one step displacement has been used to analyze the flow surrounding swimming bacteria.   

Teaching Stokesian Dynamics to Swim

We develop a generic framework for modeling the hydrodynamic self-propulsion (i.e. swimming) of bodies (e.g. microorganisms) at low Reynolds number via Stokesian Dynamics simulations. In this framework, the swimming body is composed of many spherical particles constrained to form an assembly. We map the resistance tensor describing the hydrodynamic interactions among the particles onto that for the assembly. Specifying a particular swimming gate and imposing the condition that the swimming body is force- and torque-free determines the propulsive speed. This methodology projects directly onto 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: Taylor's helical swimmer, Purcell's three-link swimmer, the Taylor/Purcell swimming toroid, pushers and pullers, flagellates, ciliates and amoeba-like bodies undergoing large-scale deformation. We also make available an open source code with which the swimming of a body of arbitrary geometry and with arbitrary swimming gate may be simulated.   

Numerical simulations of microscale steady streaming using viscous vortex particle method

Microscale steady streaming created by localized cyclic boundary deformation provides an appealing option in microfluidic systems. High-frequency oscillatory body motion creates a large-scale circulatory motion in viscous fluid that is `steady' compared to the timescale of oscillation, and this overall net flow can be used for manipulating discrete objects in a micro system. A typical steady streaming motion generated by one or more unidirectionally oscillating cylindrical probes is considered. A high-fidelity numerical approach is presented for simulating such problems using a viscous vortex particle method. By focusing on vorticity, which is confined to a narrow stokes layer surrounding each probe, the method gains computational efficiency over a typical grid-based method. In particular, the large-scale streaming motion can be computed as a post-processing step, and little additional effort is required for multiple probes. Parametric studies of varying geometric arrangement was conducted and reveal the microscale flow structures and particle transport.   

Flow shear induced cross-stream migration by a green alga

Swimming and migration characteristics of micro-organisms in shear flows has overarching implications in formation of biological thin layers in aquatic ecosystems, design of bioreactors, and cell separations. Experiments are conducted in a microfluidic channel using digital holographic microscopy. A motile micro-alga, \textit{Dunaliella primolecta,} is studied in a laminar shear flow at maximum shear rates ranging from 0.1 to 25 s$^{-1}$. It is found that \textit{D. primolecta} cells aggregate in the direction of positive vorticity when a critical local shear rate of 5 s$^{-1}$ is reached. Unlike nonmotile cells, D\textit{. primolecta} in high shear flow do not rotate along the Jeffrey orbits, neither resumes the local vorticity of flow. The torque on cell body is counter-acted by the spatial alignment of beating flagella. It is speculated that under severe viscous stresses, motile cells ``opt'' to align themselves in the direction where the least stresses are experienced on cell wall. Beating of flagella, which prevents cells from assuming local flow vorticity, consequently propel them in the span wise direction and allow them to disperse only in a thin two-dimensional layer.