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

Wiezel, Oren; Or, Yizhar

doi:10.1098/rspa.2016.0425

Cited by 33 publications

(33 citation statements)

References 43 publications

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“…like three-link flagella [22][23][24] , three-sphere swimmers 25 , squirmers 26 , necklace-like propellers 27 and undulating magnetic systems 28 . However, to date Lighthill's efficiency has been only experimentally estimated in few biological systems 29,30 , where the complex flagellar dynamics precludes the possibility to directly determine the relevant degrees of freedom of the microswimmer, and indirect methods such as optical tweezers, are used.…”

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confidence: 99%

“…In this formulation, the propulsion speed is controlled by three dimensionless parameters, the ratio between the particles sizes δ = R/L, α ind = µ 0 m n /(R 3 B x ) that compares the magnetic field induced by the ferromagnet on the paramagnetic particle with the external field, and the susceptibility χ of the paramagnetic particle. The propeller dynamics can be solved perturbatively 24,28,34 for a weak oscillating transverse field, B y /B x ≡ ε 1.…”

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Direct measurement of Lighthill's energetic efficiency of a minimal magnetic microswimmer

et al. 2019

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The realization of artificial microscopic swimmers able to propel in viscous fluids is an emergent research field of fundamental interest and vast technological applications. For certain functionalities, the efficiency of the microswimmer in converting the input power provided through an external actuation into propulsive power output can be critical. Here we use a microswimmer composed by a self-assembled ferromagnetic rod and a paramagnetic sphere and directly determine its swimming efficiency when it is actuated by a swinging magnetic field. Using fast video recording and numerical simulations we fully characterize the dynamics of the propeller and identify the two independent degrees of freedom which allow its propulsion. We then obtain experimentally the Lighthill's energetic efficiency of the swimmer by measuring the power consumed during propulsion and the energy required to translate the propeller at the same speed. Finally, we discuss how the efficiency of our microswimmer could be increased upon suitable tuning of the different experimental parameters.The realization of faster and smaller micro/nanopropellers is an active topic with direct applications in the emerging fields of drug delivery 1-3 , microsurgery 4,5 and lab-on-a-chip technol-† Electronic Supplementary Information (ESI) available: Two experimental videos (.WMF) showing the dynamics of the nanorod-colloid micropropeller played at different speed. See DOI: 00.0000/00000000. ‡These authors contributed equally to this work.Although other measures of the efficiency have been proposed 19,21 , especially to account for collective ciliary motions, the Lighthill energetic efficiency remains the standard measure to account for swimming efficiency of single propellers at the microscale. This parameter has been employed in different theoretical works to analyze the performance of simple artificial designs J o u r n a l N a me , [ y e a r ] , [ v o l . ] , 1-6 | 1 arXiv:1910.00855v1 [cond-mat.soft]

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Direct measurement of Lighthill's energetic efficiency of a minimal magnetic microswimmer

et al. 2019

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“…Moreover, due to the linearity of Stokes equations, both the forces and the velocities acting on the swimmer depend linearly on ḟ . Thus, the instantaneous power density expended by the swimmer is a quadratic form in ḟ , with coefficients depending on f (see [43] for concrete expressions using resistive force theory). We may therefore write the total energy E expended by the swimmer during the stroke as…”

Section: Power Expendedmentioning

confidence: 99%

“…We now further analyze the motion of Purcell's swimmer using leading-order terms as studied in [43]. The sinusoidal gait in (19) for the two joint angles can be rewritten as…”

Section: Optimal Gaits For Purcell's Three-link Swimmermentioning

confidence: 99%

“…That is, it has a stroke amplitude of e and phase difference of j. Using resistive force theory and the leading order expansion in e, as explained in [43], the leading-order expressions for displacement Δx and energy E in a cycle under the gait ( Therefore, for sinusoidal gaits of the form (23), our constrained optimization of minimizing the energy to cover a given displacement reduces to minimizing E/Δx. This gives an optimization problem for a scalar function of j.…”

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Energy-optimal strokes for multi-link microswimmers: Purcell's loops and Taylor's waves reconciled

et al. 2019

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Micron-scale swimmers move in the realm of negligible inertia, dominated by viscous drag forces. In this paper, we formulate the leading-order dynamics of a slender multi-link (N-link) microswimmer assuming small-amplitude undulations about its straight configuration. The energy-optimal stroke to achieve a given prescribed displacement in a given time period is obtained as the largest eigenvalue solution of a constrained optimal control problem. Remarkably, the optimal stroke is an ellipse lying within a two-dimensional plane in the (N -1)-dimensional space of joint angles, where N can be arbitrarily large. For large N, the optimal stroke is a traveling wave of bending, modulo edge effects. If the number of shape variables is small, we can consider the same problem when the prescribed displacement in one time period is large, and not attainable with small variations of the joint angles. The fully nonlinear optimal control problem is solved numerically for the cases N=3 (Purcell's three-link swimmer) and N=5 showing that, as the prescribed displacement becomes small, the optimal solutions obtained using the small-amplitude assumption are recovered. We also show that, when the prescribed displacements become large, the picture is different. For N=3 we recover the non-convex planar loops already known from previous studies. For N=5 we obtain non-planar loops, raising the question of characterizing the geometry of complex high-dimensional loops.deformation. Corresponding biological examples are, respectively, the rotary motion of helical bacterial flagellar bundles, the different shapes of cilia in the power and in the recovery part of one stroke, the beating of a eukaryotic flagellum causing the propagation of bending waves along the length of the flagellum. In fact, bending waves propagating along cilia/flagella and shape modulation during one stroke are used not only for propulsion by micro-organisms, but also for transport inside organs in humans and other higher organisms [17][18][19]. The second example is the one with more connections with the study of minimal artificial swimmers (i.e. swimmers with only two internal degrees of freedom to control shape such as the three-link swimmer of Purcell [3,20], the three-sphere swimmer of Najafi and Golestanian [21], and others). The third example is possibly the most thoroughly exploited paradigm in the fabrication of micro-swimmer prototypes, often through the action of an external magnetic field on a flexible filament [8,10,22,23].Patterns of optimal actuation have been investigated independently, for each of these three examples, with a variety of analytical and numerical methods. For the case of flagellar and ciliary propulsion, these include [2, 24-28] leading to recognizing, for example, helical shapes as the optimal ones for filaments in three dimensions, and smoothed saw-tooth traveling waves as the optimal wave forms for the planar beating of a one-dimensional flagellum or for a planar sheet. In the limit of small amplitudes, the latter reduces to Tayl...

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Path Following for Purcell’s Swimmers: An Event-Based Control Approach

et al. 2022

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

Cited by 33 publications

References 43 publications

Direct measurement of Lighthill's energetic efficiency of a minimal magnetic microswimmer

Direct measurement of Lighthill's energetic efficiency of a minimal magnetic microswimmer

Energy-optimal strokes for multi-link microswimmers: Purcell's loops and Taylor's waves reconciled

Path Following for Purcell’s Swimmers: An Event-Based Control Approach

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