Optimal feeding is optimal swimming for all P\'eclet numbers

Cells swimming in viscous fluids create flow fields which influence the transport of relevant nutrients, and therefore their feeding rate. We propose a modeling approach to the problem of optimal feeding at zero Reynolds number. We consider a simplified spherical swimmer deforming its shape tangentially in a steady fashion (so-called squirmer). Assuming that the nutrient is a passive scalar obeying an advection-diffusion equation, the optimal use of flow fields by the swimmer for feeding is determined by maximizing the diffusive flux at the organism surface for a fixed rate of energy dissipation in the fluid. The results are obtained through the use of an adjoint-based numerical optimization implemented by a Legendre polynomial spectral method. We show that, to within a negligible amount, the optimal feeding mechanism consists in putting all the energy expended by surface distortion into swimming - so-called treadmill motion - which is also the solution maximizing the swimming efficiency. Surprisingly, although the rate of feeding depends strongly on the value of the P\'eclet number, the optimal feeding stroke is shown to be essentially independent of it, which is confirmed by asymptotic analysis. Within the context of steady actuation, optimal feeding is therefore found to be equivalent to optimal swimming for all P\'eclet numbers.Comment: 14 pages, 6 figures, to appear in Physics of Fluid

Autophoretic locomotion from geometric asymmetry

Among the few methods which have been proposed to create small-scale swimmers, those relying on self-phoretic mechanisms present an interesting design challenge in that chemical gradients are required to generate net propulsion. Building on recent work, we propose that asymmetries in geometry are sufficient to induce chemical gradients and swimming. We illustrate this idea using two different calculations. We first calculate exactly the self-propulsion speed of a system composed of two spheres of unequal sizes but identically chemically homogeneous. We then consider arbitrary, small-amplitude, shape deformations of a chemically-homogeneous sphere, and calculate asymptotically the self-propulsion velocity induced by the shape asymmetries. Our results demonstrate how geometric asymmetries can be tuned to induce large locomotion speeds without the need of chemical patterning.Comment: 17 pages, 10 figure

Phoretic self-propulsion at finite P\'eclet numbers

Phoretic self-propulsion is a unique example of force- and torque-free motion on small scales. The classical framework describing the flow field around a particle swimming by self-diffusiophoresis neglects the advection of the solute field by the flow and assumes that the chemical interaction layer is thin compared to the particle size. In this paper we quantify and characterize the effect of solute advection on the phoretic swimming of a sphere. We first rigorously derive the regime of validity of the thin-interaction layer assumption at finite values of the P\'eclet number (Pe). Within this assumption, we solve computationally the flow around Janus phoretic particles and examine the impact of solute advection on propulsion and the flow created by the particle. We demonstrate that although advection always leads to a decrease of the swimming speed and flow stresslet at high values of the P\'eclet number, an increase can be obtained at intermediate values of Pe. This possible enhancement of swimming depends critically on the nature of the chemical interactions between the solute and the surface. We then derive an asymptotic analysis of the problem at small Pe allowing to rationalize our computational results. Our computational and theoretical analysis is accompanied by a parallel study of the role of reactive effects at the surface of the particle on swimming (Damk\"ohler number).Comment: 27 pages, 15 figures, to appear in J. Fluid Mec

Unsteady feeding and optimal strokes of model ciliates

The flow field created by swimming microorganisms not only enables their locomotion but also leads to advective transport of nutrients. In this paper we address analytically and computationally the link between unsteady feeding and unsteady swimming on a model microorganism, the spherical squirmer, actuating the fluid in a time-periodic manner. We start by performing asymptotic calculations at low P\'eclet number (Pe) on the advection-diffusion problem for the nutrients. We show that the mean rate of feeding as well as its fluctuations in time depend only on the swimming modes of the squirmer up to order Pe^(3/2), even when no swimming occurs on average, while the influence of non-swimming modes comes in only at order Pe^2. We also show that generically we expect a phase delay between feeding and swimming of 1/8th of a period. Numerical computations for illustrative strokes at finite Pe confirm quantitatively our analytical results linking swimming and feeding. We finally derive, and use, an adjoint-based optimization algorithm to determine the optimal unsteady strokes maximizing feeding rate for a fixed energy budget. The overall optimal feeder is always the optimal steady swimmer. Within the set of time-periodic strokes, the optimal feeding strokes are found to be equivalent to those optimizing periodic swimming for all values of the P\'eclet number, and correspond to a regularization of the overall steady optimal.Comment: 26 pages, 11 figures, to appear in Journal of Fluid Mechanic

Efficiency optimization and symmetry-breaking in a model of ciliary locomotion

A variety of swimming microorganisms, called ciliates, exploit the bending of a large number of small and densely-packed organelles, termed cilia, in order to propel themselves in a viscous fluid. We consider a spherical envelope model for such ciliary locomotion where the dynamics of the individual cilia are replaced by that of a continuous overlaying surface allowed to deform tangentially to itself. Employing a variational approach, we determine numerically the time-periodic deformation of such surface which leads to low-Reynolds locomotion with minimum rate of energy dissipation (maximum efficiency). Employing both Lagrangian and Eulerian points of views, we show that in the optimal swimming stroke, individual cilia display weak asymmetric beating, but that a significant symmetry-breaking occurs at the organism level, with the whole surface deforming in a wave-like fashion reminiscent of metachronal waves of biological cilia. This wave motion is analyzed using a formal modal decomposition, is found to occur in the same direction as the swimming direction, and is interpreted as due to a spatial distribution of phase-differences in the kinematics of individual cilia. Using additional constrained optimizations, as well as a constructed analytical ansatz, we derive a complete optimization diagram where all swimming efficiencies, swimming speeds, and amplitude of surface deformation can be reached, with the mathematically optimal swimmer, of efficiency one half, being a singular limit. Biologically, our work suggests therefore that metachronal waves may allow cilia to propel cells forward while reducing the energy dissipated in the surrounding fluid.Comment: 29 pages, 20 figure

Electro-hydrodynamic synchronization of piezoelectric flags

Hydrodynamic coupling of flexible flags in axial flows may profoundly influence their flapping dynamics, in particular driving their synchronization. This work investigates the effect of such coupling on the harvesting efficiency of coupled piezoelectric flags, that convert their periodic deformation into an electrical current. Considering two flags connected to a single output circuit, we investigate using numerical simulations the relative importance of hydrodynamic coupling to electrodynamic coupling of the flags through the output circuit due to the inverse piezoelectric effect. It is shown that electrodynamic coupling is dominant beyond a critical distance, and induces a synchronization of the flags' motion resulting in enhanced energy harvesting performance. We further show that this electrodynamic coupling can be strengthened using resonant harvesting circuits.Comment: 14 pages, 10 figures, to appear in J. Fluids Struc

Fluid-solid-electric lock-in of energy-harvesting piezoelectric flags

The spontaneous flapping of a flag in a steady flow can be used to power an output circuit using piezoelectric elements positioned at its surface. Here, we study numerically the effect of inductive circuits on the dynamics of this fluid-solid-electric system and on its energy harvesting efficiency. In particular, a destabilization of the system is identified leading to energy harvesting at lower flow velocities. Also, a frequency lock-in between the flag and the circuit is shown to significantly enhance the system's harvesting efficiency. These results suggest promising efficiency enhancements of such flow energy harvesters through the output circuit optimization.Comment: 8 pages, 8 figures, to appear in Physical Review Applie

Influence and optimization of the electrodes position in a piezoelectric energy harvesting flag

Fluttering piezoelectric plates may harvest energy from a fluid flow by converting the plate's mechanical deformation into electric energy in an output circuit. This work focuses on the influence of the arrangement of the piezoelectric electrodes along the plate's surface on the energy harvesting efficiency of the system, using a combination of experiments and numerical simulations. A weakly non-linear model of a plate in axial flow, equipped with a discrete number of piezoelectric patches is derived and confronted to experimental results. Numerical simulations are then used to optimize the position and dimensions of the piezoelectric electrodes. These optimal configurations can be understood physically in the limit of small and large electromechanical coupling.Comment: To appear in Journal of Sound and Vibratio

Synchronized flutter of two slender flags

The interactions and synchronization of two parallel and slender flags in a uniform axial flow are studied in the present paper by generalizing Lighthill's Elongated Body Theory (EBT) and Lighthill's Large Amplitude Elongated Body Theory (LAEBT) to account for the hydrodynamic coupling between flags. The proposed method consists in two successive steps, namely the reconstruction of the flow created by a flapping flag within the LAEBT framework and the computation of the fluid force generated by this nonuniform flow on the second flag. In the limit of slender flags in close proximity, we show that the effect of the wakes have little influence on the long time coupled-dynamics and can be neglected in the modeling. This provides a simplified framework extending LAEBT to the coupled dynamics of two flags. Using this simplified model, both linear and large amplitude results are reported to explore the selection of the flapping regime as well as the dynamical properties of two side-by-side slender flags. Hydrodynamic coupling of the two flags is observed to destabilize the flags for most parameters, and to induce a long-term synchronization of the flags, either in-phase or out-of-phase.Comment: 14 pages, 10 figures, to appear in J. Fluid Mec