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

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

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

Self-propulsion of chemically-active droplets

Microscopic active droplets are able to swim autonomously in viscous flows: this puzzling feature stems from solute exchanges with the surrounding fluid via surface reactions or their spontaneous solubilisation, and the interfacial flows resulting from these solutes' gradients. Contrary to asymmetric active colloids, these isotropic droplets swim spontaneously by exploiting the nonlinear coupling of solute transport with self-generated Marangoni flows, which is also responsible for secondary transitions to more complex individual and collective dynamics. Thanks to their simple design and their sensitivity to physico-chemical signals, they are fascinating physicists, chemists, biologists and fluid dynamicists alike to analyse viscous self-propulsion and collective dynamics in active matter systems, to develop synthetic cellular models or to perform targeted biomedical or engineering applications. I review here the most recent and significant developments of this rapidly-growing field, focusing on the mathematical and physical modelling of these intringuing droplets, together with its experimental design and characterisation.Comment: 26 pages, 8 figures, to appear in Annual Review of Fluid Mechanic

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

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

Flow-induced pruning of branched systems and brittle reconfiguration

Whereas most plants are flexible structures that undergo large deformations under flow, another process can occur when the plant is broken by heavy fluid-loading. We investigate here the mechanism of such possible breakage, focusing on the flow-induced pruning that can be observed in plants or aquatic vegetation when parts of the structure break under flow. By computation on an actual tree geometry, a 20-yr-old walnut tree (Juglans Regia L.) and comparison with simple models, we analyze the influence of geometrical and physical parameters on the occurrence of branch breakage and on the successive breaking events occurring in a tree-like structure when the flow velocity is increased. We show that both the branching pattern and the slenderness exponent, defining the branch taper, play a major role in the breakage scenario. We identify a criterion for branch breakage to occur before breakage of the trunk. In that case, we show that the successive breakage of peripheral branches allows the plant to sustain higher flow forces. This mechanism is therefore similar to elastic reconfiguration, and can be seen as a second strategy to overcome critical events, possibly a widespread solution in plants and benthic organisms.Comment: 9 pages, 9 figure