169 research outputs found
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
Shape of optimal active flagella
Many eukaryotic cells use the active waving motion of flexible flagella to
self-propel in viscous fluids. However, the criteria governing the selection of
particular flagellar waveforms among all possible shapes has proved elusive so
far. To address this question, we derive computationally the optimal shape of
an internally-forced periodic planar flagellum deforming as a travelling wave.
The optimum is here defined as the shape leading to a given swimming speed with
minimum energetic cost. To calculate the energetic cost though, we consider the
irreversible internal power expanded by the molecular motors forcing the
flagellum, only a portion of which ending up dissipated in the fluid. This
optimisation approach allows us to derive a family of shapes depending on a
single dimensionless number quantifying the relative importance of elastic to
viscous effects: the Sperm number. The computed optimal shapes are found to
agree with the waveforms observed on spermatozoon of marine organisms, thus
suggesting that these eukaryotic flagella might have evolved to be mechanically
optimal.Comment: 10 pages, 5 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
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
Using Surface-Motions for Locomotion of Microscopic Robots in Viscous Fluids
Microscopic robots could perform tasks with high spatial precision, such as
acting in biological tissues on the scale of individual cells, provided they
can reach precise locations. This paper evaluates the feasibility of in vivo
locomotion for micron-size robots. Two appealing methods rely only on surface
motions: steady tangential motion and small amplitude oscillations. These
methods contrast with common microorganism propulsion based on flagella or
cilia, which are more likely to damage nearby cells if used by robots made of
stiff materials. The power potentially available to robots in tissue supports
speeds ranging from one to hundreds of microns per second, over the range of
viscosities found in biological tissue. We discuss design trade-offs among
propulsion method, speed, power, shear forces and robot shape, and relate those
choices to robot task requirements. This study shows that realizing such
locomotion requires substantial improvements in fabrication capabilities and
material properties over current technology.Comment: 14 figures and two Quicktime animations of the locomotion methods
described in the paper, each showing one period of the motion over a time of
0.5 milliseconds; version 2 has minor clarifications and corrected typo
Mixing and transport by ciliary carpets: a numerical study
We use a 3D computational model to study the fluid transport and mixing due
to the beating of an infinite array of cilia. In accord with recent
experiments, we observe two distinct regions: a fluid transport region above
the cilia and a fluid mixing region below the cilia tip. The metachronal wave
due to phase differences between neighboring cilia is known to enhance the
fluid transport above the ciliary tip. In this work, we show that the
metachronal wave also enhances the mixing rates in the sub-ciliary region,
often simultaneously with the flow rate enhancement. Our results suggest that
this simultaneous enhancement in transport and mixing is due to an enhancement
in shear flow. As the flow above the cilia increases, shear rate in the fluid
increases and such shear enhances stretching, which is an essential ingredient
for mixing. Estimates of the mixing time scale indicate that, compared to
diffusion, the mixing due to the cilia beat may be significant and sometimes
dominates chemical diffusion.Comment: submitted to Journal of Fluid Mechanic
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