arXiv subjects

Jean-Luc Thiffeault

Publications and source records attributed to Jean-Luc Thiffeault.

At least 37 records · Page 2Linked to original sources

arXiv2017

Frontiers of chaotic advection

This work reviews the present position of and surveys future perspectives in the physics of chaotic advection: the field that emerged three decades ago at the intersection of fluid mechanics and nonlinear dynamics, which encompasses a range of applications with length scales ranging from micrometers to hundreds of kilometers, including systems as diverse as mixing and thermal processing of viscous fluids, microfluidics, biological flows, and oceanographic and atmospheric flows.

nlin.CD↗

arXiv2017

Lyapunov exponents for the random product of two shears

We give lower and upper bounds on both the Lyapunov exponent and generalised Lyapunov exponents for the random product of positive and negative shear matrices. These types of random products arise in applications such as fluid stirring devices. The bounds, obtained by considering invariant cones in tangent space, give excellent accuracy compared to standard and general bounds, and are increasingly accurate with increasing shear. Bounds on generalised exponents are useful for testing numerical methods, since these exponents are difficult to compute in practice.

math.DS↗

arXiv2016

Can phoretic particles swim in two dimensions?

Artificial phoretic particles swim using self-generated gradients in chemical species (self-diffusiophoresis) or charges and currents (self-electrophoresis). These particles can be used to study the physics of collective motion in active matter and might have promising applications in bioengineering. In the case of self-diffusiophoresis, the classical physical model relies on a steady solution of the diffusion equation, from which chemical gradients, phoretic flows and ultimately the swimming velocity, may be derived. Motivated by disk-shaped particles in thin films and under confinement, we examine the extension to two dimensions. Because the two-dimensional diffusion equation lacks a steady state with the correct boundary conditions, Laplace transforms must be used to study the long-time behavior of the problem and determine the swimming velocity. For fixed chemical fluxes on the particle surface, we find that the swimming velocity ultimately always decays logarithmically in time. In the case of finite Peclet numbers, we solve the full advection-diffusion equation numerically and show that this decay can be avoided by the particle moving to regions of unconsumed reactant. Finite advection thus regularizes the two-dimensional phoretic problem.

cond-mat.soft↗

arXiv2016

Microorganism Billiards

Recent experiments and numerical simulations have shown that certain types of microorganisms "reflect" off of a flat surface at a critical angle of departure, independent of the angle of incidence. The nature of the reflection may be active (cell and flagellar contact with the surface) or passive (hydrodynamic) interactions. We explore the billiard-like motion of a body with this empirical reflection law inside a regular polygon and show that the dynamics can settle on a stable periodic orbit or can be chaotic, depending on the swimmer's departure angle and the domain geometry. The dynamics are often found to be robust to the introduction of weak random fluctuations. The Lyapunov exponent of swimmer trajectories can be positive or negative, can have extremal values, and can have discontinuities depending on the degree of the polygon. A passive sorting device is proposed that traps swimmers of different departure angles into separate bins. We also study the external problem of a microorganism swimming in a patterned environment of square obstacles, where the departure angle dictates the possibility of trapping or diffusive trajectories.

nlin.CD↗

arXiv2016

Fluid transport and mixing by an unsteady microswimmer

We study the fluid drift due to a time-dependent dumbbell model of a microswimmer. The model captures important aspects of real microswimmers such as a time-dependent flagellar motion and a no-slip body. The model consists of a rigid sphere for the body and a time-dependent moving Stokeslet representing the flagella. We analyze the paths of idealized fluid particles displaced by the swimmer. The simplicity of the model allows some asymptotic calculations very near and far away from the swimmer. The displacements of particles near the swimmer diverge in a manner similar to an isolated no-slip sphere, but with a smaller coefficient due to the action of the flagellum. Far from the swimmer, the time dependence becomes negligible due to both being very fast and decaying with distance. Finally, we compute the probability distribution of particle displacements, and find that our model has fatter tails than previous steady models, due to the presence of a no-slip surface that drags particles along.

cond-mat.soft↗

arXiv2015

Distribution of particle displacements due to swimming microorganisms

The experiments of Leptos et al. [Phys. Rev. Lett. 103, 198103 (2009)] show that the displacements of small particles affected by swimming microorganisms achieve a non-Gaussian distribution, which nevertheless scales diffusively -- the 'diffusive scaling.' We use a simple model where the particles undergo repeated 'kicks' due to the swimmers to explain the shape of the distribution as a function of the volume fraction of swimmers. The net displacement is determined by the inverse Fourier transform of a single-swimmer characteristic function. The only adjustable parameter is the strength of the stresslet term in our spherical squirmer model. We give a criterion for convergence to a Gaussian distribution in terms of moments of the drift function, and show that the experimentally-observed diffusive scaling is a transient related to the slow crossover of the fourth moment from a ballistic to a linear regime with path length. We also present a simple model, with logarithmic drift function,that can be solved analytically.

cond-mat.soft↗

arXiv2015

Finite-Time Braiding Exponents

Topological entropy is a common measure of the rate of mixing in a flow. It can be computed by partition methods, or by estimating the growth rate of material lines or other material elements. This requires detailed knowledge of the velocity field, which is not always available, such as when we only know a few particle trajectories (ocean float data, for example). We propose an alternative approximation to topological entropy,applicable to two-dimensional flows, which uses only a finite number of trajectories as input data. To represent these sparse data sets, we use braids, algebraic objects that record how strands, i.e., trajectories, exchange positions with respect to a projection axis. Material curves advected by the flow are represented as simplified loop coordinates. The exponential rate at which a braid deforms loops over a finite time interval as the strands exchange places is the Finite-Time Braiding Exponent (FTBE) and serves as a proxy for topological entropy of the two-dimensional flow. We demonstrate that FTBEs are robust with respect to the value of numerical time step,details of braid representation, and choice of initial conditions inside the mixing region. We also explore how closely the FTBEs approximate topological entropy depending on the number and length of trajectories used.

nlin.CD↗

arXiv2015

Chaotic Geodesics

When a shallow layer of inviscid fluid flows over a substrate, the fluid particle trajectories are, to leading order in the layer thickness, geodesics on the two-dimensional curved space of the substrate. Since the two-dimensional geodesic equation is a two degree-of-freedom autonomous Hamiltonian system, it can exhibit chaos, depending on the shape of the substrate. We find chaotic behaviour for a range of substrates.

nlin.CD↗

arXiv2013

Scalar Decay in Chaotic Mixing

I review the local theory of mixing, which focuses on infinitesimal blobs of scalar being advected and stretched by a random velocity field. An advantage of this theory is that it provides elegant analytical results. A disadvantage is that it is highly idealised. Nevertheless, it provides insight into the mechanism of chaotic mixing and the effect of random fluctuations on the rate of decay of the concentration field of a passive scalar.

nlin.CD↗

arXiv2012

Estimating topological entropy from the motion of stirring rods

Stirring a two-dimensional viscous fluid with rods is often an effective way to mix. The topological features of periodic rod motions give a lower bound on the topological entropy of the induced flow map, since material lines must `catch' on the rods. But how good is this lower bound? We present examples from numerical simulations and speculate on what affects the 'gap' between the lower bound and the measured topological entropy. The key is the sign of the rod motion's action on first homology of the orientation double cover of the punctured disk.

nlin.CD↗

arXiv2012

Topological entropy and secondary folding

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod-stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.

math.DS↗

arXiv2012

Trajectory entanglement in dense granular materials

The particle-scale dynamics of granular materials have commonly been characterized by the self-diffusion coefficient $D$. However, this measure discards the collective and topological information known to be an important characteristic of particle trajectories in dense systems. Direct measurement of the entanglement of particle space-time trajectories can be obtained via the topological braid entropy $\Sbraid$, which has previously been used to quantify mixing efficiency in fluid systems. Here, we investigate the utility of $\Sbraid$ in characterizing the dynamics of a dense, driven granular material at packing densities near the static jamming point $ϕ_J$. From particle trajectories measured within a two-dimensional granular material, we typically observe that $\Sbraid$ is well-defined and extensive. However, for systems where $ϕ\gtrsim 0.79$, we find that $\Sbraid$ (like $D$) is not well-defined, signifying that these systems are not ergodic on the experimental timescale. Both $\Sbraid$ and $D$ decrease with either increasing packing density or confining pressure, independent of the applied boundary condition. The related braiding factor provides a means to identify multi-particle phenomena such as collective rearrangements. We discuss possible uses for this measure in characterizing granular systems.

cond-mat.soft↗

arXiv2011

Detecting coherent structures using braids

The detection of coherent structures is an important problem in fluid dynamics, particularly in geophysical applications. For instance, knowledge of how regions of fluid are isolated from each other allows prediction of the ultimate fate of oil spills. Existing methods detect Lagrangian coherent structures, which are barriers to transport, by examining the stretching field as given by finite-time Lyapunov exponents. These methods are very effective when the velocity field is well-determined, but in many applications only a small number of flow trajectories are known, for example when dealing with oceanic float data. We introduce a topological method for detecting invariant regions based on a small set of trajectories. In the method we regard the two-dimensional trajectory data as a braid in three dimensions, with time being the third coordinate. Invariant regions then correspond to trajectories that travel together and do not entangle other trajectories. We detect these regions by examining the growth of hypothetical loops surrounding sets of trajectories, and searching for loops that show negligible growth.

nlin.CD↗

arXiv2011

Moving walls accelerate mixing

Mixing in viscous fluids is challenging, but chaotic advection in principle allows efficient mixing. In the best possible scenario,the decay rate of the concentration profile of a passive scalar should be exponential in time. In practice, several authors have found that the no-slip boundary condition at the walls of a vessel can slow down mixing considerably, turning an exponential decay into a power law. This slowdown affects the whole mixing region, and not just the vicinity of the wall. The reason is that when the chaotic mixing region extends to the wall, a separatrix connects to it. The approach to the wall along that separatrix is polynomial in time and dominates the long-time decay. However, if the walls are moved or rotated, closed orbits appear, separated from the central mixing region by a hyperbolic fixed point with a homoclinic orbit. The long-time approach to the fixed point is exponential, so an overall exponential decay is recovered, albeit with a thin unmixed region near the wall.

physics.flu-dyn↗

arXiv2011

Using multiscale norms to quantify mixing and transport

Mixing is relevant to many areas of science and engineering, including the pharmaceutical and food industries, oceanography, atmospheric sciences, and civil engineering. In all these situations one goal is to quantify and often then to improve the degree of homogenisation of a substance being stirred, referred to as a passive scalar or tracer. A classical measure of mixing is the variance of the concentration of the scalar, which can be related to the $L^2$ norm of the concentration field. Recently other norms have been used to quantify mixing, in particular the mix-norm as well as negative Sobolev norms. These norms have the advantage that unlike variance they decay even in the absence of diffusion, and their decay corresponds to the flow being mixing in the sense of ergodic theory. General Sobolev norms weigh scalar gradients differently, and are known as multiscale norms for mixing. We review the applications of such norms to mixing and transport, and show how they can be used to optimise the stirring and mixing of a decaying passive scalar. We then review recent work on the less-studied case of a continuously-replenished scalar field --- the source-sink problem. In that case the flows that optimally reduce the norms are associated with transport rather than mixing: they push sources onto sinks, and vice versa.

physics.flu-dyn↗

arXiv2011

Stirring up trouble: Multi-scale mixing measures for steady scalar sources

The mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring, relative to the variance in the absence of stirring. These variances can be weighted at various spatial scales, leading to a family of multi-scale mixing measures and efficiencies. We derive a priori estimates on these efficiencies from the advection--diffusion partial differential equation, focusing on a broad class of statistically homogeneous and isotropic incompressible flows. The analysis produces bounds on the mixing efficiencies in terms of the Peclet number, a measure the strength of the stirring relative to molecular diffusion. We show by example that the estimates are sharp for particular source, sink and flow combinations. In general the high-Peclet number behavior of the bounds (scaling exponents as well as prefactors) depends on the structure and smoothness properties of, and length scales in, the scalar source and sink distribution. The fundamental model of the stirring of a monochromatic source/sink combination by the random sine flow is investigated in detail via direct numerical simulation and analysis. The large-scale mixing efficiency follows the upper bound scaling (within a logarithm) at high Peclet number but the intermediate and small-scale efficiencies are qualitatively less than optimal. The Peclet number scaling exponents of the efficiencies observed in the simulations are deduced theoretically from the asymptotic solution of an internal layer problem arising in a quasi-static model.

physics.flu-dyn↗

arXiv2011

A low-Reynolds-number treadmilling swimmer near a semi-infinite wall

We investigate the behavior of a treadmilling microswimmer in a two-dimensional unbounded domain with a semi-infinite no-slip wall. The wall can also be regarded as a probe or pipette inserted into the flow. We solve the governing evolution equations in an analytical form and numerically calculate trajectories of the swimmer for several different initial positions and orientations. We then compute the probability that the treadmilling organism can escape the vicinity of the wall. We find that many trajectories in a 'wedge' around the wall are likely to escape. This suggests that inserting a probe or pipette in a suspension of organism may push away treadmilling swimmers.

physics.bio-ph↗

arXiv2010

Measures of mixing quality in open flows with chaotic advection

We address the evaluation of mixing efficiency in experiments of chaotic mixing inside an open-flow channel. Since the open flow continuously brings new fluid into the limited mixing region, it is difficult to define relevant mixing indices, as fluid particles experience typically very different stretching and mixing histories. The repeated stretching and folding of a spot of dye leads to a persistent pattern. We propose that the normalized standard deviation of this characteristic pattern is a good measure of the mixing quality of the flow. We discuss the link between this measure and mixing of continuously-injected dye, and investigate it using an idealized map.

cond-mat.soft↗