Mixing efficiency in an excitable medium with chaotic shear flow

Control of Chemical Waves by Fluid Stretching and Compression

Oscillatory kinetics coupled to diffusion can produce traveling waves as observed in physical, chemical, and biological systems. We show experimentally that the properties of such waves can be controlled by fluid stretching and compression in a hyperbolic flow. Localized packet waves consisting in a train of parallel waves can develop due to a balance between diffusive broadening and advective compression along the unstable manifold. At a given distance from the stagnation point, the parallel waves transform into planelike waves and smeared waves where the transverse parabolic flow profile disturbs the patterns in the gap width. Once a wave packet has been obtained, it imprints a privileged direction that is maintained even if the compression rate is decreased. The width of the wave packet then scales inversely with the compression rate.

Motion, fixation probability and the choice of an evolutionary process

Seemingly minor details of mathematical and computational models of evolution are known to change the effect of population structure on the outcome of evolutionary processes. For example, birth-death dynamics often result in amplification of selection, while death-birth processes have been associated with suppression. In many biological populations the interaction structure is not static. Instead, members of the population are in motion and can interact with different individuals at different times. In this work we study populations embedded in a flowing medium; the interaction network is then time dependent. We use computer simulations to investigate how this dynamic structure affects the success of invading mutants, and compare these effects for different coupled birth and death processes. Specifically, we show how the speed of the motion impacts the fixation probability of an invading mutant. Flows of different speeds interpolate between evolutionary dynamics on fixed heterogeneous graphs and well-stirred populations; this allows us to systematically compare against known results for static structured populations. We find that motion has an active role in amplifying or suppressing selection by fragmenting and reconnecting the interaction graph. While increasing flow speeds suppress selection for most evolutionary models, we identify characteristic responses to flow for the different update rules we test. In particular we find that selection can be maximally enhanced or suppressed at intermediate flow speeds.

Whether a mutation spreads in a population or not is one of the most important questions in biology. The evolution of cancer and antibiotic resistance, for example, are mediated by invading mutants. Recent work has shown that population structure can have important consequences for the outcome of evolution. For instance, a mutant can have a higher or a lower chance of invasion than in unstructured populations. These effects can depend on seemingly minor details of the evolutionary model, such as the order of birth and death events. Many biological populations are in motion, for example due to external stirring. Experimentally this is known to be important; the performance of mutants in E. coli populations, for example, depends on the rate of mixing. Here, we focus on simulations of populations in a flowing medium, and compare the success of a mutant for different flow speeds. We contrast different evolutionary models, and identify what features of the evolutionary model affect mutant success for different speeds of the flow. We find that the chance of mutant invasion can be at its highest (or lowest) at intermediate flow speeds, depending on the order in which birth and death events occur in the evolutionary process.

Front tracking for quantifying advection-reaction-diffusion

We present an algorithm for measuring the speed and thickness of reaction fronts, and from those quantities, the diffusivity and the reaction rate of the active chemical species. This front-tracking algorithm provides local measurements suitable for statistics and requires only a sequence of concentration fields. Though our eventual goal is front tracking in advection-reaction-diffusion, here we demonstrate the algorithm in reaction-diffusion. We test the algorithm with validation data in which front speed and thickness are prescribed, as well as simulation results in which diffusivity and reaction rate are prescribed. In all tests, measurements closely match true values. We apply the algorithm to laboratory experiments using the Belousov-Zhabotinsky reaction, producing speed, diffusivity, and reaction rate measurements that are statistically more robust than in prior studies. Finally, we use thickness measurements to quantify the concentration profile of chemical waves in the reaction.

Optimal Stretching in Advection-Reaction-Diffusion Systems

We investigate growth of the excitable Belousov-Zhabotinsky reaction in chaotic, time-varying flows. In slow flows, reacted regions tend to lie near vortex edges, whereas fast flows restrict reacted regions to vortex cores. We show that reacted regions travel toward vortex centers faster as flow speed increases, but nonreactive scalars do not. For either slow or fast flows, reaction is promoted by the same optimal range of the local advective stretching, but stronger stretching causes reaction blowout and can hinder reaction from spreading. We hypothesize that optimal stretching and blowout occur in many advection-diffusion-reaction systems, perhaps creating ecological niches for phytoplankton in the ocean.

Nonperfect mixing affects synchronization on a large number of chemical oscillators immersed in a chemically active time-dependent chaotic flow

The problem of synchronization of finite-size chemical oscillators described by active inertial particles is addressed for situations in which they are immersed in a reacting nonstationary chaotic flow. Active substances in the fluid will be modeled by Lagrangian particles closely following the fluid streamlines. Their interaction with the active inertial particles as well as the properties of the fluid dynamics will result in modifying the synchronization state of the chemical oscillators. This behavior is studied in terms of the exchange rate between the Lagrangian and inertial particles, and the finite-time Lyapunov exponents characterizing the flow. The coherence of the population of oscillators is determined by means of the order parameter introduced by Kuramoto. The different dynamics observed for the inertial particles (chemical oscillators) and Lagrangian particles (describing chemicals in the flow) lead to nonlinear interactions and patches of synchronized regions within the domain.

Clustering of inertial particles in compressible chaotic flows

Clustering of inertial particles is analyzed in chaotic compressible flows. A simplified dynamical model for the motion of inertial particles in a compressible flow has been derived. Clustering enhancement has been observed for intermediate Stokes times and characterized in terms of the number of particles with negative finite-time Lyapunov exponents and the Lyapunov dimension of the model attractor. Cluster formation has been observed to depend on the nature of the flow; vortical or shear. The motion of heavy and light particles is analyzed in terms of the compressibility and correlation length of the density field.

Mixing and clustering in compressible chaotic stirred flows

The effect of compressibility on the mixing of Lagrangian tracers is analyzed in chaotic stirred flows. Mixing is studied in terms of the finite-time Lyapunov exponents. Mixing and clustering of passive tracers surrounded by Lagrangian coherent structures is observed to increase with compressibility intensity. The role of the stirring rate and compressibility on mixing and clustering has been analyzed.

Solitary Marangoni-driven convective structures in bistable chemical systems

Bistable chemical fronts can be deformed by Marangoni-driven convective flows induced by gradients of surface tension across the front. We investigate here the nonlinear dynamics of such a system by simulations of two-dimensional Navier-Stokes equations coupled to a reaction-diffusion-convection equation for a surface-active chemical species present in the bulk of the solution and involved in a bistable kinetics. We show that Marangoni flows cannot only alter the shape and speed of the front but also change the relative stability of the two stable steady states, reversing in some cases the direction of propagation of the front with regard to the pure reaction-diffusion situation. A detailed parametric study discusses the properties of the asymptotic dynamics as a function of the Marangoni number M and of a kinetic parameter d.

Chemical-wave dynamics in a vertically oscillating fluid layer

Classical Faraday experiments were conducted on the oscillatory chemical Belousov-Zhabotinsky (BZ) reaction. The vertical periodic modulation of the acceleration induces flows in the system that change the BZ dynamics, and thus the patterns exhibited. The resulting reaction-diffusion-advection system exhibits four different types of pattern for increasing stirring amplitude: deformed targets and spiral waves, filamentary patterns arranged in large-scale vortices, advection phase waves, and finally front annihilation where the medium becomes homogeneous. A wave period analysis of the forced system has been carried out. Contrary to what is expected, i.e., a continuous increase of the wave period with increasing forcing, the period changes dramatically at the boundaries between pattern domains.

Effective dimensions and chemical reactions in fluid flows

We show that chemical activity in hydrodynamical flows can be understood as the outcome of three basic effects: the stirring protocol of the flow, the local properties of the reaction, and the global folding dynamics which also depends on the geometry of the container. The essence of each of these components can be described by simple functional relations. In an ordinary differential equation approach, they determine a new chemical rate equation for the concentration, which turns out to be coupled to the dynamics of an effective fractal dimension. The theory predicts an exponential convergence to the asymptotic chemical state. This holds even in flows characterized by a linear stirring protocol where transient fractal patterns are shown to exist despite the lack of any chaotic set of the advection dynamics. In the exponential case the theory applies to flows of chaotic time dependence (chaotic flows) as well.