General theory of nonlinear flow-distributed oscillations

Pattern formation induced by a differential shear flow

Fluid flow advecting one substance while others are immobilized can generate an instability in a homogeneous steady state of a reaction-diffusion-advection system. This differential-flow instability leads to the formation of steady spatial patterns in a moving reference frame. We study the effects of shear flow on this instability by considering two layers of fluid moving independently from each other, but allowing the substances to diffuse along and across the layers. We find that shear flow can generate instabilities even if the average flow velocity is zero for both substances. These instabilities are strongly dependent on which substance is advected by the shear flow. We explain these effects using the results of Taylor dispersion, where an effective diffusivity is enhanced by shear flow.

Blocking and transmission of traveling flow-distributed-oscillation waves in an absolutely unstable flowing medium

For a flowing, self-oscillating medium, we study the competition between traveling flow-distributed-oscillation waves excited by periodic driving at the upstream boundary and bulk oscillations originating downstream from the boundary. As previously observed in the case of stationary driving, we find that there is a region in parameter space where boundary-driven traveling waves of sufficiently high amplitude can impose themselves on the entire medium despite the presence of an absolute instability, which otherwise tends to block information from upstream. For sufficiently low flow rates, however, the imposed waves are arrested at a nonlinear blocking transition. Unlike the stationary case, we find that the region of imposed waves extends well into regions where, according to the linear approximation, there should be no traveling waves at all. This suggests that the extinction of the traveling waves is analogous to a subcritical Hopf bifurcation.

Small-scale particle advection, manipulation and mixing: beyond the hydrodynamic scale

In this paper we discuss the problems of particle advection, manipulation and mixing at small scales. We start by considering reaction-advection-diffusion systems with the focus on mixing. We show how mixing advection affects the processes of reaction-diffusion and discuss mixing-induced instabilities. Further, we consider the problem of particle manipulation and discuss collective effects in systems comprising solid and compressible particles. We particularly discuss mechanisms of particle entrapment, the role of compressibility in the dynamics of bubbly liquids and nonequilibrium colloidal explosion. Finally, we address two issues related to the problem of wetting. First, we study the role of contact line motion for a sessile droplet (or a bubble) on an oscillating substrate. Second, we discuss an instability of a thin film leading to the formation of a fractal structure of droplets.

Harmonic resonant excitation of flow-distributed oscillation waves and Turing patterns driven at a growing boundary

We perform numerical studies of a reaction-diffusion system that is both Turing and Hopf unstable, and that grows by addition at a moving boundary (which is equivalent by a Galilean transformation to a reaction-diffusion-advection system with a fixed boundary and a uniform flow). We model the conditions of a recent set of experiments which used a temporally varying illumination in the boundary region to control the formation of patterns in the bulk of the photosensitive medium. The frequency of the illumination variations can select patterns from among the competing instabilities of the medium. In the usual case, the waves that are excited have frequencies (as measured at a constant distance from the upstream boundary) matching the driving frequency. In contrast to the usual case, we find that both Turing patterns and flow-distributed oscillation waves can be excited by forcing at subharmonic multiples of the wave frequencies. The final waves (with frequencies at integer multiples of the driving frequency) are formed by a process in which transient wave fronts break up and reconnect. We find ratios of response to driving frequency as high as 10.

Selection of flow-distributed oscillation and Turing patterns by boundary forcing in a linearly growing, oscillating medium

We studied the response of a linearly growing domain of the oscillatory chemical chlorine dioxide-iodide-malonic acid (CDIMA) medium to periodic forcing at its growth boundary. The medium is Hopf-, as well as Turing-unstable and the system is convectively unstable. The results confirm numerical predictions that two distinct modes of pattern can be excited by controlling the driving frequency at the boundary, a flow-distributed-oscillation (FDO) mode of traveling waves at low values of the forcing frequency f , and a mode of stationary Turing patterns at high values of f . The wavelengths and phase velocities of the experimental patterns were compared quantitatively with results from dynamical simulations and with predictions from linear dispersion relations. The results for the FDO waves agreed well with these predictions, and obeyed the kinematic relations expected for phase waves with frequencies selected by the boundary driving frequency. Turing patterns were also generated within the predicted range of forcing frequencies, but these developed into two-dimensional structures which are not fully accounted for by the one-dimensional numerical and analytical models. The Turing patterns excited by boundary forcing persist when the forcing is removed, demonstrating the bistability of the unforced, constant size medium. Dynamical simulations at perturbation frequencies other than those of the experiments showed that in certain ranges of forcing frequency, FDO waves become unstable, breaking up into harmonic waves of different frequency and wavelength and phase velocity.

Chemical pattern formation induced by a shear flow in a two-layer model

We study chemical patterns arising from instabilities in reaction-diffusion-advection systems under the influence of shear flow. Turing pattern formation without shear flow can occur in an activator-inhibitor system as long as the diffusivity of the inhibitor is larger than the diffusivity of the activator. In the presence of shear flow, a homogeneous steady state can become unstable even if this condition is not satisfied. Chemical patterns arise as a result of this instability. We study this instability in a simple system consisting of two layers moving relative to each other. We carry out a linear stability analysis showing the onset of the instability as a function of the relative speed between the layers. We solve numerically the nonlinear reaction-diffusion-advection equations to obtain these patterns. We find stationary, oscillatory, and drifting patterns extending along each layer. We also find regions of bistability that allow the formation of localized structures. The instability is analyzed in terms of Taylor dispersion.

Mixing-induced global modes in open active flow

We describe how local mixing transforms a convectively unstable active field in an open flow into absolutely unstable. Presenting the mixing region as one with a locally enhanced effective diffusion allows us to find the linear transition point to an unstable global mode analytically. We derive the critical exponent that characterizes weakly nonlinear regimes beyond the instability threshold and compare it with numerical simulations of a full two-dimensional flow problem.

Flow-distributed oscillation, flow-velocity modulation, and resonance

We examine the effects of a periodically varying flow velocity on the standing- and traveling-wave patterns formed by the flow-distributed oscillation mechanism. In the kinematic (or diffusionless) limit, the phase fronts undergo a simple, spatiotemporally periodic longitudinal displacement. On the other hand, when the diffusion is significant, periodic modulation of the velocity can disrupt the wave pattern, giving rise in the downstream region to traveling waves whose frequency is a rational multiple of the velocity perturbation frequency. We observe frequency locking at ratios of 1:1, 2:1, and 3:1, depending on the amplitude and frequency of the velocity modulation. This phenomenon can be viewed as a novel, rather subtle type of resonant forcing.

Pattern formation by boundary forcing in convectively unstable, oscillatory media with and without differential transport

Convectively unstable, open reactive flows of oscillatory media, whose phase is fixed or periodically modulated at the inflow boundary, are known to result in stationary and traveling waves, respectively. The latter are implicated in biological segmentation. The boundary-controlled pattern selection by this flow-distributed oscillator (FDO) mechanism has been generalized to include differential flow (DIFI) and differential diffusion (Turing) modes. Our present goal is to clarify the relationships among these mechanisms in the general case where there is differential flow as well as differential diffusion. To do so we analyze the dispersion relation for linear perturbations in the presence of periodic boundary forcing, and show how the solutions are affected by differential transport. We find that the DIFI and FDO modes are closely related and lie in the same frequency range, while the Turing mechanism gives rise to a distinct set of unstable modes in a separate frequency range. Finally, we substantiate the linear analysis by nonlinear simulations and touch upon the issue of competition of spatial modes.

Chemical instability induced by a shear flow

We predict a new type of instability induced by shear flow in chemical systems. A homogeneous steady state solution of a reaction-diffusion system loses stability in a Poiseuille flow. The instability appears as the speed of the flow increases beyond a certain threshold. This results in a steady pattern moving with the average fluid velocity. The chemical reaction consists of two species (activator and inhibitor) moving with identical velocities. Contrary to Turing's instability, the pattern arises when the activator has a higher diffusivity than the inhibitor.