Homogenization induced by chaotic mixing and diffusion in an oscillatory chemical reaction

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.

Topology of advective-diffusive scalar transport in laminar flows

The present study proposes a unified Lagrangian transport template for topological description of advective fluid transport and advective-diffusive scalar transport in laminar flows. The key to this unified description is the expression of scalar transport as purely advective transport by the total scalar flux. This admits generalization of the concept of transport topologies known from laminar mixing studies to scalar transport. The study is restricted to two-dimensional systems and the fluid and scalar transport topologies, as a consequence, prove to be Hamiltonian. The unified Lagrangian transport template is demonstrated and investigated for a heat-transfer problem with nonadiabatic boundaries, representing generic scalar transport with permeable boundaries. The fluid and thermal transport topologies under steady conditions both accommodate islands (constituting isolated flow and thermal regions) that undergo Hamiltonian disintegration into chaotic seas upon introducing time periodicity. The thermal transport topology invariably comprises transport conduits that connect the nonadiabatic boundaries and facilitate wall-wall and wall-fluid heat transfer. For steady conditions these transport conduits are regular; for time-periodic conditions these conduits may, depending on degree of diffusion, be regular or chaotic. Regular conduits connect nonadiabatic walls only with specific flow regions; chaotic heat conduits establish connection (and thus heat exchange) of the nonadiabatic walls with the entire flow domain.

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.

Mixing efficiency in an excitable medium with chaotic shear flow

The effect of a time-periodic chaotic shear flow on an excitable chemical medium is studied numerically. Stirring effects on pattern formation strongly depend on the shear amplitude and the ratio of the advective and chemical time scales (Damköhler number, Da). We have observed that the wave period increases with decreasing Da below some critical value, afterwards the period decreases until complete wave annihilation. In the last case, before final extinction, a set of uncorrelated, nonstationary excitable dots survive, whose number depends on the mixing rate. Insights on the nature of this critical behavior are obtained through the calculation of the mixing efficiency of the flow.

Persistent localized states for a chaotically mixed bistable reaction

We describe the evolution of a bistable chemical reaction in a closed two-dimensional chaotic laminar flow, from a localized initial disturbance. When the fluid mixing is sufficiently slow, the disturbance may spread and eventually occupy the entire fluid domain. By contrast, rapid mixing tends to dilute the initial state and so extinguish the disturbance. Such a dichotomy is well known. However, we report here a hitherto apparently unremarked intermediate case, a persistent highly localized disturbance. Such a localized state arises when the Damköhler number is great enough to sustain a "hot spot," but not so great as to lead to global spread. We show that such a disturbance is located in the neighborhood of an unstable periodic orbit of the flow, and we describe some limited aspects of its behavior using a reduced, lamellar model.

Resonant pattern formation in active media driven by time-dependent flows

The effect of a time-dependent flow in an oscillatory chemical system supporting front propagation is studied. Resonant target patterns depend on the strength and frequency of the time-dependent flow. The flow time scale needed to entrain the system to the resonant target period of oscillation depends on the closeness to the natural oscillation frequency of the medium. The flow strength needed to obtain these patterns is interpreted in terms of mixing optimization, and we give conditions for the flow that guarantee the best mixing with the Bernoulli property.