Resonant enhanced diffusion in time-dependent flow

Anomalous diffusion for inertial particles under gravity in parallel flows

We investigate the bounds between normal or anomalous effective diffusion for inertial particles transported by parallel flows. The infrared behavior of the fluid kinetic-energy spectrum, i.e., the possible presence of long-range spatiotemporal correlations, is modeled as a power law by means of two parameters, and the problem is studied as a function of these latter. Our results, obtained in the limit of weak relative inertia, extend well-known results for tracers and apply to particles of any mass density, subject to gravity and Brownian diffusion. We consider both steady and time-dependent flows, and cases of both vanishing and finite particle sedimentation.

Dynamics of light particles in oscillating cellular flows

The dynamics of light particles in chaotic oscillating cellular flows is investigated both analytically and numerically by means of Monte Carlo simulations. At level of linear analysis (in the oscillation amplitude) we determined how the known fixed points relative to the stationary cellular flow deform into closed stable trajectories. Once the latter have been analytically determined, we numerically show that they possess the dynamical role of attracting all asymptotic trajectories for a wide range of parameters values. The robustness of the attracting trajectories is tested by adding a white-noise contribution to the particle equation of motion. As a result, attracting trajectories persist up to a critical Péclet number above which an average rising velocity sets in. Possible implications of our results on the long-standing problem related to the explanation of the observed oceanic plankton patchiness will be also discussed.

Synchronization of oscillating reactions in an extended fluid system

We present experiments on the synchronization of a dynamical, chemical process in an extended, flowing, fluid system. The oscillatory Belousov-Zhabotinsky chemical reaction is the process studied, and the flow is an annular chain of counterrotating vortices. Azimuthal motion of the vortices is controlled externally, enabling us to vary the type of transport. We find that oscillations of the Belousov-Zhabotinsky reaction synchronize throughout the extended fluid system only if transport in the flow is superdiffusive, with tracers in the flow undergoing rapid, distant jumps called Lévy flights.

Front propagation and mode-locking in an advection-reaction-diffusion system

Experiments are presented on chemical front propagation in an oscillating chain of vortices in which the mixing of passive impurities is chaotic. The excitable ruthenium-catalyzed Belousov-Zhabotinsky reaction is used in these studies. Velocities of the propagating fronts are measured as a function of the frequency and amplitude of external forcing. Mode locking is observed where the front propagates an integer number of vortices in an integer number of drive periods. Arnol'd tongues are mapped out for two of the locking regimes. These two tongues are shown to form a region of overlap where the velocity of the propagating front switches erratically between two locked values. The experimental results agree with numerical predictions of mode locking in a simplified model of the flow.

Heat and mass transport in nonhomogeneous random velocity fields

The effective equation describing the transport of passive tracers in nonsolenoidal velocity fields is determined, assuming that the velocity field U(r,t) is a function of both position r and time t, albeit remaining locally random. Assuming a strong separation of scales and applying the method of homogenization, we find a Fickian constitutive relation for the coarse-grained particle flux, as the sum of a convective part, V(E)c, and a diffusive term, -D(s). Inverted Delta c, where V(E) is the Eulerian mean tracer velocity, c the average particle concentration, and D(s) the effective diffusivity. The latter can be written as D(s)(r,t)=D(0)I+D(r,r,t), where D0 is the molecular diffusivity, I the unit dyadic and D(r(1),r(2),t) the cross diffusion dyadic. Conversely, the Eulerian mean velocity V(E)(r,t) is the sum of the microscale mean tracer velocity V(r,t) and a particle drift velocity, V(d)(r,t)=-[(delta/delta r(2)).D(T)(r,r(2),t)](r(2)=r), which depends on the nonhomogeneity of the velocity field at the macroscale. The microscale mean particle velocity, in turn, is the sum of the mean fluid velocity and the ballistic tracer velocity, which is due to the local nonuniformity of the concentration field and is therefore structurally different from the tracer drift velocity. In the limit of large Peclet numbers, D(s) coincides with the self-diffusion dyadic, as it measures the local temporal growth of the mean square displacement of a tracer particle from its average position. In this case, the motion of a tracer particle is a random process in the manner of Stratonovich, where the smoothly varying mean tracer velocity equals the microscale mean tracer velocity and the fluctuating term is described through the cross diffusion dyadic D(r(1),r(2),t).

Front propagation in laminar flows

The problem of front propagation in flowing media is addressed for laminar velocity fields in two dimensions. Three representative cases are discussed: stationary cellular flow, stationary shear flow, and percolating flow. Production terms of Fisher-Kolmogorov-Petrovskii-Piskunov type and of Arrhenius type are considered under the assumption of no feedback of the concentration on the velocity. Numerical simulations of advection-reaction-diffusion equations have been performed by an algorithm based on discrete-time maps. The results show a generic enhancement of the speed of front propagation by the underlying flow. For small molecular diffusivity, the front speed V(f) depends on the typical flow velocity U as a power law with an exponent depending on the topological properties of the flow, and on the ratio of reactive and advective time scales. For open-streamline flows we find always V(f) approximately U, whereas for cellular flows we observe V(f) approximately U(1/4) for fast advection and V(f) approximately U(3/4) for slow advection.

Transport in finite size systems: An exit time approach

In the framework of chaotic scattering we analyze passive tracer transport in finite systems. In particular, we study models with open streamlines and a finite number of recirculation zones. In the nontrivial case with a small number of recirculation zones a description by means of asymptotic quantities (such as the eddy diffusivity) is not appropriate. The nonasymptotic properties of dispersion are characterized by means of the exit time statistics, which shows strong sensitivity on initial conditions. This yields a probability distribution function with long tails, making impossible a characterization in terms of a unique typical exit time. (c) 1999 American Institute of Physics.