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 $V_f \sim U$, whereas for cellular flows we observe $V_f \sim U^{1/4}$ for fast advection, and $V_f \sim U^{3/4}$ for slow advection.Comment: Enlarged, revised version, 37 pages, 14 figure

Front speed enhancement in cellular flows

The problem of front propagation in a stirred medium is addressed in the case of cellular flows in three different regimes: slow reaction, fast reaction and geometrical optics limit. It is well known that a consequence of stirring is the enhancement of front speed with respect to the non-stirred case. By means of numerical simulations and theoretical arguments we describe the behavior of front speed as a function of the stirring intensity, $U$. For slow reaction, the front propagates with a speed proportional to $U^{1/4}$, conversely for fast reaction the front speed is proportional to $U^{3/4}$. In the geometrical optics limit, the front speed asymptotically behaves as $U/\ln U$.Comment: 10 RevTeX pages, 10 included eps figure

Reaction Spreading on Graphs

We study reaction-diffusion processes on graphs through an extension of the standard reaction-diffusion equation starting from first principles. We focus on reaction spreading, i.e. on the time evolution of the reaction product, M(t). At variance with pure diffusive processes, characterized by the spectral dimension, d_s, for reaction spreading the important quantity is found to be the connectivity dimension, d_l. Numerical data, in agreement with analytical estimates based on the features of n independent random walkers on the graph, show that M(t) ~ t^{d_l}. In the case of Erdos-Renyi random graphs, the reaction-product is characterized by an exponential growth M(t) ~ e^{a t} with a proportional to ln, where is the average degree of the graph.Comment: 4 pages, 3 figure

Thin front propagation in steady and unsteady cellular flows

Front propagation in two dimensional steady and unsteady cellular flows is investigated in the limit of very fast reaction and sharp front, i.e., in the geometrical optics limit. In the steady case, by means of a simplified model, we provide an analytical approximation for the front speed, $v_{{\scriptsize{f}}}$, as a function of the stirring intensity, $U$, in good agreement with the numerical results and, for large $U$, the behavior $v_{{\scriptsize{f}}}\sim U/\log(U)$ is predicted. The large scale of the velocity field mainly rules the front speed behavior even in the presence of smaller scales. In the unsteady (time-periodic) case, the front speed displays a phase-locking on the flow frequency and, albeit the Lagrangian dynamics is chaotic, chaos in front dynamics only survives for a transient. Asymptotically the front evolves periodically and chaos manifests only in the spatially wrinkled structure of the front.Comment: 12 pages, 13 figure

Discreteness effects in a reacting system of particles with finite interaction radius

An autocatalytic reacting system with particles interacting at a finite distance is studied. We investigate the effects of the discrete-particle character of the model on properties like reaction rate, quenching phenomenon and front propagation, focusing on differences with respect to the continuous case. We introduce a renormalized reaction rate depending both on the interaction radius and the particle density, and we relate it to macroscopic observables (e.g., front speed and front thickness) of the system.Comment: 23 pages, 13 figure

Combustion dynamics in steady compressible flows

We study the evolution of a reactive field advected by a one-dimensional compressible velocity field and subject to an ignition-type nonlinearity. In the limit of small molecular diffusivity the problem can be described by a spatially discretized system, and this allows for an efficient numerical simulation. If the initial field profile is supported in a region of size l < lc one has quenching, i.e., flame extinction, where lc is a characteristic length-scale depending on the system parameters (reacting time, molecular diffusivity and velocity field). We derive an expression for lc in terms of these parameters and relate our results to those obtained by other authors for different flow settings.Comment: 6 pages, 5 figure

Mixing and reaction efficiency in closed domains

We present a numerical study of mixing and reaction efficiency in closed domains. In particular we focus our attention on laminar flows. In the case of inert transport the mixing properties of the flows strongly depend on the details of the Lagrangian transport. We also study the reaction efficiency. Starting with a little spot of product we compute the time needed to complete the reaction in the container. We found that the reaction efficiency is not strictly related to the mixing properties of the flow. In particular, reaction acts as a "dynamical regulator".Comment: 11 pages, 10 figure