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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.
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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.
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 $U/ln U$.
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.
The nonlocal Fisher equation is a diffusion-reaction equation with a nonlocal quadratic competition, which describes the reaction between distant individuals. This equation arises in evolutionary biological systems, where the arena for the dynamics is trait space, diffusion accounts for mutations and individuals with similar traits compete, resulting in partial niche overlap. It has been found that the (non-cutoff) deterministic system gives rise to a spatially inhomogeneous state for a certain class of interaction kernels, while the stochastic system produces an inhomogeneous state for small enough population densities. Here we study the problem of front propagation in this system, comparing the stochastic dynamics to the heuristic approximation of this system by a deterministic system where the linear growth term is cut off below some critical density. Of particular interest is the nontrivial pattern left behind the front. For large population density, or small cutoff, there is a constant velocity wave propagating from the populated region to the unpopulated region. As in the local Fisher equation, the spreading velocity is much lower than the Fisher velocity which is the spreading velocity for infinite population size. The stochastic simulations give approximately the same spreading velocity as the deterministic simulation with appropriate birth cutoff. When the population density is small enough, there is a different mechanism of population spreading. The population is concentrated on clusters which divide and separate. This mode of spreading has small spreading velocity, decaying exponentially with the range of the interaction kernel.
Non-equilibrium dissipative systems usually exhibit multistability, leading to the presence of propagative domain between steady states. We investigate the front propagation into an unstable state in discrete media. Based on a paradigmatic model of coupled chain of oscillators and populations dynamics, we calculate analytically the average speed of these fronts and characterize numerically the oscillatory front propagation. We reveal that different parts of the front oscillate with the same frequency but with different amplitude. To describe this latter phenomenon we generalize the notion of the Peierls-Nabarro potential, achieving an effective continuous description of the discreteness effect.
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