Chemical front propagation in periodic flows: FKPP vs G

We investigate the influence of steady periodic flows on the propagation of chemical fronts in an infinite channel domain. We focus on the sharp front arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the limit of small molecular diffusivity and fast reaction (large P\'eclet and Damk\"ohler numbers, $\mathrm{Pe}$ and $\mathrm{Da}$) and on its heuristic approximation by the G equation. We introduce a variational formulation that expresses the two front speeds in terms of periodic trajectories minimizing the time of travel across the period of the flow, under a constraint that differs between the FKPP and G equations. This formulation shows that the FKPP front speed is greater than or equal to the G equation front speed. We study the two front speeds for a class of cellular vortex flows used in experiments. Using a numerical implementation of the variational formulation, we show that the differences between the two front speeds are modest for a broad range of parameters. However, large differences appear when a strong mean flow opposes front propagation; in particular, we identify a range of parameters for which FKPP fronts can propagate against the flow while G fronts cannot. We verify our computations against closed-form expressions derived for $\mathrm{Da}\ll \mathrm{Pe}$ and for $\mathrm{Da}\gg \mathrm{Pe}$

Front propagation in cellular flows for fast reaction and small diffusivity

We investigate the influence of fluid flows on the propagation of chemical fronts arising in FKPP type models. We develop an asymptotic theory for the front speed in a cellular flow in the limit of small molecular diffusivity and fast reaction, i.e., large P\'eclet ($Pe$) and Damk\"ohler ($Da$) numbers. The front speed is expressed in terms of a periodic path -- an instanton -- that minimizes a certain functional. This leads to an efficient procedure to calculate the front speed, and to closed-form expressions for $(\log Pe)^{-1}\ll Da\ll Pe$ and for $Da\gg Pe$. Our theoretical predictions are compared with (i) numerical solutions of an eigenvalue problem and (ii) simulations of the advection--diffusion--reaction equation

Diffusion in arrays of obstacles: beyond homogenisation

We revisit the classical problem of diffusion of a scalar (or heat) released in a two-dimensional medium with an embedded periodic array of impermeable obstacles such as perforations. Homogenisation theory provides a coarse-grained description of the scalar at large times and predicts that it diffuses with a certain effective diffusivity, so the concentration is approximately Gaussian. We improve on this by developing a large-deviation approximation which also captures the non-Gaussian tails of the concentration through a rate function obtained by solving a family of eigenvalue problems. We focus on cylindrical obstacles and on the dense limit, when the obstacles occupy a large area fraction and non-Gaussianity is most marked. We derive an asymptotic approximation for the rate function in this limit, valid uniformly over a wide range of distances. We use finite-element implementations to solve the eigenvalue problems yielding the rate function for arbitrary obstacle area fractions and an elliptic boundary-value problem arising in the asymptotics calculation. Comparison between numerical results and asymptotic predictions confirm the validity of the latter