Fingering instabilities of exothermic reaction-diffusion fronts in porous media

Abstract

We consider the density fingering of exothermic autocatalytic fronts in vertically oriented Hele-Shaw cells with chemical reactions whose solutal and thermal contributions to density changes have opposite signs. Using the Darcy-Boussinesq equations we examine the influence of the competition between solutal and thermal density changes on the linear stability of traveling fronts and the fully nonlinear dynamics. Ascending fronts are characterized by standard Rayleigh-Taylor fingering dispersion curves and in the nonlinear stage of the instability they feature thermal plumes. Descending fronts on the other hand behave strikingly differently as they can feature for some values of the parameters Turing-type dispersion curves and stationary patterns with fingers of constant amplitude and wavelength.

Dispersion relations for the convective instability of an acidity front in Hele-Shaw cells

“…13 As the CT reaction is exothermic, heat release can contribute to the density change. 7,8,16 The total density variation across the front is therefore the sum of a solutal ⌬ S and thermal ⌬ T contribution, i.e., ⌬ ϭ⌬ S ϩ⌬ T . The isothermal ⌬ S does not depend on the gap width but varies with chemical composition.…”

“…Previous studies have shown that varying ␦ ͑Ref. 13͒ and taking explicitly into account the evolution equation for the temperature 8,16 can have an influence on dispersion curves so that further work to understand the influence of these effects on the 2D versus 3D description would be of interest. Our quantitative comparison with experimental data depends also on choices made on the numerical values of the kinetic constant ␥ and of ⌬ T , this last parameter being strongly dependent on chemical concentrations 9 and gap width.…”

“…To model the chemical reactions, we use two different points of view, i.e., the full reaction-diffusion system or the eikonal relation. Assuming that all chemical species diffuse with the same diffusion coefficient D, the reaction-diffusion-advection model for the isothermal CT reaction is 8,13,16 …”

“…7,8 On the basis of Darcy's law coupled to an evolution equation for both the concentration and the temperature, it has been shown theoretically that heat effects can qualitatively change the characteristics of the dispersion curves. 8,16 This enlightens the fact that, even in very thin Hele-Shaw cells for which Darcy's law is valid, the coupling between solutal and thermal effects can affect the nature of dispersion curves. In this context, it is necessary to check the influence of the hydrodynamic description of the fluid velocity on the dispersion curves.…”

On the classification of buoyancy-driven chemo-hydrodynamic instabilities of chemical fronts

“…Numerous modeling and theoretical analyses have been performed on fronts that exhibit simple convection. [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] Fewer analytical work have focused on double-diffusive convection of fronts 25,48,49 and it is only recently that heat effects have been shown to allow for new convective instabilities to come into play. 50,51 A comparison between the stability domains of pure fluids vs those of chemical fronts in the parameter space spanned by the solutal and thermal Rayleigh numbers associated with ⌬ c and ⌬ T shows that chemical reactions profoundly change the stability properties.…”