A. De Wit

Miscible density fingering of chemical fronts in porous media: Nonlinear simulations

Nonlinear interactions between chemical reactions and Rayleigh–Taylor type density fingering are studied in porous media or thin Hele-Shaw cells by direct numerical simulations of Darcy’s law coupled to the evolution equation for the concentration of a chemically reacting solute controlling the density of miscible solutions. In absence of flow, the reaction-diffusion system features stable planar fronts traveling with a given constant speed v and width w. When the reactant and product solutions have different densities, such fronts are buoyantly unstable if the heavier solution lies on top of the lighter one in the gravity field. Density fingering is then observed. We study the nonlinear dynamics of such fingering for a given model chemical system, the iodate-arsenious acid reaction. Chemical reactions profoundly affect the density fingering leading to changes in the characteristic wavelength of the pattern at early time and more rapid coarsening in the nonlinear regime. The nonlinear dynamics of the syst...

Nonlinear evolution equations for thin liquid films with insoluble surfactants

The dynamics of a free‐liquid film with insoluble surfactants is followed until film rupture with a simple model based on three nonlinear evolution equations for the film thickness, the surfactants concentration and the tangential velocity of the fluid in the film. This model is derived asymptotically from the full Navier–Stokes equations for free films and incorporates the effect of van der Waals attraction, capillary forces and Marangoni forces due to gradients of surface tension. Different stability regimes are observed numerically for periodic and fixed boundary conditions and several initial conditions. Furthermore, the role of the relevant parameters (Hamaker constant, tension, Marangoni number) on the rupture time is assessed and comparison is made with the flow dynamics for a liquid film with insoluble surfactants on a solid substrate.

Viscous Fingering in reaction-diffusion systems

The problem of viscous fingering is studied in the presence of simultaneous chemical reactions. The flow is governed by the usual Darcy equations, with a concentration-dependent viscosity. The concentration field in turn obeys a reaction–convection–diffusion equation in which the rate of chemical reaction is taken to be a function of the concentration of a single chemical species and admits two stable equilibria separated by an unstable one. The solution depends on four dimensionless parameters: R, the log mobility ratio, Pe, the Peclet number, α, the Damkohler number or dimensionless rate constant, and d, the dimensionless concentration of the unstable equilibrium. The resulting nonlinear partial differential equations are solved by direct numerical simulation over a reasonably wide range of Pe, α, and d. We find new mechanisms of finger propagation that involve the formation of isolated regions of either less or more viscous fluid in connected domains of the other. Both the mechanism of formation of the...

Viscous fingering of miscible slices

Viscous fingering of a miscible high viscosity slice of fluid displaced by a lower viscosity fluid is studied in porous media by direct numerical simulations of Darcy’s law coupled to the evolution equation for the concentration of a solute controlling the viscosity of miscible solutions. In contrast with fingering between two semi-infinite regions, fingering of finite slices is a transient phenomenon due to the decrease in time of the viscosity ratio across the interface induced by fingering and dispersion processes. We show that fingering contributes transiently to the broadening of the peak in time by increasing its variance. A quantitative analysis of the asymptotic contribution of fingering to this variance is conducted as a function of the four relevant parameters of the problem, i.e., the log-mobility ratio R, the length of the slice l, the Peclet number Pe, and the ratio between transverse and axial dispersion coefficients e. Relevance of the results is discussed in relation with transport of visc...

Correlation structure dependence of the effective permeability of heterogeneous porous media

A theory is given in which the effective permeability tensor Keff of heterogeneous porous media is derived by a perturbation expansion of Darcy’s law in the variance σ2 of the log‐permeability ln[κ(ub;;‐45rubx)]. The only assumption is that the spatially varying permeability κ(ub;;‐45rubx) is a expressed in terms of the moments of the distribution of ln[κ(ub;;‐45rubx)], i.e. Keff can formally be computed for any given distribution of the fluctuations of the log‐permeability. The explicit dependence of Keff on multi‐point statistics is given for non‐gaussian log‐permeability fluctuations up to order σ6. As a special case of the theory, we examine Keff for a normal distribution function for both isotropic and anisotropic media. In the case of three‐dimensional isotropic porous media, a conjecture has been made in the past according to which the scalar effective permeability κeff=KGexp[σ2/6] where KG is the geometric mean of the log‐permeability. It is shown here that this conjecture is incorrect as the σ6‐o...

Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations

We study nonlinear viscous fingering in heterogeneous media through direct numerical simulation. A pseudospectral method is developed and applied to our spatially periodic model introduced in Paper I [J. Chem. Phys. 107, 9609 (1997)]. The problem involves several parameters, including the Peclet number, Pe, the magnitude and wave numbers of the heterogeneity, σ, nx, ny, respectively, and the log of the viscosity ratio R. Progress is made by fixing R at 3.0 and then working first with layered systems nx=0 and finally with “checkerboard” systems in which both wave numbers are nonzero. Strongly nonlinear finger dynamics are compared and contrasted with those occurring in the homogeneous case. For layered systems, it is found that very low levels of heterogeneity leads to an enhancement of the growth rate of the fingered zones, and that both harmonic and subharmonic resonances between the intrinsic scale of nonlinear fingering and those of the heterogeneity occur. We also find that the fingering regime of lay...

Rayleigh–Taylor instability of reaction-diffusion acidity fronts

We consider the buoyancy driven Rayleigh–Taylor instability of reaction-diffusion acidity fronts in a vertical Hele–Shaw cell using the chlorite–tetrathionate (CT) reaction as a model system. The acid autocatalysis of the CT reaction coupled to molecular diffusion yields isothermal planar reaction-diffusion fronts separating the two miscible reactants and products solutions. The reaction is triggered at the top of the Hele–Shaw cell and the resulting front propagates downwards, invading the fresh reactants, leaving the product of the reaction behind it. The density of the product solution is higher than that of the reactant solution, and hence a hydrodynamic instability develops due to unfavorable density stratification. We examine the linear stability of the isothermal traveling wavefront with respect to disturbances in the spanwise direction and demonstrate the existence of a preferred wavelength for the developed fingering instability. Our linear stability analysis is in excellent agreement with two-di...

Buoyancy-driven instabilities of miscible two-layer stratifications in porous media and Hele-Shaw cells

Buoyancy-driven instabilities of a horizontal interface between two miscible solutions in the gravity field are theoretically studied in porous media and Hele-Shaw cells (two glass plates separated by a thin gap). Beyond the classical Rayleigh-Taylor (RT) and double diffusive (DD) instabilities that can affect such two-layer stratifications right at the initial time of contact, diffusive-layer convection (DLC) as well as delayed-double diffusive (DDD) instabilities can set in at a later time when differential diffusion effects act upon the evolving density profile starting from an initial step-function profile between the two miscible solutions. The conditions for these instabilities to occur can therefore be obtained only by considering time evolving base-state profiles. To do so, we perform a linear stability analysis based on a quasi-steady-state approximation (QSSA) as well as nonlinear simulations of a diffusion-convection model to classify and analyse all possible buoyancy-driven instabilities of a stratification of a solution of a given solute A on top of another miscible solution of a species B. Our theoretical model couples Darcys law to evolution equations for the concentration of species A and B ruling the density of the miscible solutions. The parameters of the problem are a buoyancy ratio R quantifying the ratio of the relative contribution of B and A to the density as well as δ, the ratio of diffusion coefficients of these two species. We classify the region of RT, DD, DDD and DLC instabilities in the (R, δ) plane as a function of the elapsed time and show that, asymptotically, the unstable domain is much larger than the one captured on the basis of linear base-state profiles which can only obtain stability thresholds for the RT and DD instabilities. In addition the QSSA allows one to determine the critical time at which an initially stable stratification of A above B can become unstable with regard to a DDD or DLC mechanism when starting from initial step function profiles. Nonlinear dynamics are also analysed by a numerical integration of the full nonlinear model in order to understand the influence of R and δ on the dynamics.

Competition in ramped Turing structures

Abstract Stationary pattern selection and competition in the uniform Brusselator in two (2D) and three (3D) dimensions are reviewed, including reentrant hexagonal and striped zig-zag phases. Influences of linear or chain-like profiles of the pool chemicals on this selection are presented in the form of numerical experiments. The relation with the recent experimental patterns obtained with the CIMA reaction is discussed.

Nonlinear interactions of chemical reactions and viscous fingering in porous media

Nonlinear interactions of chemical reactions and viscous fingering are studied in porous media by direct numerical simulations of Darcy’s law coupled to the evolution equation for the concentration of a chemically reacting solute controlling the viscosity of miscible solutions. Chemical kinetics introduce important topological changes in the fingering pattern: new robust pattern formation mechanisms such as droplet formation and enhanced tip splitting are evidenced and analyzed.