Density Fingering Research Articles

ALIGNMENT OF FINGERING PATTERN INDUCED BY AN INTERFACIAL NEUTRALIZATION REACTION

In this paper, we numerically study buoyancy-driven chemoconvective instabilities in a vertically oriented Hele-Shaw cell. At the beginning, the cavity is filled with a homogeneous aqueous solution of a strong inorganic alkali, and this system is statically stable. We assume that the upper boundary is free, and a fixed value of the concentration of a strong inorganic acid is set there. After bringing the reactants into contact, density stratification quickly becomes unstable due to a neutralization reaction, resulting in density fingering. The mathematical model includes the effect of producing a new amount of solvent (water). We show that this effect is a reason for the sudden alignment of salt fingers that we previously observed in experiments. In this work, we carry out a formal parametric study of the system with a change in the dimensionless parameter responsible for the intensity of water production. One can interpret this variation as a sequential consideration of substances from the homologous series of alkalis. We show that there exists a critical value of the parameter, crossing which one can observe a spontaneous transition from an irregular fingering pattern to cellular chemoconvection. The latter looks like a system of fingers tightly pressed against each other with tips aligned along a horizontal line. The lower boundary of the vortices uniformly moves down. We found that instead of the usual coarsening of the structure, one observes an increase in the aspect ratio of vortices. We investigate the dynamics of the lower boundary of the fingering pattern and changes in the pattern wavelength. Complex rearrangements of the reaction front, which include the processes of plume creation and coalescence, are illustrated using space-time diagrams.

Pore-scale study of miscible density instability with viscosity contrast in porous media

The transport of miscible fluids in porous media is a prevalent phenomenon that occurs in various natural and industrial contexts. However, this fundamental phenomenon is usually coupled with interface instabilities (e.g., viscous/density fingering), which has yet to be thoroughly investigated. In this paper, a multiple-relaxation-time lattice Boltzmann method is applied to study the displacement between two miscible fluids in porous media at the pore scale, with the coexistence of density difference (Rayleigh number Ra), viscosity contrast (R), and injection velocity (Utop). A parametric study is conducted to evaluate the impact of Ra, R, and Utop on the flow stability. For a fixed Ra that can trigger density fingering, the increase in R or Utop is found to suppress density fingering. Consequently, under a large Utop and a moderate R, the density fingering is fully stabilized and the flow follows a stabile pattern. Furthermore, as both R and Utop grow to a sufficiently high level, they can jointly trigger viscous fingering. In addition, the increasing Ra shows an enhancing effect on both density fingering and viscous fingering. Finally, by quantitatively analyzing the fingering length (lm) and the fingering propagation time (te), five different flow patterns are classified as viscosity-suppressed (I), viscosity-enhanced (II), viscosity-unstable (III), displacement-suppressed (IV), and stable (V) regimes. In a three-dimensional parameter space spanned by Ra, R, and Utop, the parameter ranges of the five regimes are determined according to lm and te. These findings hold a significant value in providing guidance for controlling the flow stability by selecting appropriate operating conditions.

Lattice Boltzmann Simulation of Multicomponent Porous Media Flows With Chemical Reaction

Flows with chemical reactions in porous media are fundamental phenomena encountered in many natural, industrial, and scientific areas. For such flows, most existing studies use continuum assumptions and focus on volume-averaged properties on macroscopic scales. Considering the complex porous structures and fluid–solid interactions in realistic situations, this study develops a sophisticated lattice Boltzmann (LB) model for simulating reactive flows in porous media on the pore scale. In the present model, separate LB equations are built for multicomponent flows and chemical species evolutions, source terms are derived for heat and mass transfer, boundary schemes are formulated for surface reaction, and correction terms are introduced for temperature-dependent density. Thus, the present LB model offers a capability to capture pore-scale information of compressible/incompressible fluid motions, homogeneous reaction between miscible fluids, and heterogeneous reaction at the fluid–solid interface in porous media. Different scenarios of density fingering with homogeneous reaction are investigated, with effects of viscosity contrast being clarified. Furthermore, by introducing thermal flows, the solid coke combustion is modeled in porous media. During coke combustion, fluid viscosity is affected by heat and mass transfer, which results in unstable combustion fronts.

Interface Fingering Instability Triggered by a Density-Coupled Oscillatory Chemical Reaction via Precipitation.

A density fingering hydrodynamic instability is triggered by a chemical reaction at the interface between two fluids. The density instability is controlled by the density gradient between both solutions, while the excitability of the bubble-free Belousov-Zhabotinsky-1,4-cyclohexanedione (BZ-CHD) oscillatory chemical reaction controls the importance of the chemistry in the system. Both parameters are thoroughly analyzed, and the mechanism underlying the instability is unveiled. The experimental observations lead us to modify the existing and accepted models for the BZ-CHD reaction within this context. The important role played by precipitation is considered in this context and included into the model. The modified kinetic model once coupled with fluid dynamics along with the precipitation mechanism was able to reproduce the experimental observations.

Pore-scale study of dissolution-driven density instability with reaction A+B→C in porous media

Dissolution-driven density instability (DI) occurs when a species A dissolves into a host fluid and introduces a buoyantly unstable stratification. Such an instability has positive effects in related applications and may be affected if species A reacts with solute B in the host fluid. In this paper, the lattice Boltzmann (LB) method is employed to simulate the dynamics of such an instability coupled with reaction A + B → C in porous media at the pore scale. Numerical simulations in homogeneous media have demonstrated that six types of dissolution-driven DI can be classified based on the Rayleigh numbers of three chemical species Ra r (ratio of buoyancy to viscous forces), and reaction can accelerate, delay or even trigger the development of DI. Then, a parametric study has indicated that, increasing Δ Ra C B ( Ra C − Ra B ) can intensify density instability and reaction, promote the diffusion of species A , and also introduce either stabilizing or destabilizing effects of reaction. Besides, the increase of initial reactant concentration or/and Damkohler number Da (ratio of flow time to chemical time) can enhance the influence of chemistry. Finally, simulations are carried out in three types of heterogeneous media HE1–HE3, and six groups of fingering scenarios can also be observed in each medium. However, compared with the homogeneous case, heterogeneous media HE1 with randomly distributed solid grains can introduce deeper advancing position and rougher density fingering, and media HE2 and HE3 with vertical variations of pore spaces can affect the developing speed of fingering obviously. In terms of the storage of species A in the host fluid, medium HE2 with large pore size in the top layer is favorable. The present study is of significant importance for applications such as carbon capture and storage.

Experimental analysis of density fingering instability modified by precipitation.

We analyze the effect of precipitate formation on the development of density induced hydrodynamic instabilities. In this case, the precipitate is BaCO3, obtained by reaction of CO2 with aqueous BaCl2. CO2(g) dissolution increases the local density of the aqueous phase, triggering Rayleigh-Taylor instabilities and BaCO3 formation. It was observed that at first the precipitate was formed at the finger front. As the particles became bigger, they began to fall down from the front. These particles were used as tracers using PIV technique to visualize the particle streamlines and to obtain the velocity of that movement. This falling produced a downward flow that might increase the mixing zone. Contrary to expectations, it was observed that the finger length decreased, indicating that for the mixing zone development, the consumption of CO2 to form the precipitate is more important than the downward flow. The mixing zone length was recovered by increasing the availability of the reactant (higher CO2 partial pressure), compensating the CO2 used for BaCO3 formation. Mixing zone development rates reached constant values at shorter times when the precipitate is absent than when it is present. An analysis of the nonlinear regime with and without the precipitate is performed.

Development of Concentration-Dependent Diffusion Instability in Reactive Miscible Fluids Under Influence of Constant or Variable Inertia

In this work, we focus on the processes which accompany a frontal neutralization reaction occurring between two miscible fluids filling a vertical Hele-Shaw cell. We have found that chemically-induced changes of reagent concentrations coupled with concentration- dependent diffusion (CDD) can produce spatially localized low density areas which are sensitive to the external inertial field. In the case of static gravity we have demonstrated both experimentally and theoretically that it can give rise to the development of perfectly periodic convective structure. This scenario is strikingly different from the irregular density fingering, which is typically observed in the miscible systems. When the system is under the influence of the periodic low-frequency vibrations perpendicular to the reaction front, we found numerically the excitation of a mixed-mode instability combining the double-diffusion instabilities and the Rayleigh-Taylor mechanism of the convection within the low density areas.

Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models

We present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua. In the framework of mixture theory, we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases. The model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture (barycentric) velocities, phase pressure, and saturation. We construct an accurate, robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume-element discretization of the mass (or transport) equations. The scheme is devised for unstructured meshes and relies on mixed Brezzi–Douglas–Marini approximations of phase and total velocities, on piecewise constant elements for the approximation of phase or total pressures, as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest (such as volume fraction, total density, saturation, etc.). As the discretization scheme is derived for a general formulation of multicontinuum physical systems, it can be readily applied to a large class of simplified multiphase models; on the other, the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate. An extensive set of numerical test cases involving two- and three-dimensional porous media are presented to demonstrate the accuracy of the method (displaying an optimal convergence rate), the physics-preserving properties of the mixed–primal scheme, as well as the robustness of the method (which is successfully used to simulate diverse physical phenomena such as density fingering, Terzaghi's consolidation, deformation of a cantilever bracket, and Boycott effects). The applicability of the method is not limited to flow in porous media, but can also be employed to describe many other physical systems governed by a similar set of equations, including e.g. multi-component materials.

On Rayleigh-Bénard Mechanism of Alignment of Salt Fingers in Reactive Immiscible Two-Layer Systems

The stability of a front of chemoconvective finger structures being spontaneously formed in a two-layer system of fluids filling a vertical Hele-Shaw cell, each of them containing a reactant of an exothermic A + B → S reaction is examined. If the configuration consists of more dense acid (or salt) on top of less dense base in the presence of gravity, the development of the Rayleigh- Taylor instability leads to the standard scenario of density fingering. Despite the widespread perception of fingering as an irregular process, we show that, at least in some cases, the exact balance between the instabilities involved can result quasi-regular fingering pattern formation. In the case of immiscible fluids, we demonstrate that the Rayleigh-Benard mechanism associated with intensive heat release during the reaction performs fine-tuning of the envelope of salt fingers. The mathematical model we develop consists in a set of reaction-diffusion-convection equations governing the evolution of concentrations and temperature coupled to Navier-Stokes and energy equations, written in a Hele-Shaw approximation. The results of linear stability analysis and direct numerical simulations of the fully nonlinear system are presented.

Tuning density fingering by changing stoichiometry in the chlorite–tetrathionate reaction

We show that buoyancy-driven spatiotemporal pattern formation can be conveniently tuned by varying the ratio of the initial concentrations of the reactants. The increase in the stoichiometric excess of chlorite even leads to the reversal of the sign of the net density change during the reaction and hence the originally unstable downward propagating reaction diffusion fronts can be stabilized. The dispersion curves quantitatively characterizing the density fingering are in good agreement with the observed behavior.

Influence of a simple magnetic bar on buoyancy-driven fingering of traveling autocatalytic reaction fronts

Magnetic fields have been shown experimentally to modify convective dynamics developing around traveling chemical fronts in presence of unfavorable density gradients. To understand the conditions in which such magnetic fields affect autocatalytic fronts, we study theoretically the influence of a simple magnetic bar on buoyancy-driven density fingering of a chemical front by numerical simulations of a reaction-diffusion-convection system. The model couples Darcy's law for the flow velocity to an evolution equation for the concentration of the autocatalytic product, which affects both the density of the solution and the magnetic force. The solutions of both products and reactants are assumed to be diamagnetic (i.e., negative magnetic susceptibility) and the magnetization is oriented perpendicularly to the plane in which the front travels. We show that, when aligned along the direction of front propagation, the magnetic force is able to suppress or enhance the convective instability depending on the value of the magnetic Rayleigh number of the problem. If the magnetic force is oriented transversely to the front propagation direction, tilted drifting convective patterns are obtained.

Convection and reaction in a diffusive boundary layer in a porous medium: Nonlinear dynamics

We study numerically the nonlinear interactions between chemical reaction and convective fingering in a diffusive boundary layer in a porous medium. The reaction enhances stability by consuming a solute that is unstably distributed in a gravitational field. We show that chemical reaction profoundly changes the dynamics of the system, by introducing a steady state, shortening the evolution time, and altering the spatial patterns of velocity and concentration of solute. In the presence of weak reaction, finger growth and merger occur effectively, driving strong convective currents in a thick layer of solute. However, as the reaction becomes stronger, finger growth is inhibited, tip-splitting is enhanced and the layer of solute becomes much thinner. Convection enhances the mass flux of solute consumed by reaction in the boundary layer but has a diminishing effect as reaction strength increases. This nonlinear behavior has striking differences to the density fingering of traveling reaction fronts, for which stronger chemical kinetics result in more effective finger merger owing to an increase in the speed of the front. In a boundary layer, a strong stabilizing effect of reaction can maintain a long-term state of convection in isolated fingers of wavelength comparable to that at onset of instability.

Hot spots in density fingering of exothermic autocatalytic chemical fronts

Measurements of two-dimensional (2D) temperature fields are performed by an interferometric method during density fingering of the autocatalytic chlorite-tetrathionate reaction in a Hele-Shaw cell. These measures confirm that, because of heat losses through the glass walls of the reactor, the temperature profile across the front is a pulse rather than a front. Moreover, the full 2D temperature field shows the presence in the reactive zone of hot spots where the temperature exceeds the maximum temperature measured in a stable planar front. We investigate here experimentally the increase of temperature in the hot spots when the composition of the reactants is varied to increase the exothermicity of the reaction. We back up these experimental observations by nonlinear simulations of a reaction-diffusion-convection model which show that the maximum temperature reached in the system depends on the intensity of convection.

Rayleigh–Taylor instability in nonlinear Schrödinger flow

The Rayleigh–Taylor instability (RTI) is a fundamental fluid instability that occurs when a light fluid is accelerated into a heavier one. While techniques for observing the RTI in classical fluids continue to improve, the instability has not been demonstrated in quantum fluids. Here, we exploit the formal equivalence between condensed matter and coherent nonlinear optics to observe the superfluid-like instability directly in the optical system. For the RTI, an initial refractive index gradient sets the acceleration, while self-induced nonlinear interactions lead to velocity differences and shear. The experimental observations show that density fingering is always accompanied by vortex generation, with perturbation modes following a hybrid dynamics: horizontal modes (along the interface) propagate as an incompressible fluid, but the vertical length scale (mixing length) is set by compressible shock dynamics. The growth rate, obtained analytically, shows that inhibition due to diffraction has the same spectral form as viscosity and diffusion, despite the fact that the system is dispersive rather than dissipative. This gives rigorous support for the observation that turbulence in quantum fluids has the same scaling as turbulence in normal fluids. The results hold for any Schrödinger flow, e.g. superfluids and quantum plasma, and introduce a new class of fluid-inspired instabilities in nonlinear optics.

Stability of reactive interfaces in saturated porous media under gravity in the presence of transverse flows

AbstractThe stability of a horizontal interface between a solution of reactant $A$ on top of another solution of reactant $B$ is analysed. A chemical product $C$ is generated at the interface as a result of a bimolecular chemical reaction $A+ B\ensuremath{\rightarrow} C$. In general, all chemical components are assumed to have different densities and viscosities, and a transverse velocity is introduced parallel to the interface between the reactants. Although the transverse flow is known for its stabilizing effect in viscously unstable non-reactive systems in the presence of an injection velocity, it is shown here that it can actually destabilize an initially stable reactive front. An expression for the critical transverse velocity beyond which an initially stable interface is destabilized is derived in the case of an initial sharp interface for reactants of the same viscosity. The analysis is extended to a diffused profile, and purely buoyancy-driven flows are analysed first in the absence of viscosity contrast and then in the presence of transverse flows and viscosity contrast. Various possible density fingering scenarios are determined based on the relative contribution of each chemical component to the density profile. It is found that the chemical reaction can destabilize a buoyancy-stable initial interface by generating a non-monotonic density profile. Unlike the viscous fingering of a reactive interface, a symmetry in the stability characteristics with respect to density increase or decrease by chemical reaction product is observed in the case of chemically buoyancy-driven flows for identical reactants.

Density fingering in spatially bimodulated Hele-Shaw cells

Density fingering in the chlorite-tetrathionate reaction has been studied experimentally in Hele-Shaw cells with spatially modulated gapwidth. The periodic modulation parallel and perpendicular to the front propagation is introduced in the form of round depressions in the inner wall of the reaction vessel. Both the initial and the long time behavior have been studied while varying the chemical composition, the gapwidth, and the periodicity of the modulation. We have shown that although the initial pattern formation is unaffected, an enhanced tip splitting accompanies the general coarsening of the structures during the long time behavior.

Viscosity-Change-Induced Density Fingering in Polyelectrolytes

We have studied the density fingering of an acid-catalyzed autocatalytic reaction in the presence of carboxylate containing polyelectrolyte. The decrease in viscosity as a result of the change in the ionic character of the polymer due to the pH-change during the reaction is the major driving force for the spatiotemporal pattern formation. The front evolution is quantitatively characterized by dispersion curves.

Erratum: “Scaling law of stable single cells in density fingering of chemical fronts” [J. Chem. Phys. 128, 144509 (2008)

Erratum: “Scaling law of stable single cells in density fingering of chemical fronts” [J. Chem. Phys. 128, 144509 (2008)

Scaling law of stable single cells in density fingering of chemical fronts

Stable single fingers evolving in a thin layer of hydrodynamically unstable reactive solution are investigated in moderately wide Hele-Shaw cells. The mixing length associated with the final pattern is shown to be proportional to the width of the container. The scaling law, which is best illustrated by the overlay of temporal average front profiles, is demonstrated for two chemical compositions and boundaries of different thermal conductances. The observed single finger is found to be independent of the orientation of the reaction vessel for a wide range of tilt angles.

Density fingering in spatially modulated Hele-Shaw cells

Density fingering of the chlorite-tetrathionate reaction has been studied experimentally in a periodically heterogeneous Hele-Shaw cell where the heterogeneity is introduced in the form of spatial modulation of gap width along the front. Depending on the spatial wavelength, gap width, and chemical composition, three types of cellular structures have been observed. The initial evolution is characterized by dispersion curves, while the long time behavior is described by the change in the autocorrelation function of the front profile and in the mixing length of the patterns.