Viscous fingering in reaction-diffusion systems

Effect of sinusoidal injection velocity on miscible viscous fingering of a finite sample: Nonlinear simulation

The effect of a sinusoidal injection on the fingering instability in a miscible displacement in the application of liquid chromatography, pollutant contamination in aquifers, etc., is investigated. The injection velocity, U ( t ) is characterized by its amplitude of Γ and time-period of T . The solute transport, flow in porous media, and mass conservation in a two-dimensional porous media is modeled by the convection-diffusion equation, Darcy's equation, and the continuity equation, respectively. The numerical simulation is performed in COMSOL Multiphysics utilizing a finite-element based approach. The fingering dynamics for various time-period have been studied for two scenarios namely, injection-extraction ( Γ > 1 ) and extraction-injection ( Γ < - 1 ). The onset of fingers and vigorous mixing is observed for Γ > 1 , whereas for Γ < - 1 , the onset gets delayed. The viscosity contrast between the sample and the surrounding fluid is characterized by the log-mobility ratio R . When R > 0 the rare interface becomes unstable, while for R < 0 the frontal interface deformed. In the case of R < 0 , the extraction-injection process attenuates the fingering dynamics, which is beneficial in chromatographic separations or pollutant dispersion in underground aquifers. The injection-extraction process is observed to have a longer mixing length, indicating early interaction between both interfaces. The degree of mixing χ ( t ) is more pronounced for injection-extraction scenario and least for extraction-injection R < 0 , Γ = - 2 . The average convective forces are more dominant for Γ > 1 , R = 2 till the deformed rare interface interact with diffusive frontal interface. The average diffusive forces are significant for Γ < - 1 , R = - 2 which can be helpful in separation of chemicals in chromatography. This study therefore provided new insights into the role of alternate injection-extraction injections in altering the fingering dynamics of the miscible sample.

Simulation of Nonlinear Viscous Fingering in a Reactive Flow Displacement: A Multifractal Approach

fractal analysis of viscous fingering of a reactive miscible flow displacement in homogeneous porous media is investigated and multifractal spectrum, and fractal dimension are introduced as two essential features to characterize the irregularity of finger patterns. The Reaction of the two reactant fluids generates a miscible chemical product C in the contact zone. Considering the similarity between chemical products and coastline, monofractal and multifractal analyzes are performed. In monofractal analysis, the box-counting method is implemented on binary images and in multifractal analysis, due to the image processing; the fractal characteristics of viscous fingering instability are analyzed by means of fractal quantities such as Holder exponent, multifractal spectrum, f (α)-image and fractal dimension dynamics. Fractal analysis shows that the fractal dimension increases with time. Also, by considering five different nonlinear simulations, the results show that in the case both sides of the chemical product C are unstable, the multifractal spectrum curve has the highest peak, which means the more complex finger patterns lead to more values of fractal dimension. In addition, a comparison between different values of Ar is conducted and the results show similar behavior. However, small value of aspect ratio leads to a broader width of the multifractal spectrum curve. Furthermore, f (α)-images of concentration contour were investigated for different precisions and some undetectable finger patterns were observed in these images. It can be concluded that the use of f (α)-image represents more detailed image than concentration contours.

Spatial programming of self-organizing chemical systems using sustained physicochemical gradients from reaction, diffusion and hydrodynamics

Living organisms employ chemical self-organization to build structures, and inspire new strategies to design synthetic systems that spontaneously take a particular form, via a combination of integrated chemical reactions, assembly pathways and physicochemical processes. However, spatial programmability that is required to direct such self-organization is a challenge to control. Thermodynamic equilibrium typically brings about a homogeneous solution, or equilibrium structures such as supramolecular complexes and crystals. This perspective addresses out-of-equilibrium gradients that can be driven by coupling chemical reaction, diffusion and hydrodynamics, and provide spatial differentiation in the self-organization of molecular, ionic or colloidal building blocks in solution. These physicochemical gradients are required to (1) direct the organization from the starting conditions (e.g. a homogeneous solution), and (2) sustain the organization, to prevent it from decaying towards thermodynamic equilibrium. We highlight four different concepts that can be used as a design principle to establish such self-organization, using chemical reactions as a driving force to sustain the gradient and, ultimately, program the characteristics of the gradient: (1) reaction-diffusion coupling; (2) reaction-convection; (3) the Marangoni effect and (4) diffusiophoresis. Furthermore, we outline their potential as attractive pathways to translate chemical reactions and molecular/colloidal assembly into organization of patterns in solution, (dynamic) self-assembled architectures and collectively moving swarms at the micro-, meso- and macroscale, exemplified by recent demonstrations in the literature.

A bottom-up approach to construct or deconstruct a fluid instability

Fluid instabilities have been the subject of study for a long time. Despite all the extensive knowledge, they still constitute a serious challenge for many industrial applications. Here, we experimentally consider an interface between two fluids with different viscosities and analyze their relative displacement. We designed the contents of each fluid in such a way that a chemical reaction takes place at the interface and use this reaction to suppress or induce a fingering instability at will. This process describes a road map to control viscous fingering instabilities in more complex systems via interfacial chemical reactions.

Nanocatalytic chemohydrodynamic instability: Deposition effects

Due to the high surface area to volume ratio of nanoparticles, nanocatalytic reactive flows are widely utilized in various applications, such as water purification, fuel cell, energy storage, and biodiesel production. The implementation of nanocatalysts in porous media flow, such as oil recovery and contaminant transport in soil, can trigger or modify the interfacial instabilities called viscous fingering. These instabilities grow at the interface of the fluids when a less viscous fluid displaces a high viscous one in porous media. Here the flow dynamics and the total amount of chemical product are investigated when two reactive miscible fluids meet in a porous medium while undergoing A+B+n → C+n reaction. Nanocatalysts (n) are dispersed in the displacing fluid and deposited gradually with time. Four generic regimes are observed over time as a result of the particle deposition: (1) the initial diffusive regime, where the flow is stable with decreasing production rate, (2) the mixing-dominant fingering regime, where the flow is unstable and the production rate generally increases, (3) the transition regime, where the production rate generally decreases regardless of whether the system is stable or unstable, and (4) the final zero-production regime, where the product diffuses and fades away in the channel. Although the general trend shows a decreasing reaction rate with nanocatalysts deposition, there is a period in which the production rate increases due to the moderate deposition rates. Such an increase of production, however, is not observed in two groups: first, those systems in which the nanocatalysts do not change the viscosity of the base fluid and, second, a subgroup of the systems that are stable before and after the reaction in the absence of deposition.

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.

Elastic fingering in three dimensions

Recent studies on quasi-two-dimensional (2D) fluid flows in Hele-Shaw cells revealed the emergence of the so-called elastic fingering phenomenon. This pattern-forming process takes place when a reaction occurs at the fluid-fluid interface, transforming it into an elastic gel-like boundary. The interplay of viscous and elastic forces leads to the development of pattern morphologies significantly different from those seen in the conventional, purely hydrodynamic viscous fingering problem. In this work, we investigate the occurrence of elastic fingering for radial fluid displacements in a 3D uniform porous medium. A perturbative third-order mode-coupling approach is employed to examine how the combined action of viscous and elastic effects influences the linear stability of the interface, and the weakly nonlinear pattern formation in such a 3D environment. In addition, a variational method is used to determine how to minimize the growth of interfacial perturbation amplitudes via a time-dependent injection rate scheme.

Flow instabilities due to the interfacial formation of surfactant-fatty acid material in a Hele-Shaw cell

We present an experimental study of pattern formation during the penetration of an aqueous surfactant solution into a liquid fatty acid in a Hele-Shaw cell. When a solution of the cationic surfactant cetylpyridinium chloride is injected into oleic acid, a wide variety of fingering patterns are observed as a function of surfactant concentration and flow rate, which are strikingly different than the classic Saffman-Taylor (ST) instability. We observe evidence of interfacial material forming between the two liquids, causing these instabilities. Moreover, the number of fingers decreases with increasing flow rate Q, while the average finger width increases with Q, both trends opposite to the ST case. Bulk rheology on related mixtures indicates a gel-like state. Comparison of experiments using other oils indicates the importance of pH and the carboxylic head group in the formation of the surfactant-fatty acid material.

The effect of a crosslinking chemical reaction on pattern formation in viscous fingering of miscible fluids in a Hele-Shaw cell

Viscous fingering can occur in fluid motion whenever a high mobility fluid displaces a low mobility fluid in a Darcy type flow. When the mobility difference is primarily attributable to viscosity (e.g., flow between the two horizontal plates of a Hele-Shaw cell), viscous fingering (VF) occurs, which is sometimes termed the Saffman-Taylor instability. Alternatively, in the presence of differences in density in a gravity field, buoyancy-driven convection can occur. These instabilities have been studied for decades, in part because of their many applications in pollutant dispersal, ocean currents, enhanced petroleum recovery, and so on. More recent interest has emerged regarding the effects of chemical reactions on fingering instabilities. As chemical reactions change the key flow parameters (densities, viscosities, and concentrations), they may have either a destabilizing or stabilizing effect on the flow. Hence, new flow patterns can emerge; moreover, one can then hope to gain some control over flow instabilities through reaction rates, flow rates, and reaction products. We report effects of chemical reactions on VF in a Hele-Shaw cell for a reactive step-growth cross-linking polymerization system. The cross-linked reaction product results in a non-monotonic viscosity profile at the interface, which affects flow stability. Furthermore, three-dimensional internal flows influence the long-term pattern that results.

Fingering instability and mixing of a blob in porous media

The curvature of the unstable part of the miscible interface between a circular blob and the ambient fluid in two-dimensional homogeneous porous media depends on the viscosity of the fluids. The influence of the interface curvature on the fingering instability and mixing of a miscible blob within a rectilinear displacement is investigated numerically. The fluid velocity in porous media is governed by Darcy's law, coupled with a convection-diffusion equation that determines the evolution of the solute concentration controlling the viscosity of the fluids. Numerical simulations are performed using a Fourier pseudospectral method to determine the dynamics of a miscible blob (circular or square). It is shown that for a less viscous circular blob, there exist three different instability regions without any finite R-window for viscous fingering, unlike the case of a more viscous circular blob. Critical blob radius for the onset of instability is smaller for a less viscous blob as compared to its more viscous counterpart. Fingering enhances spreading and mixing of miscible fluids. Hence a less viscous blob mixes with the ambient fluid quicker than the more viscous one. Furthermore, we show that mixing increases with the viscosity contrast for a less viscous blob, while for a more viscous one mixing depends nonmonotonically on the viscosity contrast. For a more viscous blob mixing depends nonmonotonically on the dispersion anisotropy, while it decreases monotonically with the anisotropic dispersion coefficient for a less viscous blob. We also show that the dynamics of a more viscous square blob is qualitatively similar to that of a circular one, except the existence of the lump-shaped instability region in the R-Pe plane. We have shown that the Rayleigh-Taylor instability in a circular blob (heavier or lighter than the ambient fluid) is independent of the interface curvature.

Chemo-hydrodynamic patterns in porous media

Chemical reactions can interplay with hydrodynamic flows to generate chemo-hydrodynamic instabilities affecting the spatio-temporal evolution of the concentration of the chemicals. We review here such instabilities for porous media flows. We describe the influence of chemical reactions on viscous fingering, buoyancy-driven fingering in miscible systems, convective dissolution as well as precipitation patterns. Implications for environmental systems are discussed.This article is part of the themed issue 'Energy and the subsurface'.

Fingering dynamics driven by a precipitation reaction: Nonlinear simulations

A fingering instability can develop at the interface between two fluids when the more mobile fluid is injected into the less-mobile one. For example, viscous fingering appears when a less viscous (i.e., more mobile) fluid displaces a more viscous (and hence less mobile) one in a porous medium. Fingering can also be due to a local change in mobility arising when a precipitation reaction locally decreases the permeability. We numerically analyze the properties of the related precipitation fingering patterns occurring when an A+B→C chemical reaction takes place, where A and B are reactants in solution and C is a solid product. We show that, similarly to reactive viscous fingering patterns, the precipitation fingering structures differ depending on whether A invades B or vice versa. This asymmetry can be related to underlying asymmetric concentration profiles developing when diffusion coefficients or initial concentrations of the reactants differ. In contrast to reactive viscous fingering, however, precipitation fingering patterns appear at shorter time scales than viscous fingers because the solid product C has a diffusivity tending to zero which destabilizes the displacement. Moreover, contrary to reactive viscous fingering, the system is more unstable with regard to precipitation fingering when the high-concentrated solution is injected into the low-concentrated one or when the faster diffusing reactant displaces the slower diffusing one.

Interface evolution during radial miscible viscous fingering

We study experimentally the miscible radial displacement of a more viscous fluid by a less viscous one in a horizontal Hele-Shaw cell. For the range of tested injection rates and viscosity ratios we observe two regimes for the evolution of the fluid-fluid interface. At early times the interface length increases linearly with time, which is typical of the Saffman-Taylor instability for this radial configuration. However, as time increases, the interface growth slows down and scales as ∼t(1/2), as one expects in a stable displacement, indicating that the overall flow instability has shut down. Surprisingly, the crossover time between these two regimes decreases with increasing injection rate. We propose a theoretical model that is consistent with our experimental results, explains the origin of this second regime, and predicts the scaling of the crossover time with injection rate and the mobility ratio. The key determinant of the observed scalings is the competition between advection and diffusion time scales at the displacement front, suggesting that our analysis can be applied to other interfacial-evolution problems such as the Rayleigh-Bénard-Darcy instability.

Diffusive fingering in a precipitation reaction driven by autocatalysis

The interaction of an autocatalytic reaction with a fast precipitation reaction is shown to produce a permanent precipitate pattern where the major driving force is differential diffusion. The final structure emerges from the leading transient cellular front, the cusps of which evolve into precipitate free zones. The experimental observations are reproduced by a simple model calculation based on the empirical rate-law of the reaction.

CO2 sequestration in a radial Hele-Shaw cell via an interfacial chemical reaction

In this manuscript, experimental data for the displacement of a finite volume of aqueous Ca(OH)(2) using CO(2) gas in a radial Hele-Shaw cell will be presented. This chemical reaction is known to generate CaCO(3) precipitate along the gas-liquid interface and we seek to understand the influence of the reactive process on fluid displacement. The reactive experiment is compared with the non-reactive case to determine if there are any measurable differences between the two in the range of parameters: CO(2) pressures (1%-10% of an atmosphere measured in gage pressure), liquid volumes (either 50 or 70 μl), and Ca(OH)(2) concentrations (0, 10, or 20 mM) studied. Analysis is performed by measuring the displacing fluid area A(gas) and total fluid area A(tot) to determine several quantities (gas expansion rate, quasi-equilibrium film rate and value, and presence of fingering instability) used to distinguish the experiments. In general there appears to be little effect of the chemical reaction on most of the measured quantities.

Experimental evidence of reaction-driven miscible viscous fingering

An experimental demonstration of reaction-driven viscous fingering developing when a more viscous solution of a reactant A displaces a less viscous miscible solution of another reactant B is presented. In the absence of reaction, such a displacement of one fluid by another less mobile one is classically stable. However, a simple A+B→C reaction can destabilize this interface if the product C is either more or less viscous than both reactant solutions. Using the pH dependence of the viscosity of some polymer solutions, we provide experimental evidence of both scenarios. We demonstrate quantitatively that reactive viscous fingering results from the buildup in time of nonmonotonic viscosity profiles with patterns behind or ahead of the reaction zone, depending on whether the product is more or less viscous than the reactants. The experimental findings are backed up by numerical simulations.

Quantifying mixing in viscously unstable porous media flows

Viscous fingering is a well-known hydrodynamic instability that sets in when a less viscous fluid displaces a more viscous fluid. When the two fluids are miscible, viscous fingering introduces disorder in the velocity field and exerts a fundamental control on the rate at which the fluids mix. Here we analyze the characteristic signature of the mixing process in viscously unstable flows, by means of high-resolution numerical simulations using a computational strategy that is stable for arbitrary viscosity ratios. We propose a reduced-order model of mixing, which, in the spirit of turbulence modeling and in contrast with previous approaches, recognizes the fundamental role played by the mechanical dissipation rate. The proposed model captures the nontrivial interplay between channeling and creation of interfacial area as a result of viscous fingering.

A bidirectional interface growth model for cranial interosseous suture morphogenesis

Interosseous sutures exhibit highly variable patterns of interdigitation and corrugation. Recent research has identified fundamental molecular mechanisms of suture formation, and computer models have been used to simulate suture morphogenesis. However, the role of bone strain in the development of complex sutures is largely unknown, and measuring suture morphologies beyond the evaluation of fractal dimensions remains a challenge. Here we propose a morphogenetic model of suture formation, which is based on the paradigm of Laplacian interface growth. Computer simulations of suture morphogenesis under various boundary conditions generate a wide variety of synthetic sutural forms. Their morphologies are quantified with a combination of Fourier analysis and principal components analysis, and compared with natural morphological variation in an ontogenetic sample of human interparietal suture lines. Morphometric analyses indicate that natural sutural shapes exhibit a complex distribution in morphospace. The distribution of synthetic sutures closely matches the natural distribution. In both natural and synthetic systems, sutural complexity increases during morphogenesis. Exploration of the parameter space of the simulation system indicates that variation in strain and/or morphogen sensitivity and viscosity of sutural tissue may be key factors in generating the large variability of natural suture complexity.

Miscible viscous fingering induced by a simple A+B-->C chemical reaction

Viscous fingering (VF) is a hydrodynamical instability that occurs in porous media when a less viscous fluid displaces a more viscous one. We investigate here numerically how such an instability can be triggered by a simple A+B-->C reaction when a solution of one reactant is displacing linearly a miscible solution of another reactant of same viscosity producing a more viscous product C at the interface. The properties of the fingering pattern observed in the zone where the less viscous reactant pushes more viscous products are studied as a function of the relevant parameters of the problem. These are the Damköhler number, the viscosity contrast between reactants and product, the ratio of initial concentrations of A and B , and the diffusion coefficients of each species. Our study shows that the fingering pattern can in some cases be different whether A displaces B or vice versa, enlightening recent experimental observations of such asymmetries in micellar systems. In particular, we show that in this asymmetric case, VF is more intense when the invading chemical solution is either the less concentrated one for equal diffusivities or when it contains the slower diffusing reactant for fixed equimolar initial concentrations.

Miscible viscous fingering with a chemical reaction involving precipitation

We experimentally investigated the effects of a chemical reaction involving precipitation on the miscible viscous fingering pattern formed in a Hele-Shaw cell. The precipitation concentration, the ratio of the reactant concentrations initially included in the more- and less-viscous liquids, and the Péclet number were varied. For a Péclet number at the stoichiometric ratio the precipitation had significant effects on the fingering pattern when its concentration exceeded a threshold value. Interestingly, the type of effect of the precipitation on the pattern depended on its concentration. At moderate concentration, a straight-shaped finger was observed. At high concentration, the finger was bent in an almost perpendicular direction. The effect of precipitation on the pattern also depended on the ratio of reactant concentrations.

Viscous fingering in a horizontal flow through a porous medium induced by chemical reactions under isothermal and adiabatic conditions

In this work we analyze the viscous fingering instability induced by an autocatalytic chemical reaction in a liquid flowing horizontally through a porous medium. We have analyzed the behavior of the system for isothermal as well as adiabatic conditions. The kinetics of the reaction is chosen so that the rate depends on the concentration of only a single species. Since the reaction is autocatalytic the system admits a traveling wave solution. For endothermic reactions the concentration wave and temperature wave are mirror images, whereas for an exothermic reaction they are similar or parallel. The viscosity of the fluid is assumed to depend strongly on the concentration of the product and temperature of the medium. The dependence of viscosity on concentration (decrease with concentration) can destabilize the traveling wave resulting in the formation of viscous fingers. We have performed a linear stability analysis to determine the stability of the base traveling wave solution. The stability predictions have been confirmed by nonlinear simulations of the governing equations based on a finite difference scheme. We observe that including the temperature dependency of viscosity stabilizes the flow for an endothermic reaction, i.e., regions which exhibited viscous fingering now demonstrate stable displacement. For exothermic systems, however, the system exhibits less stable behavior under adiabatic conditions, i.e., it is destabilized by both concentration and temperature dependencies of viscosity.

Propagation velocities of chemical reaction fronts advected by Poiseuille flow

Poiseuille flow between parallel plates advects chemical reaction fronts, distorting them and altering their propagation velocities. Analytical solutions of the cubic reaction-diffusion-advection equation resolve the chemical concentration for narrow gaps, wide gaps, and small-amplitude flow. Numerical solutions supply a general description for fluid flow in the direction of propagation of the chemical reaction front, and for flow in the opposite direction. Empirical relations for the velocity agree with numerical solutions to within a few percent, and agree exactly with the analytical limits. Applications to nonlinear fingering are discussed.

Nonlinear fingering dynamics of reaction-diffusion acidity fronts: Self-similar scaling and influence of differential diffusion

Nonlinear interactions between chemical reactions and buoyancy-driven Rayleigh-Taylor instability of reaction-diffusion acidity fronts of the chlorite-tetrathionate (CT) reaction are studied theoretically in a vertical Hele-Shaw cell or a porous medium. To do so, we perform a numerical integration of a two-variable reaction-diffusion model of the CT system coupled through an advection term to Darcy's law ruling the evolution of the velocity field of the fluid. The fingering dynamics of these chemical fronts is characterized by the appearance of several fingers at onset. These fingers then undergo coarsening and eventually merge to form one single symmetric finger. We study this asymptotic dynamics as a function of the three dimensionless parameters of the problem, i.e., the Damkohler number Da, the diffusivity ratio delta of the two chemical species, and the Rayleigh number Ra constructed here on the basis of the width L(y) of the system. For moderate values of Ra, the asymptotic single finger is shown to have self-similar scaling properties while above a given value of Ra, which depends on the other values of the parameters, tip splitting comes into play. Increasing the difference of diffusivities of the two chemical species (i.e., increasing delta) leads to more efficient coarsening and smaller asymptotic fingers. Experimental procedures to verify our predictions are proposed.

Structural and dynamical characterization of Hele-Shaw viscous fingering

Viscous fingering occurs in the interfacial zone between two fluids confined between two plates with a narrow gap (Hele-Shaw geometry) when a highly viscous fluid is displaced by a fluid with relatively low viscosity. Using a mesoscopic approach--the lattice Boltzmann method--we investigate the dynamics of spatially extended Hele-Shaw flow under conditions corresponding to various experimental systems by tuning the 'surface tension' and the reactivity between the two fluids. We discuss the onset of the fingering instability (dispersion relation), analyse the structural properties (characterization of the interface) and the dynamical properties (growth of the mixing zone) of the Hele-Shaw systems, and show the effect of reactive processes on the structure of the interfacial zone.

Hydrodynamics of bacterial colonies: a model

We propose a hydrodynamic model for the evolution of bacterial colonies growing on soft agar plates. This model consists of reaction-diffusion equations for the concentrations of nutrients, water, and bacteria, coupled to a single hydrodynamic equation for the velocity field of the bacteria-water mixture. It captures the dynamics inside the colony as well as on its boundary and allows us to identify a mechanism for collective motion towards fresh nutrients, which, in its modeling aspects, is similar to classical chemotaxis. As shown in numerical simulations, our model reproduces both usual colony shapes and typical hydrodynamic motions, such as the whirls and jets recently observed in wet colonies of Bacillus subtilis. The approach presented here could be extended to different experimental situations and provides a general framework for the use of advection-reaction-diffusion equations in modeling bacterial colonies.

Fingering of chemical fronts in porous media

The influence of chemical reactions on the hydrodynamical fingering instability is analyzed for miscible systems in porous media. Using a realistic reaction scheme, it is shown that the stability of chemical fronts towards density fingering crucially depends on the width and the speed of the front which are functions of chemical parameters. The major difference between the pure and chemically driven fingering is that, in the presence of chemical reactions, the dispersion curves do not vary in time which has important practical experimental consequences. Good agreement with recent experimental data is found.

Growth rates of the buoyancy-driven instability of an autocatalytic reaction front in a narrow cell

Experimental studies were performed on the buoyancy-driven instability of an autocatalytic reaction front in a quasi-2D cell. The unstable density stratification at an ascending front leads to convection that results in a fingerlike front deformation. The growth rates of the spatial modes of the instability are determined at the initial stage. A stabilization is found at higher wave numbers, while the system is unstable against low wave number perturbations. Whereas comparison with a reported model governed by Hele-Shaw flow fails, a two-dimensional Navier-Stokes model yields more satisfactory results. Still, present deviations suggest the presence of an additional mechanism that suppresses the growth.