Experimental evidence of reaction-driven miscible viscous fingering

Elastic fingering in a rotating Hele-Shaw cell

We consider the steady-state fingering instability of an elastic membrane separating two fluids of different density under external pressure in a rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are considered, and the role of membrane tension is detailed in each case. Both systems exhibit a centrifugally driven Rayleigh-Taylor-like instability when the density of the inner fluid exceeds that of the outer one, and this instability competes with the restoring forces arising from curvature and tension, thereby setting the finger scale. Numerical continuation is used to compute not only strongly nonlinear primary finger states up to the point of self-contact, but also secondary branches of mixed modes and circumferentially localized folds as a function of the rotation rate and the externally imposed pressure. Both reflection-symmetric and symmetry-broken chiral states are computed. The results are presented in the form of bifurcation diagrams. The ratio of system scale to the natural length scale is found to determine the ordering of the primary bifurcations from the unperturbed circle state as well as the solution profiles and onset of secondary bifurcations.

Stability of a layered reactive channel flow

We analyse the linear stability of a reactive plane Poiseuille flow, where a reactant fluid A overlies another reactant B in a layered fashion within a two-dimensional channel. Both reactants are miscible and have the same viscosity, while upon reaction, they produce either a less or more-viscous product fluid C . The reaction kinetics is of simple A + B → C type, and the production of C occurs across the initial contact line of reactants A and B in a mixed zone of small and finite width. All three fluids have the same density. We demonstrate the effects of various controlling parameters such as the log-mobility ratio, Damköhler number, Schmidt number, Reynolds number, position and thicknesses of the reactive zone on the stability characteristics. We show that a tiny viscosity stratification by the reaction destabilizes the flow at a moderate (10–1000) and even at low Reynolds numbers (0.01–1). The maximum growth occurs for shorter waves than for the Tollmien–Schlichting eigenmode, and the ranges of unstable wavenumbers are wider than that known for non-reactive channel flow systems. In most cases, the instability occurs due to the overlap of the critical layer with the viscosity-stratified layer. Surprisingly for some parameters, it is observed that the reaction can make σ M decrease with increasing Reynolds number.

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.

Stabilization of periodically forced Hele-Shaw flows by means of a nonmonotonic viscosity profile

The onset of viscous fingering in the presence of a viscosity profile is investigated theoretically for two immiscible fluids undergoing a time-dependent injection. Here, we show that the presence of a positive viscosity gradient at the interface between both fluids stabilizes the interface facilitating the spread of the perturbation. This effect is much more pronounced in the case of sinusoidal injection flows. The influence of the viscosity gradient on the dispersion relation is analyzed. Numerical simulations of the Navier-Stokes equation confirm the linear stability analysis.

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.

Anomalous patterns of Saffman-Taylor fingering instability during a metastable phase separation

Phase separation is important in biology, biochemistry, industry, and other areas and is divided into two types: a spinodal decomposition type and a nucleation and growth type. The spinodal decomposition type phase separation occurs under the thermodynamically unstable conditions, and the nucleation and growth type phase separation occurs under thermodynamically metastable conditions. On the other hand, when a less viscous fluid displaces a more viscous one in porous media, the interface of the two fluids becomes hydrodynamically unstable and forms a finger-like pattern. The coupling of the hydrodynamic instability with the thermodynamic instability has been studied. It is reported that the hydrodynamic instability under thermodynamically unstable conditions, where spinodal decomposition type phase separation occurs, creates multiple moving droplets with a radius of 3-4 mm because of the spontaneous convection induced by the Korteweg force, which is driven by a compositional gradient during phase separation. However, the hydrodynamic instability under metastable conditions, where the phase separation of nucleation and growth type occurs, is still unrevealed. In this study, we applied fingering instability (hydrodynamic instability) under the metastable conditions, where the patterns are changed from fingering or droplets to anomalous patterns such as tip-widening or needle-like (top-pointed) fingering patterns when the initial concentration is metastable, which is considered near a binodal curve. These patterns are ubiquitous in nature, similar to dendrite crystals (snowflakes) or our body's cells. Thus, the patterns created can be controlled through hydrodynamic conditions such as the injection flow and thermodynamic conditions such as spinodal decomposition (thermodynamically unstable conditions) and metastable conditions.

Fluid Morphologies Governed by the Competition of Viscous Dissipation and Phase Separation in a Radial Hele-Shaw Flow

The displacement of a less viscous fluid by a more viscous fluid in a radial Hele-Shaw cell makes a circular pattern because the interface is hydrodynamically stable in this condition. Very recently, it has been experimentally reported that the hydrodynamically stable displacement in a partially miscible system induces fingering patterns while stable circular patterns are made at fully miscible and immiscible systems. The fingering instability in the partially miscible system results from complex and entangled elements involving viscous dissipation, molecular diffusion, and phase separation. The analyzing mechanism requires a quantitative relationship between the hydrodynamic interfacial fingering patterns and underlying physicochemical properties. Here, we experimentally investigated the change in fluid patterns formed by the progression of phase separation in the partially miscible systems and categorized them into three patterns: finger-like pattern, annular-like pattern, and circular pattern. Moreover, we propose the mechanism of the pattern formation by an interfacial tension measurement and evaluate the patterns by modified capillary number and newly defined body force ratio, Bf. Our analysis revealed that the deformation index of the pattern can be expressed as a function of Bf on a single curve regardless of the miscibility.

Interaction between rarefaction wave and viscous fingering in a Langmuir adsorbed solute

The evolution of dissolved species in a porous medium is determined by its adsorption on the porous matrix through the classical advection-diffusion processes. The extent to which the adsorption affects the solute propagation in applications related to chromatography and contaminant transport is largely dependent upon the adsorption isotherm. Here, we examine the influence of a nonlinear Langmuir adsorbed solute on its propagation dynamics. Interfacial deformations can also be induced by classical viscous fingering (VF) instability that develops when a less viscous fluid displaces a more viscous one. We present numerical simulations of an initially step-up concentration profile of the solute that capture a rarefaction/diffusive wave solution due to the nonlinearity introduced through Langmuir adsorption and variety of pattern-forming behaviors of the solute dissolved in the displaced fluid. The fluid velocity is governed by Darcy's law, coupled with the advection-diffusion equation that determines the evolution of the solute concentration controlling the viscosity of the fluids. Numerical simulations are performed using the Fourier pseudospectral method to investigate and illustrate the role played by VF and Langmuir adsorption in the development of the patterns of the interface. We show that the solute transport proceeds by the formation of a rarefaction wave results in the enhanced spreading of the solute. Interestingly we obtained a nonmonotonic behavior in the onset of VF, which depends on the adsorption parameters and existence of an optimal value of such adsorption constant is obtained near b=1, for which the most delayed VF is observed. Hence, it can be concluded that the rarefaction wave formation stands out to be an effective tool for controlling the VF dynamics.

Unpredictable Dynamics of Polymeric Reacting Flow by Comparison between Pre- and Post-Reaction Fluid Properties: Hydrodynamics Involving Molecular Diagnosis via ATR-FTIR Spectroscopy

In reacting flows, changes in fluid properties induced by the chemical reaction can alter the flow dynamics. Generally, these changes in fluid properties are evaluated by comparison between their pre- and post-reaction properties. If a fluid property such as viscosity decreases between pre- and post-reaction, we expect a decrease in viscosity to occur in the reacting flow. However, this study demonstrates a reacting polymeric liquid flow where a remarkable increase in the viscoelasticity temporally occurs despite the viscosity slightly decreasing after the reaction. We elucidated the underlying reaction mechanism, which involves a structural change in the side functional group (carboxyl) in polyacrylamide at ultrahigh molecular weights ( M_w > 10⁶) with ultralow concentrations ([polymer] < 1 wt %) by using ATR-FTIR spectroscopy. This study demonstrates the existence of a reacting flow in which examination of microscopic molecular structure is required to understand the macroscopic flow dynamics. The findings will be valuable not only for industrial application such as reactor designs and rheology control but also for opening a new research area: chemically reacting flow involving the diagnosis of molecule structure.

Reaction-driven oscillating viscous fingering

Localized oscillations can develop thanks to the interplay of reaction and diffusion processes when two reactants A and B of an oscillating reaction are placed in contact, meet by diffusion, and react. We study numerically the properties of such an A+B→ oscillator configuration using the Brusselator model. The influence of a hydrodynamic viscous fingering instability on localized concentration oscillations is next analyzed when the oscillating chemical reaction changes the viscosity of the solutions involved. Nonlinear simulations of the related reaction-diffusion-convection equations with the fluid viscosity varying with the concentration of an intermediate oscillatory species show an active coupling between the oscillatory kinetics and the viscously driven instability. The periodic oscillations in the concentration of the intermediate species induce localized changes in the viscosity, which in turn can affect the fingering instability. We show that the oscillating kinetics can also trigger viscous fingering in an initially viscously stable displacement, while localized changes in the viscosity profile can induce oscillations in an initially nonoscillating reactive system.

Viscous Fingering Induced by a pH-Sensitive Clock Reaction

A pH-changing clock chemical system, also known to induce changes in viscosity, is shown experimentally to induce a viscous fingering instability during the displacement of reactive solutions in a Hele-Shaw cell. Specifically, a low-viscosity solution of formaldehyde is displaced by a more viscous solution of sulfite and of a pH-sensitive poly(acrylic acid) polymer. The pH change triggered by the formaldehyde-sulfite clock reaction in the reactive contact zone between the two solutions affects the polymer and induces a local increase of the viscosity that destabilizes the displacement via a viscous fingering instability. The influence of changes in the chemical parameters on this fingering instability is analyzed using different techniques and the results are compared with numerical simulations.

Influence of microscopic precipitate structures on macroscopic pattern formation in reactive flows in a confined geometry

Thanks to the coupling between chemical precipitation reactions and hydrodynamics, new dynamic phenomena may be obtained and new types of materials can be synthesized.

Spontaneous and Flow-Driven Interfacial Phase Change: Dynamics of Microemulsion Formation at the Pore Scale

The dynamic behavior of microemulsion-forming water-oil-amphiphiles mixtures is investigated in a 2.5D micromodel. The equilibrium phase behavior of such mixtures is well-understood in terms of macroscopic phase transitions. However, what is less understood and where experimental data are lacking is the coupling between the phase change and the bulk flow. Herein, we study the flow of an aqueous surfactant solution-oil mixture in porous media and analyze the dependence of phase formation and spatial phase configurations on the bulk flow rate. We find that a microemulsion forms instantaneously as a boundary layer at the initial surface of contact between the surfactant solution and oil. The boundary layer is temporally continuous because of the imposed convection. In addition to the imposed flow, we observe spontaneous pulsed Marangoni flows that drag the microemulsion and surfactant solution into the oil stream, forming large (macro)emulsion droplets. The formation of the microemulsion phase at the interface distinguishes the situation from that of the more common Marangoni flow with only two phases present. Additionally, an emulsion forms via liquid-liquid nucleation or the Ouzo effect (i.e., spontaneous emulsification) at low flow rates and via mechanical mixing at high flow rates. With regard to multiphase flow, contrary to the common belief that the microemulsion is the wetting liquid, we observe that the minor oil phase wets the solid surface. We show that a layered flow pattern is formed because of the out-of-equilibrium phase behavior at high volumetric flow rates (order of 2 m/day) where advection is much faster than the diffusive interfacial mass transfer and transverse mixing, which promote equilibrium behavior. At lower flow rates (order of 30 cm/day), however, the dynamic and equilibrium phase behaviors are well-correlated. These results clearly show that the phase change influences the macroscale flow behavior.

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.

Capillary and geometrically driven fingering instability in nonflat Hele-Shaw cells

The usual viscous fingering instability arises when a fluid displaces another of higher viscosity in a flat Hele-Shaw cell, under sufficiently large capillary number conditions. In this traditional framing, the reverse flow case (more viscous fluid displacing a less viscous one) and the viscosity-matched situation (fluids of equal viscosities) are stable. We revisit this classical fluid dynamic problem, now considering flow in a nonflat Hele-Shaw cell. For a specific nonflat environment, we show that both the reverse and the viscosity-matched flows can become unstable, even at low capillary number. This peculiar fluid fingering instability is driven by the combined action of capillary effects and geometric properties of the nonflat Hele-Shaw cell. Our theoretical results indicate that the Hele-Shaw cell geometry significantly impacts the linear stability and nonlinear pattern-forming dynamics of the system. This suggests that the geometry of the medium plays an important role in favoring the occurrence of fingering patterns in nonflat, confined fluid flows.

Temporal viscosity modulations driven by a pH sensitive polymer coupled to a pH-changing chemical reaction

Conformational changes of a PAA molecule linked to a pH-changing reaction can produce a temporal viscosity modulation.

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'.

Localized stationary and traveling reaction-diffusion patterns in a two-layer A+B→ oscillator system

When two solutions containing separate reactants A and B of an oscillating reaction are put in contact in a gel, localized spatiotemporal patterns can develop around the contact zone thanks to the interplay of reaction and diffusion processes. Using the Brusselator model, we explore analytically the deployment in space and time of the bifurcation diagram of such an A+B→ oscillator system. We provide a parametric classification of possible instabilities as a function of the ratio of the initial reactant concentrations and of the reaction intermediate species diffusion coefficients. Related one-dimensional reaction-diffusion dynamics are studied numerically. We find that the system can spatially localize waves and Turing patterns as well as induce more complex dynamics such as zigzag spatiotemporal waves when Hopf and Turing modes interact.

Development of tip-splitting and side-branching patterns in elastic fingering

Elastic fingering supplements the already interesting features of the traditional viscous fingering phenomena in Hele-Shaw cells with the consideration that the two-fluid separating boundary behaves like an elastic membrane. Sophisticated numerical simulations have shown that under maximum viscosity contrast the resulting patterned shapes can exhibit either finger tip-splitting or side-branching events. In this work, we employ a perturbative mode-coupling scheme to get important insights into the onset of these pattern formation processes. This is done at lowest nonlinear order and by considering the interplay of just three specific Fourier modes: a fundamental mode n and its harmonics 2n and 3n. Our approach further allows the construction of a morphology diagram for the system in a wide range of the parameter space without requiring expensive numerical simulations. The emerging interfacial patterns are conveniently described in terms of only two dimensionless controlling quantities: the rigidity fraction C and a parameter Γ that measures the relative strength between elastic and viscous effects. Visualization of the rigidity field for the various pattern-forming structures supports the idea of an elastic weakening mechanism that facilitates finger growth in regions of reduced interfacial bending rigidity.

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.

Genericity of confined chemical garden patterns with regard to changes in the reactants

Typical patterns emerging during the growth of chemical gardens in a confined geometry when the concentration of the reactants are changed. These patterns are robust to changes in the reactant ions.

Hydrodynamic fingering instability induced by a precipitation reaction

We experimentally demonstrate that a precipitation reaction at the miscible interface between two reactive solutions can trigger a hydrodynamic instability due to the buildup of a locally adverse mobility gradient related to a decrease in permeability. The precipitate results from an A+B→C type of reaction when a solution containing one of the reactants is injected into a solution of the other reactant in a porous medium or a Hele-Shaw cell. Fingerlike precipitation patterns are observed upon displacement, the properties of which depend on whether A displaces B or vice versa. A mathematical modeling of the underlying mobility profile confirms that the instability originates from a local decrease in mobility driven by the localized precipitation. Nonlinear simulations of the related reaction-diffusion-convection model reproduce the properties of the instability observed experimentally. In particular, the simulations suggest that differences in diffusivity between A and B may contribute to the asymmetric characteristics of the fingering precipitation patterns.

Interfacial elastic fingering in Hele-Shaw cells: a weakly nonlinear study

We study a variant of the classic viscous fingering instability in Hele-Shaw cells where the interface separating the fluids is elastic, and presents a curvature-dependent bending rigidity. By employing a second-order mode-coupling approach we investigate how the elastic nature of the interface influences the morphology of emerging interfacial patterns. This is done by focusing our attention on a conventionally stable situation in which the fluids involved have the same viscosity. In this framework, we show that the inclusion of nonlinear effects plays a crucial role in inducing sizable interfacial instabilities, as well as in determining the ultimate shape of the pattern-forming structures. Particularly, we have found that the emergence of either narrow or wide fingers can be regulated by tuning a rigidity fraction parameter. Our weakly nonlinear findings reinforce the importance of the so-called curvature weakening effect, which favors the development of fingers in regions of lower rigidity.

Manipulation of the Saffman-Taylor instability: a curvature-dependent surface tension approach

A variant of the classic Saffman-Taylor instability problem is reported, in which the surface tension at the fluid-fluid interface depends on the interfacial curvature. We show that the interplay between the variable surface tension and three-dimensional effects connected to the contact angle significantly modifies the scenario of instability formation. This allows the manipulation of the Saffman-Taylor instability, leading to the stabilization (destabilization) of conventionally unstable (stable) situations. This is done analytically through a perturbative mode-coupling approach, providing relevant information about both linear and weakly nonlinear regimes of interface evolution.

Nonlinear dynamics of thin liquid films consisting of two miscible components

Recently, we systematically derived a system of two coupled conservation equations governing a thin liquid layer with a deformable surface composed of two completely miscible components [Phys. Fluids 22, 104102 (2010)]. One equation describes the location of the free surface and the second one the evolution of the mean concentration. This lubrication model was investigated previously in linearized form. The study is now extended to the fully nonlinear case of thin liquid films of a binary mixture (in one and two horizontal spatial dimensions) with and without heat transport. For an initially flat and motionless film heated from below, we analyze the component separation induced by the Soret effect. Nonlinear simulations show that the Soret effect can cause a multitude of interesting behaviors, such as oscillatory patterns and solitonlike structures (localized traveling drops or holes). A stronger component separation induced by stronger Soret effects favors faster-moving localized structures. For isothermal systems, we study the fusion and the mixing of two thin liquid films of different but perfectly miscible liquids. Marangoni-driven forces can cause delayed coalescence, ripple formation, and fingering patterns at the borderline between the two liquid layers. A systematic analysis for ripple pattern formation and finger instabilities at different diffusion constants shows that these phenomena appear more pronounced for lower diffusion in the system.

Introduction to the focus issue: chemo-hydrodynamic patterns and instabilities

Pattern forming instabilities are often encountered in a wide variety of natural phenomena and technological applications, from self-organization in biological and chemical systems to oceanic or atmospheric circulation and heat and mass transport processes in engineering systems. Spatio-temporal structures are ubiquitous in hydrodynamics where numerous different convective instabilities generate pattern formation and complex spatiotemporal dynamics, which have been much studied both theoretically and experimentally. In parallel, reaction-diffusion processes provide another large family of pattern forming instabilities and spatio-temporal structures which have been analyzed for several decades. At the intersection of these two fields, "chemo-hydrodynamic patterns and instabilities" resulting from the coupling of hydrodynamic and reaction-diffusion processes have been less studied. The exploration of the new instability and symmetry-breaking scenarios emerging from the interplay between chemical reactions, diffusion and convective motions is a burgeoning field in which numerous exciting problems have emerged during the last few years. These problems range from fingering instabilities of chemical fronts and reactive fluid-fluid interfaces to the dynamics of reaction-diffusion systems in the presence of chaotic mixing. The questions to be addressed are at the interface of hydrodynamics, chemistry, engineering or environmental sciences to name a few and, as a consequence, they have started to draw the attention of several communities including both the nonlinear chemical dynamics and hydrodynamics communities. The collection of papers gathered in this Focus Issue sheds new light on a wide range of phenomena in the general area of chemo-hydrodynamic patterns and instabilities. It also serves as an overview of the current research and state-of-the-art in the field.