Differences in miscible viscous fingering of finite width slices with positive or negative log-mobility ratio

Trade Off between Hydrodynamic and Thermodynamic Forces at the Liquid-Liquid Interface

Viscous fingering (VF) instability has been investigated in the case of a partially miscible binary system by nonlinear numerical simulations. Partially miscible fluid systems offer the possibility of phase separation coupled with VF instability. The thermodynamics of such systems are governed by the Margules parameter (interaction parameter) as well as the fluid concentrations. Kinetics of the decomposition is also influenced by dynamical parameters such as the viscosity of the fluid, which incidentally also affects the hydrodynamic forces. Here, we explore the effects of concentration and Margules parameter in order to ascertain the trade-offs incurred between hydrodynamic and thermodynamic effects at the interface as well as the thermodynamics of the bulk. Based on the Gibb's free energy versus concentration curve, we select concentrations (i) outside spinodal and binodal regions, (ii) within binodal but outside the spinodal, and (iii) within the spinodal curve. We solve the modified Cahn-Hilliard-Hele-Shaw equation employing the COMSOL Multiphysics software. Applying high-resolution numerical simulations, we show a strong dependence of the thermodynamic forces on the concentration of the mixtures. Rapid phase separation and hence a faster rate of droplet formation have been found when the concentration lies inside the spinodal region. Further, we have investigated the correlation between the fractal dimension and dynamics of the system. The spatiotemporal studies presented in this work clearly illustrate the competition between hydrodynamic and thermodynamic forces and provide insights on the kinetics of decomposition and growth of interfacial instabilities.

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.

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.

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.

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.

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.

Nonmodal linear stability analysis of miscible viscous fingering in porous media

The nonmodal linear stability of miscible viscous fingering in a two-dimensional homogeneous porous medium has been investigated. The linearized perturbed equations for Darcy's law coupled with a convection-diffusion equation is discretized using a finite difference method. The resultant initial value problem is solved by a fourth-order Runge-Kutta method, followed by a singular value decomposition of the propagator matrix. Particular attention is given to the transient behavior rather than the long-time behavior of eigenmodes predicted by the traditional modal analysis. The transient behaviors of the response to external excitations and the response to initial conditions are studied by examining the ε-pseudospectra structures and the largest energy growth function, respectively. With the help of nonmodal stability analysis we demonstrate that at early times the displacement flow is dominated by diffusion and the perturbations decay. At later times, when convection dominates diffusion, perturbations grow. Furthermore, we show that the dominant perturbation that experiences the maximum amplification within the linear regime lead to the transient growth. These two important features were previously unattainable in the existing linear stability methods for miscible viscous fingering. To explore the relevance of the optimal perturbation obtained from nonmodal analysis, we performed direct numerical simulations using a highly accurate pseudospectral method. Furthermore, a comparison of the present stability analysis with existing modal and initial value approach is also presented. It is shown that the nonmodal stability results are in better agreement than the other existing stability analyses, with those obtained from direct numerical simulations.

Enhanced mixing via alternating injection in radial Hele-Shaw flows

Mixing at low Reynolds numbers, especially in the framework of confined flows occurring in Hele-Shaw cells, porous media, and microfluidic devices, has attracted considerable attention lately. Under such circumstances, enhanced mixing is limited due to the lack of turbulence, and absence of sizable inertial effects. Recent studies, performed in rectangular Hele-Shaw cells, have demonstrated that the combined action of viscous fluid fingering and alternating injection can dramatically improve mixing efficiency. In this work, we revisit this important fluid mechanical problem, and analyze it in the context of radial Hele-Shaw flows. The development of radial fingering instabilities under alternating injection conditions is investigated by intensive numerical simulations. We focus on the impact of the relevant physical parameters of the problem (Péclet number Pe, viscosity contrast A, and injection time interval Δt) on fluid mixing performance.

Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell

The influence of fluid dispersion on the Saffman-Taylor instability in miscible fluids has been investigated in both the linear and the nonlinear regimes. The convective characteristic scales are used for the dimensionless formulation that incorporates the Péclet number (Pe) into the governing equations as a measure for the fluid dispersion. A linear stability analysis (LSA) is performed in a similarity transformation domain using the quasi-steady-state approximation. LSA results confirm that a flow with a large Pe has a higher growth rate than a flow with a small Pe. The critical Péclet number (Pec) for the onset of instability for all possible wave numbers and also a power-law relation of the onset time and most unstable wave number with Pe are observed. Unlike the radial source flow, Pec is found to vary with t0. A Fourier spectral method is used for direct numerical simulations (DNS) of the fully nonlinear system. The power-law dependence of the onset of instability ton on Pe is obtained from the DNS and found to be inversely proportional to Pe and it is in good agreement with that obtained from the LSA. The influence of the anisotropic dispersion is analyzed in both the linear and the nonlinear regimes. The results obtained confirm that for a weak transverse dispersion merging happens slowly and hence the small wave perturbations become unstable. We also observ that the onset of instability sets in early when the transverse dispersion is weak and varies with the anisotropic dispersion coefficient, ε, as ∼√[ε], in compliance with the LSA. The combined effect of the Korteweg stress and Pe in the linear regime is pursued. It is observed that, depending on various flow parameters, a fluid system with a larger Pe exhibits a lower instantaneous growth rate than a system with a smaller Pe, which is contrary to the results when such stresses are absent.

Synergetic fluid mixing from viscous fingering and alternating injection

We study mixing of two fluids of different viscosity in a microfluidic channel or porous medium. We show that the synergetic action of alternating injection and viscous fingering leads to a dramatic increase in mixing efficiency at high Péclet numbers. Based on observations from high-resolution simulations, we develop a theoretical model of mixing efficiency that combines a hyperbolic mixing model of the channelized region ahead and a mixing-dissipation model of the pseudosteady region behind. Our macroscopic model quantitatively reproduces the evolution of the average degree of mixing along the flow direction and can be used as a design tool to optimize mixing from viscous fingering in a microfluidic channel.

Scaling and unified characterization of flow instabilities in layered heterogeneous porous media

The physics of miscible flow displacements with unfavorable mobility ratios through horizontal layered heterogeneous media is investigated. The flow model is solved numerically, and the effects of various physical parameters such as the injection velocity, diffusion, viscosity, and the heterogeneity length scale and variance are examined. The flow instability is characterized qualitatively through concentration contours as well as quantitatively through the mixing zone length and the breakthrough time. This characterization allowed us to identify four distinct regimes that govern the flow displacement. Furthermore, a scaling of the model resulted in generalized curves of the mixing zone length for any flow scenario in which the first three regimes of diffusion, channeling, and lateral dispersion superpose into a single unifying curve and allowed us to clearly identify the onset of the fourth regime. A critical effective Péclet number w_{c} based on the layers' width is proposed to identify flows where heterogeneity effects are expected to be important and those where the flow can be safely treated as homogeneous. A similar scaling of the breakthrough time was obtained and allowed us to identify two optimal effective Péclet numbers w_{opt} that result in the longest and shortest breakthrough times for any flow displacement.

Interface dynamics of immiscible two-phase lattice-gas cellular automata: a model with random dynamic scatterers and quenched disorder in two dimensions

We use a lattice gas cellular automata model in the presence of random dynamic scattering sites and quenched disorder in the two-phase immiscible model with the aim of producing an interface dynamics similar to that observed in Hele-Shaw cells. The dynamics of the interface is studied as one fluid displaces the other in a clean lattice and in a lattice with quenched disorder. For the clean system, if the fluid with a lower viscosity displaces the other, we show that the model exhibits the Saffman-Taylor instability phenomenon, whose features are in very good agreement with those observed in real (viscous) fluids. In the system with quenched disorder, we obtain estimates for the growth and roughening exponents of the interface width in two cases: viscosity-matched fluids and the case of unstable interface. The first case is shown to be in the same universality class of the random deposition model with surface relaxation. Moreover, while the early-time dynamics of the interface behaves similarly, viscous fingers develop in the second case with the subsequent production of bubbles in the context of a complex dynamics. We also identify the Hurst exponent of the subdiffusive fractional Brownian motion associated with the interface, from which we derive its fractal dimension and the universality classes related to a percolation process.

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.

Fluid mixing from viscous fingering

Mixing efficiency at low Reynolds numbers can be enhanced by exploiting hydrodynamic instabilities that induce heterogeneity and disorder in the flow. The unstable displacement of fluids with different viscosities, or viscous fingering, provides a powerful mechanism to increase fluid-fluid interfacial area and enhance mixing. Here we describe the dissipative structure of miscible viscous fingering, and propose a two-equation model for the scalar variance and its dissipation rate. Our analysis predicts the optimum range of viscosity contrasts that, for a given Péclet number, maximizes interfacial area and minimizes mixing time. In the spirit of turbulence modeling, the proposed two-equation model permits upscaling dissipation due to fingering at unresolved scales.

Influence of double diffusive effects on miscible viscous fingering

Miscible viscous fingering classically occurs when a less viscous fluid displaces a miscible more viscous one in a porous medium. We analyze here how double diffusive effects between a slow diffusing S and a fast diffusing F component, both influencing the viscosity of the fluids at hand, affect such fingering, and, most importantly, can destabilize the classically stable situation of a more viscous fluid displacing a less viscous one. Various instability scenarios are classified in a parameter space spanned by the log-mobility ratios R(s) and R(f) of the slow and fast component, respectively, and parametrized by the ratio of diffusion coefficients δ. Numerical simulations of the full nonlinear problem confirm the existence of the predicted instability scenarios and highlight the influence of differential diffusion effects on the nonlinear fingering dynamics.

Hydrodynamic instability in the transport of miscible reactive slices through porous media

Miscible displacement of a slice of a pollutant solution by a carrier solution in homogeneous porous media is examined. The carrier solution reacts with the slice solution to generate a chemical product, and as a result of differences in viscosities of the three species, a hydrodynamic instability known as viscous fingering is observed. The dynamics of the instability and the rate of consumption as well as spread of the pollutant are examined through numerical simulations. The study shows that the rate of consumption of the pollutant is the highest when the chemical product is the most or the least viscous solution in the system. It was also found that displacements in which the pollutant viscosity is the smallest or the largest of all three species lead to the widest spread of the pollutant in the porous media. In addition, the most complex finger structures are observed when the carrier solution has the smallest or largest viscosity in the flow. Furthermore, a mechanism of channeling whereby the carrier is able to break through the slice, therefore bypassing the pollutant, is found in cases where the chemical product is more viscous than the carrier solution. The dynamics of the displacement are analyzed and physical interpretations of their development are presented.

Noise-induced multi-decrease and multi-increase of net voltage in Josephson junctions

In this paper, we investigate the net voltage in the Josephson junction subject to a thermal noise, a dc constant bias electric current and an ac time-periodic electric drive for the overdamped case and underdamped case simultaneously. It is shown that, by increasing the thermal noise strength, the net voltage can be decreased and increased several times as a function of the driving frequency. In addition, noise-enhanced stability is found for the net voltage of the junction.