Fingering Instability Research Articles

Fingering instability in Marangoni spreading on a deep layer of polymer solution

Spreading on the free surface of a complex fluid is ubiquitous in nature and industry, such as drug delivery, oil spill, and surface treatment with patterns. Here, we report on a fingering instability that develops during Marangoni spreading on a deep layer of the polymer solution. In particular, the wavelength depends on the molecular weight and concentration of the polymer solution. We use the transmission lattice method to characterize the free surface morphology during spreading and the finger height at the micron scale. We use the Maxwell model to explain the spreading radius, which is dominated by elasticity at small time scales and by viscous dissipation at large time scales. In a viscous regime, with consideration of shear thinning, the spreading radius follows the universal 3/4 power law. Our model suggests a more generalized law of the spreading radius than the previous laws for Newtonian fluids. Furthermore, we give a physical explanation on the origin of the fingering instability as due to normal stresses at high shear rates generating a high contact angle, providing a necessary condition for the fingering instability. The normal stress also generates the elastic deformation at the leading edge and so selects the wavelength of the fingering instability. Understanding the spreading mechanism on a layer of viscoelastic fluid has a particular implication in airway drug delivery and surface coating.

NUMERICAL ANALYSIS OF VISCOUS FINGERING INSTABILITY DUE TO MISCIBLE DISPLACEMENT

In this study, miscible viscous fingering instability is examined numerically by using the two-phase Darcy’s law and transport equations. The effects of the viscosity ratio, anisotropic permeability, and porosity on instabilities are investigated. The finger patterns and their splitting and spreading in the domain are discussed. An image processing algorithm is applied to concentration contours to quantify instability parameters, such as the breakthrough time, efficiency, and fractal dimension. It is revealed that more complex fingers are obtained as the viscosity ratio increases, while the efficiency and the breakthrough time decrease. It is demonstrated that high permeability perpendicular to the flow direction leads to instability intensification and to an increase in the fractal dimension, whereas changing the porosity does not have any considerable impact on viscous fingering instability.

Phase separation effects on a partially miscible viscous fingering dynamics

Classical viscous fingering (VF) instability, the formation of finger-like interfacial patterns, occurs when a less viscous fluid displaces a more viscous one in porous media in immiscible and fully miscible systems. However, the dynamics in partially miscible fluid pairs, exhibiting a phase separation due to its finite solubility into each other, has not been largely understood so far. This study has succeeded in experimentally changing the solution system from immiscible to fully miscible or partially miscible by varying the compositions of the components in an aqueous two-phase system (ATPS) while leaving the viscosities relatively unchanged at room temperature and atmospheric pressure. Here, we have experimentally discovered a new topological transition of VF instability by performing a Hele-Shaw cell experiment using the partially miscible system. The finger formation in the investigated partially miscible system changes to the generation of spontaneously moving multiple droplets. Through additional experimental investigations, we determine that such anomalous VF dynamics is driven by thermodynamic instability such as phase separation due to spinodal decomposition and Korteweg convection induced by compositional gradient during such phase separation. We perform the numerical simulation by coupling hydrodynamics with such chemical thermodynamics and the spontaneously moving droplet dynamics is obtained, which is in good agreement with the experimental investigations of the ATPS. This numerical result strongly supports our claim that the origin of such anomalous VF dynamics is thermodynamic instability.

Criterion for Fingering Instabilities in Colloidal Gels.

We sandwich a colloidal gel between two parallel plates and induce a radial flow by lifting the upper plate at a constant velocity. Two distinct scenarios result from such a tensile test: (i)stable flows during which the gel undergoes a tensile deformation without yielding, and (ii)unstable flows characterized by the radial growth of air fingers into the gel. We show that the unstable regime occurs beyond a critical energy input, independent of the gel's macroscopic yield stress. This implies a local fluidization of the gel at the tip of the growing fingers and results in the most unstable wavelength of the patterns exhibiting the characteristic scalings of the classical viscous fingering instability. Our work provides a quantitative criterion for the onset of fingering in colloidal gels based on a local shear-induced yielding in agreement with the delayed failure framework.

Combustion in miscible displacement for high-pressure air injection

This paper describes miscible displacement upon air injection in a porous medium saturated with oil corresponding to conditions of high-pressure air injection (HPAI). We assume that injection fluids and produced fluids are fully miscible with the oil at the prevailing high pressure. We use three pseudo-components, viz., oxygen, oil, and an inert component, which includes nitrogen, carbon dioxide, etc. To model the fingering instabilities, we follow a similar procedure as proposed by Koval (SPE J. 3(02):145–154, 1963) and include the reaction between oxygen and oil in the Koval model. The equations are solved numerically, using a finite element software package (COMSOL). The results show that a combustion wave is formed. We study the performance at low and high viscosities and show that the reaction improves the speed and degree of recovery at later times.

Numerical Simulation of Unstable Preferential Flow during Water Infiltration into Heterogeneous Dry Soil

Water infiltration and unsaturated flow through heterogeneous soil control the distribution of soil moisture in the vadose zone and the dynamics of groundwater recharge, providing the link between climate, biogeochemical soil processes and vegetation dynamics. Infiltration into dry soil is hydrodynamically unstable, leading to preferential flow through narrow wet regions (fingers). In this paper we use numerical simulation to study the interplay between fingering instabilities and soil heterogeneity during water infiltration. We consider soil with heterogeneous intrinsic permeability. Permeabilities are random, with point Gaussian statistics, and vary smoothly in space due to spatial correlation. The key research question is whether the presence of moderate or strong heterogeneity overwhelms the fingering instability, recovering the simple stable displacement patterns predicted by most simplified model of infiltration currently used in hydrological models from the Darcy to the basin scales. We perform detailed simulations of constant-rate infiltration into soils with isotropic and anisotropic intrinsic permeability fields. Our results demonstrate that soil heterogeneity does not suppress fingering instabilities, but it rather enhances its effect of preferential flow and channeling. Fingering patterns strongly depend on soil structure, in particular the correlation length and anisotropy of the permeability field. While the finger size and flow dynamics are only slightly controlled by correlation length in isotropic fields, layering leads to significant finger meandering and bulging, changing arrival times and wetting efficiencies. Fingering and soil heterogeneity need to be considered when upscaling the constitutive relationships of multiphase flow in porous media (relative permeability and water retention curve) from the finger to field and basin scales. While relative permeabilities remain unchanged upon upscaling for stable displacements, the inefficient wetting due to fingering leads to relative permeabilities at the field scale that are significantly different from those at the Darcy scale. These effective relative permeability functions also depend, although less strongly, on heterogeneity and soil structure.

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.

Influence of the Péclet number on reactive viscous fingering

Depending on the value of the P\'eclet number, a chemical reaction decreasing the viscosity locally may either reduce or enhance a viscous fingering instability. The stabilization and destabilization at low and high P\'eclet numbers, respectively, are related to changes in viscosity gradients that develop around the miscible interface due to the reaction. In particular, at low values of the P\'eclet number, the buildup of a local viscosity minimum improves the sweep efficiency in comparison to the nonreactive case, paving the way to a chemical control of viscous fingering.

Immiscible liquid-liquid displacement flows in a Hele-Shaw cell including shear thinning effects

We experimentally studied the viscous fingering instability considering a Newtonian oil displacing viscoelastic shear-thinning liquids and vice versa. The non-Newtonian liquids are aqueous solutions of polyacrylamide and xanthan gum, i.e., flexible and rigid polymers, respectively. A rectangular Hele-Shaw cell, connected by two plenum chambers, was developed to evaluate the displacement of a fixed volume. The experiment consists of analyzing the interface time evolution through a digital camera as a function of the geometric, dynamic, and rheological parameters. The displacement efficiency was determined through image processing in order to identify the formation of fingers or plugs. Unlike the Newtonian case, the transition does not occur when the viscosity ratio is roughly equal to one, but nevertheless, it was observed that the stability of the interface depends on the viscosity ratio. Specifically, more branches are observed at low viscosity ratios. Furthermore, a higher stability is observed when the Newtonian liquid displaces the shear thinning liquid, especially when the polymer is more rigid. When the Newtonian liquid is being displaced, elastic effects favor the displacing efficiency.

Numerical Analysis of Viscous Fingering Instability due to Miscible Displacement

Numerical Analysis of Viscous Fingering Instability due to Miscible Displacement

Gas-Phase Induced Marangoni Flow Causes Unstable Drop Merging.

The merging of drops plays a key role in many processes from simple rain to complex coating applications. In technical applications, often liquids with different surface tensions merge on a substrate like inkjet printing. For a suitable set of surface tensions, one drop can form a stable wetting film that is covering the other drop. Here, we explore the dynamics of driven wetting films and show a route toward their instability when these wetting films are driven by an external source of energy, which is Marangoni stress in our case. The wetting becomes unstable via a fingering instability and can be observed in various liquid combinations. The vapor of the liquid with the lower surface tension induces a Marangoni driven flow inside the other drop that pulls the wetting film. The concentration of the driving vapor can be controlled through the spreading velocity of the corresponding drop. We use this dependence to map out the characteristics of the instability. For very high or very low spreading velocities, no instability is observed. This is summarized in a stability diagram, which has three different regimes. A detailed analysis reveals a strong coupling of the characteristics of the fingering instability to the spreading velocity. The use of the spreading velocity as a control parameter is justified by a simplified 1D model that motivates how the spreading velocity controls the concentration profile of the second liquid vapor before and at contact. The strength of the Marangoni flow that drives the instability depends on the exposure time of the sitting drop to the vapor concentration profile around the spreading drop.

On the miscible thermo-viscous fingering instability of non-Newtonian fluids in heterogeneous porous media

In this paper, the thermo-viscous fingering instability in the displacement of Newtonian fluid by a shear-thinning non-Newtonian fluid in the heterogeneous porous media is studied. Here, the heterogeneity of medium is assumed to be layered. Two different types of displacements are considered: a cold low viscous fluid displaces a hot high viscous one and vice versa. The numerical solution is obtained using a spectral method based on Hartley transforms and fourth-order Adams–Bashforth technique. By interoperation of concentration and temperature contours, transversely averaged profiles, mixing length, and sweep efficiency, it is shown that heterogeneity of porous media has a significant effect on the thermo-viscous fingering instability. The decreasing thermal-lag coefficient leads to more unstable flow and Lewis number has different effects on displacements. When a cold less viscous fluid is displacing fluid, this parameter has destabilizing effect while for displacing hot more viscous fluid, and it acts in favor of stability.

Elasticity suppresses viscous fingering in a reactive Hele‐Shaw cell

AbstractWe consider a system of two liquids in a Hele‐Shaw cell that form a viscoelastic material layer between them by a chemical reaction. Is has been demonstrated that this system leads to a fingering instability, supposedly by the elasticity of the newly created material phase. To explore this mechanism we employ a novel phase‐field model [1], which can handle fluid‐structure interaction involving growing elastic materials. Assuming the newly created phase to be a viscoelastic Kelvin‐Voigt material, we find that this mechanism alone does not explain the experimentally observed fingering. Imposing a viscosity contrast between the materials, we even find the presence of elasticity to suppress any viscous fingering.

Boundary effects on the onset of miscible viscous fingering in a Hele-Shaw flow

Linear stability analysis and nonlinear simulations confirm that the onset of viscous fingering (VF) instability is delayed in the miscible VF model with the absorbing boundary as compared to the reflective one. A threshold log mobility ratio is found for which such delayed onset is maximum.

Empirical mode decomposition of multiphase flows in porous media: characteristic scales and speed of convergence

We apply a proper orthogonal decomposition (POD) to data stemming from numerical simulations of a fingering instability in a multiphase flow passing through obstacles in a porous medium, to study water injection processes in the production of hydrocarbon reservoirs. We show that the time evolution of a properly defined flow correlation length can be used to identify the onset of the fingering instability. Computation of characteristic lengths for each of the modes resulting from the POD provides further information on the dynamics of the system. Finally, using numerical simulations with different viscosity ratios, we show that the convergence of the POD depends non-trivially on whether the fingering instability develops or not. This result has implications on proposed methods to decrease the dimensionality of the problem by deriving reduced dynamical systems after truncating the system’s governing equations to a few POD modes.

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.

Nonlinear Simulation of Viscoelastic Fingering Instability in Miscible Displacement through Homogeneous and Heterogeneous Porous Media

AbstractThe miscible displacement of a Newtonian fluid pushed by a nonlinear viscoelastic fluid (viscous fingering instability) has been investigated via a pseudospectral method and Hartley transfo...

A meniscus fingering instability in viscoelastic fluids

We report experiments where a viscoelastic fingering instability develops at the free interface between air and a model viscoelastic fluid confined in a Hele-Shaw cell. The fluid is symmetrically stretched with constant velocity along two opposite directions, leading to the inflation of a two-dimensional air bubble growing from a millimetric centered hole. The instability is observed when the circumferential stretch of the inflating bubble reaches a threshold that depends on the viscoelastic properties of the fluid through the ratio of the material relaxation time to the time elapsed before the fingers start to develop. The critical stretch of the bulk fingering instability of a stretchable elastic solid [B. Saintyves, O. Dauchot, and E. Bouchaud, “Bulk elastic fingering instability in Hele-Shaw cells,” Phys. Rev. Lett. 111, 047801 (2013)] is recovered for large values of this ratio.

Stability of lubricated viscous gravity currents. Part 2. Global analysis and stabilisation by buoyancy forces

The novel viscous fingering instability recently found in the experiments of Kowal & Worster (J. Fluid Mech., vol. 766, 2015, pp. 626–655), involving two superposed currents of viscous fluid, has been shown to originate at the lubrication front when the fluids are of equal density. However, when the densities are unequal, additional buoyancy forces associated with the underlying layer act to suppress this instability and are largest at the lubrication front, which is where the instability originates. In this paper, we investigate the interaction between the mechanism of the instability and the stabilising influence of these buoyancy forces by performing a global and fully time-dependent analysis, which does not use the frozen-time approximation. We determine a critical condition for instability in terms of the viscosity ratio and the density difference between the two layers. Consistently with the local analysis of the companion paper, instabilities occur when the jump in hydrostatic pressure gradient across the lubrication front is negative, or, equivalently, when the intruding fluid is less viscous than the overlying fluid, provided the two fluids are of equal densities. Once there is a non-zero density difference, these driving buoyancy forces suppress the instability for large wavelengths, giving rise to wavelength selection. As the density difference increases, the instability criterion requires higher viscosity ratios for any instability to occur, and the band of unstable wavenumbers becomes bounded. Large enough density differences suppress the instability completely.

Stability of lubricated viscous gravity currents. Part 1. Internal and frontal analyses and stabilisation by horizontal shear

A novel viscous fingering instability, involving a less viscous fluid intruding underneath a current of more viscous fluid, was recently observed in the experiments of Kowal & Worster (J. Fluid Mech., vol. 766, 2015, pp. 626–655). We examine the origin of the instability by asking whether the instability is an internal instability, arising from internal dynamics, or a frontal instability, arising from viscous intrusion. We find it is the latter and characterise the instability criterion in terms of viscosity difference or, equivalently, the jump in hydrostatic pressure gradient at the intrusion front. The mechanism of this instability is similar to, but contrasts with, the Saffman–Taylor instability, which occurs as a result of a jump in dynamic pressure gradient across the intrusion front. We focus on the limit in which the two viscous fluids are of equal density, in which a frontal singularity, arising at the intrusion, or lubrication, front, becomes a jump discontinuity, and perform a local analysis in an inner region near the lubrication front, which we match asymptotically to the far field. We also investigate the large-wavenumber stabilisation by transverse shear stresses in two dynamical regimes: a regime in which the wavelength of the perturbations is much smaller than the thickness of both layers of fluid, in which case the flow of the perturbations is resisted dominantly by horizontal shear stresses; and an intermediate regime, in which both vertical and horizontal shear stresses are important.