Manoranjan Mishra

Miscible viscous fingering with linear adsorption on the porous matrix

Viscous fingering between miscible fluids of different viscosities can affect the dispersion of finite samples in porous media. In some applications, as typically in chromatographic separations or pollutant dispersion in underground aquifers, adsorption onto the porous matrix of solutes (the concentration of which rules the viscosity of the solution) can affect the fingering dynamics. Here, we investigate theoretically the influence of such an adsorption on the stability and nonlinear properties of viscous samples displaced in a two-dimensional system by a less viscous and miscible carrying fluid. The model is based on Darcy’s law for the evolution of the fluid velocity coupled to a diffusion-convection equation for the concentration of a solute in the mobile phase inside the porous medium. The adsorption-desorption dynamics of the solute onto the stationary phase is assumed to be at equilibrium, to follow a linear isotherm and is characterized by a retention parameter κ′ equal to the adsorption-desorptio...

Linear stability analysis of Korteweg stresses effect on miscible viscous fingering in porous media

Viscous fingering (VF) is an interfacial hydrodynamic instability phenomenon observed when a fluid of lower viscosity displaces a higher viscous one in a porous media. In miscible viscous fingering, the concentration gradient of the undergoing fluids is an important factor, as the viscosity of the fluids are driven by concentration. Diffusion takes place when two miscible fluids are brought in contact with each other. However, if the diffusion rate is slow enough, the concentration gradient of the two fluids remains very large during some time. Such steep concentration gradient, which mimics a surface tension type force, called the effective interfacial tension, appears in various cases such as aqua-organic, polymer-monomer miscible systems, etc. Such interfacial tension effects on miscible VF is modeled using a stress term called Korteweg stress in the Darcys equation by coupling with the convection-diffusion equation of the concentration. The effect of the Korteweg stresses at the onset of the instabil...

Influence of miscible viscous fingering of finite slices on an adsorbed solute dynamics

Viscous fingering (VF) between miscible fluids of different viscosities can affect the dispersion of finite width samples in porous media. We investigate here the influence of such VF due to a difference between the viscosity of the displacing fluid and that of the sample solvent on the spatiotemporal dynamics of the concentration of a passive solute initially dissolved in the injected sample and undergoing adsorption on the porous matrix. Such a three component system is modeled using Darcy’s law for the fluid velocity coupled to mass-balance equations for the sample solvent and solute concentrations. Depending on the conditions of adsorption, the spatial distribution of the solute concentration can either be deformed by VF of the sample solvent concentration profiles or disentangle from the fingering zone. In the case of deformation by fingering, a parametric study is performed to analyze the influence of parameters such as the log-mobility ratio, the ratio of dispersion coefficients, the sample length,...

Experimental study of dispersion and miscible viscous fingering of initially circular samples in Hele-Shaw cells

Experimental studies are conducted to analyze dispersion and miscible viscous fingering of initially circular samples of a given solution displaced linearly at constant speed U by another solution in horizontal Hele-Shaw cells (two glass plates separated by a thin gap). In the stable case of a dyed water sample having the same viscosity as that of displacing nondyed water, we analyze the transition between dispersive and advective transport of the passive scalar displaced linearly. At low displacement speed and after a certain time, the length of the sample increases as a square root of time allowing to compute the value of a dispersion coefficient. At larger injection speed, the displacement remains advective for the duration of the experiment, with a length of the sample increasing linearly in time. A parametric study allows to gain insight into the switch from one regime to another as a function of the gap width of the cell. In the unstable case of viscous glycerol samples displaced by dyed water, the ...

Combined influences of viscous fingering and solvent effect on the distribution of adsorbed solutes in porous media

The displacement of two fluids in a porous medium can be affected by a viscous fingering instability (VF) that arises at the interface between the fluids when their viscosities are different. In parallel, one of the fluids may contain solutes that reversibly adsorb on the porous matrix at a rate that depends on the composition of the two-fluid mixture, a so-called solvent strength effect. In some systems encountered for instance in liquid chromatographic columns or in underground flows in environmental applications, both VF and solvent strength effects may combine to influence the spatio-temporal distribution of solutes. Here, a computational investigation of such dynamics is performed. The distribution of the solute in the porous medium is affected by the combined effects of VF and solvent strength. A three component system (displacing fluid, sample solvent and solute) is modeled using Darcys law for the fluid flow velocity coupled to a convection–diffusion equation for the sample solvent and a mass balance equation for the solute in the mobile and stationary phases. The sample solvent is assumed to have a larger solvent strength than the displacing fluid, in which the retention parameter due to the linear adsorption of the solute depends exponentially on the concentration of the sample solvent. A parametric study of the influences of the factors controlling the VF and solvent strength effects on the displacement velocity of the fronts of solute zone and on its width along the porous medium has been performed by direct numerical simulation of the governing equations. While each of the two effects (VF and solvent strength effects) distorts and significantly increases the broadening of the solute zone, the simulations reveal that, when they are acting in combination, these solute zone perturbations are reduced.

Influence of a strong sample solvent on analyte dispersion in chromatographic columns

In chromatographic columns, when the eluting strength of the sample solvent is larger than that of the carrier liquid, a deformation of the analyte zone occurs because its frontal part moves at a relatively high velocity due to a low retention factor in the sample solvent while the rear part of the analyte zone is more retained in the carrier liquid and hence moves at a lower velocity. The influence of this solvent strength effect on the separation of analytes is studied here theoretically using a mass balance model describing the spatio-temporal evolution of the eluent, the sample solvent and the analyte. The viscosity of the sample solvent and carrier fluid is supposed to be the same (i.e. no viscous fingering effects are taken into account). A linear isotherm adsorption with a retention factor depending upon the local concentration of the liquid phase is considered. The governing equations are numerically solved by using a Fourier spectral method and parametric studies are performed to analyze the effect of various governing parameters on the dispersion and skewness of the analyte zone. The distortion of this zone is found to depend strongly on the difference in eluting strength between the mobile phase and the sample solvent as well as on the sample volume.

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.

Influence of viscosity contrast on buoyantly unstable miscible fluids in porous media

The influence of viscosity contrast on buoyantly unstable miscible fluids in a porous medium is investigated through a linear stability analysis (LSA) as well as direct numerical simulations (DNS). The linear stability method implemented in this paper is based on an initial value approach, which helps to capture the onset of instability more accurately than the quasi-steady state analysis. In the absence of displacement, we show that viscosity contrast delays the onset of instability in buoyantly unstable miscible fluids. Further, it is observed that suitably choosing the viscosity contrast and injection velocity a gravitationally unstable miscible interface can be stabilized completely. Through LSA we draw a phase diagram, which shows three distinct stability regions in a parameter space spanned by the displacement velocity and the viscosity contrast. DNS are performed corresponding to parameters from each regime and the results obtained are in accordance with the linear stability results. Moreover, the conversion from a dimensionless formulation to the other and its essence to compare between two different type of flow problems associated with each dimensionless formulation are discussed.

Viscosity scaling of fingering instability in finite slices with Korteweg stress

We perform linear stability analyses (LSA) and direct numerical simulations (DNS) to investigate the influence of the dynamic viscosity on viscous fingering (VF) instability in miscible slices. Selecting the characteristic scales appropriately the importance of the magnitude of the dynamic viscosity of individual fluids on VF in miscible slice has been shown in the context of the transient interfacial tension. Further, we have confirmed this result for immiscible fluids and manifest the similarities between VF in immiscible and miscible slices with transient interfacial tension. In a more general setting, the findings of this letter will be very useful for multiphase viscous flow, in which the momentum balance equation contains an additional stress term free from the dynamic viscosity.

Coupled effect of viscosity and density gradients on fingering instabilities of a miscible slice in porous media

Miscible displacements in porous media exhibit interesting spatio-temporal patterns. A deeper understanding of the physical mechanisms of these emergent patterns is relevant in a number of physicochemical processes. Here, we have numerically investigated the instabilities in a miscible slice in vertical porous media. Depending on the viscosity and density gradients at the two interfaces, four distinct flow configurations are obtained, which are partitioned into two different groups, each containing a pair of equivalent flows until the interaction between the two interfaces. An analysis of the pressure drop around the respective unstable interface(s) supports numerical results. We classify the stabilizing and destabilizing scenarios in a parameter space spanned by the log-mobility ratio (R) and the displacement velocity (U). When the viscosity and density gradients are unstably stratified at the opposite interfaces, the stability characteristics are very complex. The most notable findings of this paper are...