Satyajit Pramanik

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...

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...

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 Darcys 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.

Viscosity Scaling in Hydrodynamic Instabilities in Porous Media

The importance of viscosity scaling in the context of viscous fingering in a finite slice with viscosity dependent diffusivity is investigated theoretically. Choosing the characteristic viscosity classically as either the displacing or displaced fluid viscosity for both more and less viscous slice leads to inappropriate theoretical predictions, which do not support the physics. With an appropriate choice of the characteristic viscosity, we show that the onset of instability and the initial dynamics of the finger patterns are the same for both more and less viscous slices. Our analysis will be helpful in the theoretical understanding of buoyancy-driven convection in a variable viscosity layer in vertical porous media or VF with non-monotonic viscosity profiles.

A General Approach to the Linear Stability Analysis of Miscible Viscous Fingering in Porous Media

We analyse the linear growth of the viscous fingering instability for miscible, non-reactive, neutral buoyant fluids using the non-modal analysis (NMA). The onset of instability is obscured due to the continually changing base state, and the normal mode analysis is not applicable to the non-autonomous linearized perturbed equations. Commonly used techniques such as frozen time method or amplification theory approach with random initial condition using transient amplifications yield substantially different results for the threshold of instability. We present the classical non-modal methods in the short-time limit using singular value decomposition of the propagator matrix. Using the non-modal approach we characterize the existence of a transition region between a domain exhibiting strong convection and a domain where initial perturbations are damped due to diffusion. Further, at the early times the algebraic growth of perturbations is possible which suggest that NMA could play an important role in describing the onset of instability in the physical phenomenon involving VF.

Confinement effects in premelting dynamics

We examine the effects of confinement on the dynamics of premelted films driven by thermomolecular pressure gradients. Our approach is to modify a well-studied setting in which the thermomolecular pressure gradient is driven by a temperature gradient parallel to an interfacially premelted elastic wall. The modification treats the increase in viscosity associated with the thinning of films, studied in a wide variety of materials, using a power law and we examine the consequent evolution of the confining elastic wall. We treat (1) a range of interactions that are known to underlie interfacial premelting and (2) a constant temperature gradient wherein the thermomolecular pressure gradient is a constant. The difference between the cases with and without the proximity effect arises in the volume flux of premelted liquid. The proximity effect increases the viscosity as the film thickness decreases thereby requiring the thermomolecular pressure driven flux to be accommodated at higher temperatures where the premelted film thickness is the largest. Implications for experiment and observations of frost heave are discussed.