arXiv subjects

Satyajit Pramanik

Publications and source records attributed to Satyajit Pramanik.

At least 19 recordsLinked to original sources

arXiv2026

On the selection of Saffman-Taylor fingers in a tapered Hele-Shaw cell

We present an analytical study for predicting the finger width of the Saffman-Taylor finger in a tapered Hele-Shaw cell. We consider a rectilinear geometry with a constant depth gradient and apply analytical techniques of singular perturbation analysis and WKB approximation to derive an expression for the finger selection mechanism for such tapered Hele-Shaw cells with small depth gradients. We establish \[ Λ- \frac{1}{2} \sim f(α) Ca_m^{2/3} \quad \mbox{as} \quad Ca_m \rightarrow 0, \;\;\; \mbox{and} \;\;\; \lvert α\rvert \ll 1.\] Here, $Λ$ is the dimensionless finger width, $Ca_m$ denotes the modified Capillary parameter, and $f(α)$ is a linear function of the gap gradient $α$, such that $f(α= 0) = 1$ recovering the results of parallel Hele-Shaw cell (Hong and Langer \cite{hong1986analytic}, Combescot \emph{et al.} \cite{Combescot1986}, Shraiman \cite{shraiman1986velocity}). Our findings indicate that the Hele-Shaw cell gap gradient plays a crucial role in determining $Λ$, allowing for control over fingering instabilities such that the single-finger steady state can be stabilised or destabilised depending on the sign of the gradient, compared to the standard Hele-Shaw cell. The theoretical estimates reveal excellent agreement with experimental finger-width data and predictions from linear stability analyses.

cond-mat.soft↗

arXiv2026

HOC simulations of miscible viscous fingering of a finite slice: A new insight

We investigate the dynamics of viscous fingering (VF) in miscible slices in homogeneous, isotropic porous media. The fluid flow is governed by incompressible Darcy's law, whereas the solute transport is described using an advection-diffusion equation. The viscosity of the miscible system depends on the solute concentration, creating a viscosity contrast between the displacing fluid and the finite sample. When expressed in terms of stream function, the flow is described by a system of nonlinear, two-way coupled advection-diffusion type equations. We consider three types of boundary conditions: (a) periodic, (b) impermeable (zero normal velocity) and no-flux (solute), and (c) permeable (allowing non-zero normal velocity) and no diffusive flux (solute) transverse boundaries. This initial boundary value problem is solved numerically using a fourth-order compact finite difference method, while the Crank-Nicolson technique is used for time integration. Although the onset of viscous fingering and early time behavior are independent of the choice of boundary types, long-time behavior, solute mixing and spreading depend on the boundary conditions. In particular, it is observed that the permeable boundaries allow solute mass to increase, leading to stronger fingering instabilities, larger mixing lengths and non-trivial evolution of interfacial lengths. The findings of this study have implications in chromatography separation.

physics.flu-dyn↗

arXiv2026

Existence and uniqueness of solutions of unsteady Darcy-Brinkman problem for modelling miscible reactive flows in porous media

In this work, we investigate a model describing flow through porous media with permeability heterogeneity, combining an advection-reaction-diffusion equation for solute concentration with an unsteady Darcy-Brinkman equation with Korteweg stresses in the presence of external body forces for the flow field. Such models are appropriate in describing flows in fractured karst reservoirs, mineral wool, industrial foam, coastal mud, etc. These equations are coupled with Neumann boundary conditions for the solute concentration and no-flow conditions for the fluid velocity. For a broad class of initial data, we proved the existence of weak solutions. In the presence of a second-order nonlinear reaction, we show that the long-time behaviour of the solution depends on the initial concentration $C_0$. More precisely, the solution exists for all time if $0\leq C_0\leq 1$, and blows up at finite time if $C_0>1$. Furthermore, the uniqueness of the solution is proved for a two-dimensional domain. Finally, numerical simulations based on the finite element method have been presented that illustrate non-negativity of the concentration, long-time decay, and finite-time blow-up in agreement with theoretical estimates.

math.AP↗

arXiv2025

Dissolution-driven transport in a rotating horizontal cylinder

We study the combined effects of natural convection and rotation on the dissolution of a solute in a solvent-filled circular cylinder. The density of the fluid increases with the increasing concentration of the dissolved solute, and we model this using the Oberbeck-Boussinesq approximation. The underlying moving-boundary problem has been modelled by combining the Navier-Stokes equations with the advection-diffusion equation and a Stefan condition for the evolving solute-fluid interface. We use highly resolved numerical simulations to investigate the flow regimes, dissolution rates, and mixing of the dissolved solute for $Sc = 1$, $Ra \in [10^5, 10^8]$ and $Ω\in [0, 2.5]$. In the absence of rotation and buoyancy, the distance of the interface from its initial position follows a square root relationship with time ($r_d \propto \sqrt{t}$), which ceases to exist at a later time due to the finite-size effect of the liquid domain. We then explore the rotation parameter, considering a range of rotation frequency -- from smaller to larger, relative to the inverse of the buoyancy-induced timescale -- and Rayleigh number. We show that the area of the dissolved solute varies nonlinearly with time depending on $Ra$ and $Ω$. The symmetry breaking of the interface is best described in terms of $Ra/Ω^2$.

physics.flu-dyn↗

arXiv2025

Towards enhanced mixing of a high viscous miscible blob in porous media

In this study, we investigate the rectilinear displacement and deformation of a highly viscous, miscible circular blob influenced by a less viscous fluid within a homogeneous porous medium featuring physically realistic no-flux boundaries. We utilize a fourth-order accurate compact finite difference scheme for the spatial discretization of the nonlinear partial differential equations that govern this phenomenon. The resulting semi-discrete equations are then integrated using the second-order Crank-Nicolson (CN) method. We conduct numerical simulations for a Péclet number ($Pe \leq 3000$) and a log-mobility ratio $0 \leq R \leq 7$, which reveal three distinct pattern formations: comet-shape, lump-shape, and viscous fingering instability. Our results demonstrate that the deformation, spreading, and mixing of the blob vary non-ideally with both $Pe$ and $R$, a behavior attributed to the blob's initial curvature. Consequently, enhanced mixing can be achieved at intermediate values of $Pe$ and $R$, suggesting the existence of an optimal mixing condition. These findings have significant implications for fields such as oil recovery, CO$_2$ sequestration, pollution remediation, and chromatography separation.

physics.flu-dyn↗

arXiv2025

A novel time integration scheme for linear parabolic PDEs

This paper presents a class of Crank-Nicolson (CN) type schemes enhanced by radial basis function (RBF) interpolation for the time integration of linear parabolic partial differential equations (PDEs). The resulting RBF-CN schemes preserve the structural simplicity of the classical CN method while enabling higher-order temporal accuracy through the optimization of the shape parameter. Consistency analysis shows that the schemes are always at least second-order accurate and a simple choice of the shape parameter increases the accuracy by two orders over the standard CN scheme. Further optimization can reduce the next leading error term to attain even higher-order accuracy. A von Neumann stability analysis confirms that the stability conditions are essentially the same as those of the standard CN scheme. Several numerical experiments in 1D are carried out to verify the theoretical results under different boundary conditions. Particular focus is given to the startup stage, where several strategies for computing the initial steps are examined, and Gauss-Legendre implicit Runge-Kutta (IRK) methods are found to be the most effective. The experiments further demonstrate that, with optimal initialization, the proposed schemes deliver accuracy comparable to implicit Runge-Kutta methods while achieving nearly two orders of magnitude reduction in computational cost.

math.NA↗

arXiv2025

Solute transport due to Periodic Loading in a Soft Porous Material

In soft porous media, deformation drives solute transport via the intrinsic coupling between flow of the fluid and rearrangement of the pore structure. Solute transport driven by periodic loading, in particular, can be of great relevance in applications including the geomechanics of contaminants in the subsurface and the biomechanics of nutrient transport in living tissues and scaffolds for tissue engineering. However, the basic features of this process have not previously been systematically investigated. Here, we fill this hole in the context of a 1D model problem. We do so by expanding the results from a companion study, in which we explored the poromechanics of periodic deformations, by introducing and analysing the impact of the resulting fluid and solid motion on solute transport. We first characterise the independent roles of the three main mechanisms of solute transport in porous media -- advection, molecular diffusion, and hydrodynamic dispersion -- by examining their impacts on the solute concentration profile during one loading cycle. We next explore the impact of the transport parameters, showing how these alter the relative importance of diffusion and dispersion. We then explore the loading parameters by considering a range of loading periods -- from slow to fast, relative to the poroelastic timescale -- and amplitudes -- from infinitesimal to large. We show that solute spreading over several loading cycle increases monotonically with amplitude, but is maximised for intermediate periods because of the increasing poromechanical localisation of the flow and deformation near the permeable boundary as the period decreases.

physics.flu-dyn↗

arXiv2023

Flow and deformation due to periodic loading in a soft porous material

Soft porous materials, such as biological tissues and soils, are exposed to periodic deformations in a variety of natural and industrial contexts. The detailed flow and mechanics of these deformations have not yet been systematically investigated. Here, we fill this gap by identifying and exploring the complete parameter space associated with periodic deformations in the context of a 1D model problem. We use large-deformation poroelasticity to consider a wide range of loading periods and amplitudes. We identify two distinct mechanical regimes, distinguished by whether the loading period is slow or fast relative to the characteristic poroelastic timescale. We develop analytical solutions for slow loading at any amplitude and for infinitesimal amplitude at any period. We use these analytical solutions and a full numerical solution to explore the localisation of the deformation near the permeable boundary as the period decreases and the emergence of nonlinear effects as the amplitude increases. We show that large deformations lead to asymmetry between the loading and unloading phases of each cycle in terms of the distributions of porosity and fluid flux.

physics.flu-dyn↗

arXiv2021

A homogenised model for flow, transport and sorption in a heterogeneous porous medium

A major challenge in flow through porous media is to better understand the link between microstructure and macroscale flow and transport. For idealised microstructures, the mathematical framework of homogenisation theory can be used for this purpose. Here, we consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium. We use homogenisation via the method of multiple scales to systematically upscale a novel problem involving cells of varying area to obtain effective continuum equations for macroscale flow and transport. The equations are characterised by the local porosity, a local anisotropic flow permeability, an effective local anisotropic solute diffusivity, and an effective local adsorption rate. These macroscale properties depend nontrivially on the two degrees of microstructural geometric freedom in our problem: obstacle size, and obstacle spacing. We exploit this dependence to construct and compare scenarios where the same porosity profile results from different combinations of obstacle size and spacing. We focus on a simple example geometry comprising circular obstacles on a rectangular lattice, for which we numerically determine the macroscale permeability and effective diffusivity. We investigate scenarios where the porosity is spatially uniform but the permeability and diffusivity are not. Our results may be useful in the design of filters, or for studying the impact of deformation on transport in soft porous media.

physics.flu-dyn↗

arXiv2019

The breakdown of Darcy's law in a soft porous material

We perform direct numerical simulations of the flow through a model of a deformable porous medium. Our model is a two-dimensional hexagonal lattice, with defects, of soft elastic cylindrical pillars, with elastic shear modulus $G$, immersed in a liquid. We use a two-phase approach: the liquid phase is a viscous fluid and the solid phase is modeled as an incompressible viscoelastic material, whose complete nonlinear structural response is considered. We observe that the Darcy flux ($q$) is a nonlinear function -- steeper than linear -- of the pressure-difference ($ΔP$) across the medium. Furthermore, the flux is larger for a softer medium (smaller $G$). We construct a theory of this super-linear behavior by modelling the channels between the solid cylinders as elastic channels whose walls are made of a material with a linear constitutive relation but can undergo large deformation. Our theory further predicts that the flow permeability is a universal function of $ΔP/G$, which is confirmed by the present simulations.

physics.flu-dyn↗

arXiv2019

Influence of Langmuir adsorption and viscous fingering on transport of finite size samples in porous media

We examine the transport in a homogeneous porous medium of a finite slice of a solute which adsorbs on the porous matrix following a Langmuir adsorption isotherm and can influence the dynamic viscosity of the solution. In the absence of any viscosity variation, the Langmuir adsorption induces the formation of a shock layer wave at the frontal interface and of a rarefaction wave at the rear interface of the sample. For a finite width sample, these waves interact after a given time that varies nonlinearly with the adsorption properties to give a triangle-like concentration profile in which the mixing efficiency of the solute is larger in comparison to the linear or no-adsorption cases. In the presence of a viscosity contrast such that a less viscous carrier fluid displaces the more viscous finite slice, viscous fingers are formed at the rear rarefaction interface. The fingers propagate through the finite sample to preempt the shock layer at the viscously stable front. In the reverse case i.e. when the shock layer front features viscous fingering, the fingers are unable to intrude through the rarefaction zone and the qualitative properties of the expanding rear wave are preserved. A non-monotonic dependence with respect to the Langmuir adsorption parameter $b$ is observed in the onset time of interaction between the nonlinear waves and viscous fingering. The coupled effect of viscous fingering at the rear interface and of Langmuir adsorption provides a powerful mechanism to enhance the mixing efficiency of the adsorbed solute.

physics.flu-dyn↗

arXiv2019

Control of radial miscible viscous fingering

We investigate the stability of radial viscous fingering (VF) in miscible fluids. We show that the instability is decided by an interplay between advection and diffusion during initial stages of flow. Using linear stability analysis and nonlinear simulations, we demonstrate that this competition is a function of the radius $r_0$ of the circular region initially occupied by the less viscous fluid in the porous medium. For each $r_0$, we further determine the stability in terms of Péclet number ($Pe$) and log-mobility ratio ($M$). The $Pe-M$ parameter space is divided into stable and unstable zones--the boundary between the two zones is well approximated by $M =α(r_0) Pe^{-0.55}$. In the unstable zone, the instability is reduced (enhanced) with an increase (decrease) in $r_0$. Thus, a natural control measure for miscible radial VF in terms of $r_0$ is established. Finally, the results are validated by performing experiments which provide a good qualitative agreement with our numerical study. Implications for observations in oil recovery and other fingering instabilities are discussed.

physics.flu-dyn↗

arXiv2019

Confinement induced control of similarity solutions in premelting dynamics and other thin film problems

We study the combined effects of nonlocal elasticity and confinement induced ordering on the dynamics of thermomolecular pressure gradient driven premelted films bound by an elastic membrane. The confinement induced ordering is modeled using a film thickness dependent viscosity. When there is no confinement induced ordering, we recover the similarity solution for the evolution of the elastic membrane, which exhibits an infinite sequence of oscillations. However, when the confinement induced viscosity is comparable to the bulk viscosity, the numerical solutions of the full system reveal the conditions under which the oscillations and similarity solutions vanish. Implications of our results for general thermomechanical dynamics, frost heave observations and cryogenic cell preservation are discussed. Finally, through its influence on the viscosity, the confinement effect implicitly introduces a new universal length scale into the volume flux. Thus, there are a host of thin film problems, from droplet breakup to wetting/dewetting dynamics, whose properties (similarity solutions, regularization, and compact support) will change under the action of the confinement effect. Therefore, our study suggests revisiting the mathematical structure and experimental implications of a wide range of problems within the framework of the confinement effect.

physics.flu-dyn↗

arXiv2018

A numerical study on reaction-induced radial fingering instability

The dynamics of $A + B \rightarrow C$ fronts is analyzed numerically in a radial geometry. We are interested to understand miscible fingering instabilities when the simple chemical reaction changes the viscosity of the fluid locally and a non-monotonic viscosity profile with a global maximum or minimum is formed. We consider viscosity-matched reactants $A$ and $B$ generating a product $C$ having different viscosity than the reactants. Depending on the effect of $C$ on the viscosity relative to the reactants, different viscous fingering (VF) patterns are captured which are in good qualitative agreement with the existing radial experiments. We have found that for a given chemical reaction rate, an unfavourable viscosity contrast is not always sufficient to trigger the instability. For every fixed $Pe$, these effects of chemical reaction on VF are summarized in the $Da-R_c$ parameter space that exhibits a stable region separating two unstable regions corresponding to the cases of more and less viscous product. Fixing $Pe$, we determine $Da$-dependent critical log-mobility ratios $R_c^+$ and $R_c^-$ such that no VF is observable whenever $R^-_c \leq R_c \leq R^+_c$. The effect of geometry is observable on the onset of instability, where we obtain significant differences from existing results in the rectilinear geometry.

physics.flu-dyn↗

arXiv2017

Stability of miscible Rayleigh-Taylor fingers in porous media with non-monotonic density profiles

We study miscible Rayleigh-Taylor (RT) fingering instability in two-dimensional homogeneous porous media, in which the fluid density varies non-monotonically as a function of the solute concentration such that the maximum density lies in the interior of the domain, thus creating a stable and an unstable density gradients that evolve in time. With the help of linear stability analysis (LSA) as well as nonlinear simulations the effects of density gradients on the stability of RT fingers are investigated. As diffusion relaxes the concentration gradient, a non-monotonic density profile emerges in time. Our simple mathematical treatment addresses the importance of density gradients on the stability of miscible Rayleigh-Taylor fingering. In this process we identify that RT fingering instabilities are better understood combining Rayleigh number (Ra) with the density gradients--hence defining a new dimensionless group, gradient Rayleigh number (Ra$_g$). We show that controlling the stable and unstable density gradients, miscible Rayleigh-Taylor fingers are controllable.

physics.flu-dyn↗

arXiv2017

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.

physics.flu-dyn↗

arXiv2015

Non-modal linear stability analysis of miscible viscous fingering in porous media

The non-modal 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 finite difference method. The resultant initial value problem is solved by 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. With the help of non-modal 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 non-modal analysis, we performed direct numerical simulations using a highly accurate pseudo-spectral method. Further, a comparison of the present stability analysis with existing modal and initial value approach is also presented. It is shown that the non-modal stability results are in better agreement, than the other existing stability analyses, with those obtained from direct numerical simulations.

physics.flu-dyn↗

arXiv2015

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.

physics.flu-dyn↗