arXiv subjects

Manoranjan Mishra

Publications and source records attributed to Manoranjan Mishra.

15 recordsLinked to original sources

arXiv2026

Global Well-Posedness and Numerical Approximation of a Coupled Darcy-Convection-Diffusion System with Exponential Nonlinearity

This paper investigates density driven flow in porous media, focusing on the roles of viscosity contrast, density contrast, and linear adsorption. In this setup, the fluid on top is heavier and more viscous than the fluid below. Under the effect of gravity, this system becomes unstable, and finger-like structures appear. The phenomenon is described mathematically by coupling Darcy's law with a convection-diffusion reaction equation. The nonlinearity in this model arises mainly from the concentration dependence of viscosity and the convective transport term. The existence of a unique pair of weak solutions is shown using the Galerkin approximation method and truncation technique. Moreover, an application of the maximum principle shows non-negativity of the concentration. Additionally, we analyze the long-time behavior of the solution and prove that the concentration converges exponentially to zero in the $L^p$-norm for all $1 \le p \le \infty$ as $t \to \infty.$ To complement the theoretical analysis, we perform numerical simulations based on a pressure formulation. By tracking total kinetic energy and mixing measures over time, we discuss the instability and the mixing efficiency, respectively. The present study reveals that although increasing the density contrast amplifies the total kinetic energy, the marginal impact diminishes with successive increments of density contrast. Similarly, while adsorption acts to suppress mixing, its efficiency in doing so tends to saturate with further increases. These behavior are consistent with the numerical simulations.

math.AP↗

arXiv2026

Global Existence and Finite-Time Blow-Up for a Coupled Darcy-Forchheimer-Brinkman System with Quadratic Reaction Dynamics

We study a nonlinear system coupling the Darcy-Forchheimer-Brinkman equations with a convection-diffusion-reaction equation, arising in reactive transport through porous media. The model features a nonlinear viscosity coupling, Forchheimer inertial drag, convective transport, and a quadratic reaction term. We establish the existence of local-in-time weak solutions for general initial data. Under the physically relevant condition on initial data $0 \leq c_0 \leq 1$, a maximum principle for the concentration is proved, yielding global existence and uniqueness of weak solutions in two and three space dimensions. For higher regular initial data, we obtain the existence, uniqueness, and continuous dependence of strong solutions. In this regime, the concentration decays exponentially to zero in $L^p$-norm for all $1 \leq p \leq \infty$ with a uniform decay rate. In contrast, if $c_0 > 1$, we demonstrate the occurrence of finite-time blow-up of solutions and derive an explicit upper bound for the blow-up time. Finally, numerical simulations based on the finite element method are presented to illustrate both the decay behavior and finite-time blow-up predicted by the theory.

math.AP↗

arXiv2025

Well-posedness and Fingering Patterns in $A + B \rightarrow C$ Reactive Porous Media Flow

The convection-diffusion-reaction system governing incompressible reactive fluids in porous media is studied, focusing on the $ A + B \to C $ reaction coupled with density-driven flow. The time-dependent Brinkman equation describes the velocity field, incorporating permeability variations modeled as an exponential function of the product concentration. Density variations are accounted for using the Oberbeck-Boussinesq approximation, with density as a function of reactants and product concentration. The existence and uniqueness of weak solutions are established via the Galerkin approach, proving the system's well-posedness. A maximum principle ensures reactant nonnegativity with nonnegative initial conditions, while the product concentration is shown to be bounded, with an explicit upper bound derived in a simplified setting. Numerical simulations are performed using the finite element method to explore reactive fingering instabilities and illustrate the effects of density stratification, differential product mobility, and two- or three-dimensionality. Two cases with initial flat and elliptic interfaces further demonstrate the theoretical result that solutions continuously depend on initial and boundary conditions. These theoretical and numerical findings provide a foundation for understanding chemically induced fingering patterns and their implications in applications such as carbon dioxide sequestration, petroleum migration, and rock dissolution in karst reservoirs.

math.AP↗

arXiv2024

Existence and Uniqueness of Solution to Unsteady Darcy-Brinkman Problem with Korteweg Stress for Modelling Miscible Porous Media Flow

The work investigates a model that combines a convection-diffusion-reaction equation for solute concentration with an unsteady Darcy-Brinkman equation for the flow field, including the Kortweg stress. Additionally, the flow field experiences an external body force term while the permeability fluctuates with solute concentration. Such models are used to describe flows in porous mediums such as fractured karst reservoirs, mineral wool, industrial foam, coastal mud, etc. The system of equations has Neumann boundary conditions for the solute concentration and no-flow conditions for the velocity field, and the well-posedness of the model is discussed for a wide range of initial data. The proofing techniques remain applicable in establishing the well-posedness of non-reactive and homogeneous porous media flows under the specified simplifications.

math.AP↗

arXiv2021

Effect of Serum Starvation on Rheology of Cell Monolayers

The rheological properties of cells and tissues are central to embryonic development and homoeostasis in adult tissues and organs and are closely related to their physiological activities. In this work, we present our study of rheological experiments on cell monolayer under serum starvation compared to that of healthy cell monolayer with full serum. The normal functioning of cells depends on the micronutrient supply provided by the serum in the growth media. Serum starvation is one of the most widely used procedures in cell biology. Serum deficiency may lead to genomic instability, variation in protein expression, chronic diseases, and some specific types of cancers. However, the effect of deprivation of serum concentration on the material properties of cells is still unknown. Therefore, we performed the macro-rheology experiments to investigate the effect of serum starvation on a fully confluent Madin Darby Canine Kidney (MDCK) cell monolayer. The material properties such as storage modulus (G') and loss modulus (G''), of the monolayer, were measured using oscillatory shear experiments under serum-free (0% FBS) and full serum (10% FBS) conditions. Additionally, the step strain experiments were performed to gain more insights into the viscoelastic properties of the cell monolayer. Our results indicate that without serum, the loss and storage moduli decrease and do not recover fully even after small deformation. This is because of the lack of nutrients, which may result in many permanent physiological changes. Whereas, the healthy cell monolayer under full serum condition, remains strong & flexible, and can fully recover even from a large deformation at higher strain.

physics.bio-ph↗

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↗

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↗

arXiv2018

Nonmodal stability analysis of miscible viscous fingering with non-monotonic viscosity profiles

A non-modal linear stability analysis (NMA) of the miscible viscous fingering in a porous medium is studied for a toy model of non-monotonic viscosity variation. The onset of instability and its physical mechanism are captured in terms of the singular values of the propagator matrix corresponding to the non-autonomous linear equations. We discuss two types of non-monotonic viscosity profiles, namely, with unfavorable (when a less viscous fluid displaces a high viscous fluid) and with favorable (when a more viscous fluid displaces a less viscous fluid) end-point viscosities. A linear stability analysis yields instabilities for such viscosity variations. In addition, we also show that to understand the spatiotemporal evolution of the perturbations it is necessary to analyse the viscosity gradient with respect to the concentration and the location of the maximum concentration $c_m$. For the favorable end-point viscosities, a weak transient instability is observed when the viscosity maximum moves close to the pure invading or defending fluid. This instability is attributed to an interplay between the sharp viscosity gradient and the favorable end-point viscosity contrast. We demonstrate the dissimilarity between the quasi-steady-state approach and NMA in finding the correct perturbation structure and the onset, for both the favorable and unfavorable viscosity profiles. The correctness of the linear perturbation structure obtained from the non-modal stability analysis is validated through nonlinear simulations. We have found that the nonlinear simulations and NMA results are in good agreement. In summary, a non-monotonic variation of the viscosity of a miscible fluid pair is seen to have a larger influence on the onset of fingering instabilities, than the corresponding Arrhenius type relationship.

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↗

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↗

arXiv2015

Onset of fingering instability in a finite slice of adsorbed solute

The effect of a linear adsorption isotherm on the onset of fingering instability in a miscible displacement in the application of liquid chromatography, pollutant contamination in aquifers etc. is investigated. Such fingering instability on the solute dynamics arise due to the miscible viscus fingering (VF) between the displacing fluid and sample solvent. We use a Fourier pseudo-spectral method to solve the initial value problem appeared in the linear stability analysis. The present linear stability analysis is of generic type and it captures the early time diffusion dominated region which was never expressible through the quasi-steady state analysis (QSSA). In addition, it measures the onset of instability more accurately than the QSSA methods. It is shown that the onset time depends non-monotonically on the retention parameter of the solute adsorption. This qualitative influence of the retention parameter on the onset of instability resemblances with the results obtained from direct numerical simulations of the nonlinear equations. Moreover, the present linear stability method helps for an appropriate characterisation of the linear and the nonlinear regimes of miscible VF instability and also can be useful for the fluid flow problems with the unsteady base-state.

physics.flu-dyn↗

arXiv2015

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.

physics.flu-dyn↗

arXiv2014

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

Influence of fluid dispersion on the Saffman-Taylor instability in miscible fluids has been investigated both in the linear and nonlinear regimes. The convective characteristic scales are used for the dimensionless formulation that incorporates Péclet number (Pe) into the governing equations as a measure for the fluid dispersion. A linear stability analysis (LSA) has been performed in a similarity transformation domain using the quasi-steady-state approximation. LSA results show that systems with large Pe become more unstable and the onset of instability occurs earlier compared to the case when Pe is smaller. Variations of the most unstable wave number and the cut-off wave number with Pe have been analyzed. Fourier spectral method has been used for the numerical simulations of the fully nonlinear system. The results indicate that the wave numbers of the unstable modes increase with Pe. Influence of the anisotropic dispersion on the onset in both the linear and nonlinear regimes has been analyzed. Large transverse diffusivity increases the length scale of the fingers quickly and merges the fingers to generate coarser fingers. Finally the combined effect of the Korteweg stress and Pe in the linear regime has been perused. In the presence of the Korteweg stresses and depending upon various flow parameters, a fluid system with larger Pe exhibits smaller instantaneous growth rate than with smaller Pe, which is a counter-intuitive result.

physics.flu-dyn↗