## What is/are Porous Medium?

Porous Medium - Thermophysical properties of water-copper nanoliquid as a function of properties of water as base liquid, copper as nanoparticle and 30% glass fiber reinforced polycarbonate as porous medium are obtained from either phenomenological laws or mixture theory.^{[1]}In order to predict the evolution of solute concentration in a porous medium, the macroscopic continuum models (CMs) are commonly employed.

^{[2]}particulate flow in high aspect ratio microfluidic channels, in a porous medium or in microfluidic devices with repetitive structures).

^{[3]}The flexible point cloud format can represent the geometry and flow distribution of different aneurysms before and after FD stent (represented by porous medium layer) placement with high resolution.

^{[4]}It was shown that a decrease in permeability of the porous medium (in other words, increase in its porosity) causes an increase in flow stability.

^{[5]}38, the particles can match through the porous medium.

^{[6]}The results show that an increase in pore density leads to an increase in the temperature difference between the irradiated face and the rear face of the absorber, this occurs because when pore density increases the concentrated energy no longer penetrates in the deepest space of the absorber and energy is absorbed in areas close to the surface; therefore, temperature gradients are created within the porous medium.

^{[7]}This work provides an analytical approach to characterize and determine solutions to a porous medium system of equations with views in applications to invasive-invaded biological dynamics.

^{[8]}Furthermore, the PPG concentration of 1750 ppm resulted in a significant entrapment in the porous medium.

^{[9]}Previous studies have shown that this transition is highly sensitive to fluid flow rate, capillarity, and the structural properties of the porous medium.

^{[10]}It is named the fractional differential scheme and is based on a fractional integral formulation of the DS for a porous medium considered as a two-component composite.

^{[11]}Due to importance of squeezing flow, in this paper, an unsteady squeezing flow of a viscous magnetohydrodynamic (MHD) fluid which is passing through porous medium has been modeled and analyzed with and without slip effects at the boundaries.

^{[12]}In this work, the impact of a magnetic field on the onset of the Jeffrey fluid convection through a porous medium is investigated theoretically.

^{[13]}A porous medium with a high specific surface area can be used to promote hydrate formation.

^{[14]}A computational study has been carried out to assess the effectiveness of a porous medium as a passive control device suitable for reducing the drag in a normal-shock-wave/boundary-layer interaction at transonic speeds with a view toward application in aircraft wings.

^{[15]}The fixed bed represents a porous medium where Darcy-type flow conditions can be assumed.

^{[16]}Aim of this study is to investigate the properties of mono-atomic gas flow through the porous medium by using Event-Driven Molecular Dynamics (EDMD) simulation in the transition regime.

^{[17]}Our simulations show that the protein matrix acts as a porous medium for the transport of water molecules in and out during the electron transfer process.

^{[18]}Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm.

^{[19]}A conceptual exploration is conducted to analyze the numerical experiment of the pattern hydrodynamic slip flow control and thermo-fluidic transport features coupled with the influence of the combined electromagnetohydrodynamic (EMHD) effect in a wavy microchannel through the porous medium.

^{[20]}A porous medium of varying permeability was employed to simulate the sclerosant effect on the nidus haemodynamics.

^{[21]}It is found that the stability characteristics of plane Couette flow in a porous medium in the presence of throughflow are remarkably different from no throughflow and non-porous cases.

^{[22]}Bearing in mind such exhilarating features of hybrid nanofluid, our intention in current research is to examine the heat and flow transfer rates in water-based hybrid nanofluid with suspension of hybrid nanoparticles (Cu– $$\hbox {Fe}_{3}\hbox {O}_{4}$$ ) past a vertical cone enclosed in a porous medium.

^{[23]}In various branches of applied sciences, porous medium equations exist where this basic model occurs in a natural fashion.

^{[24]}Unlike for a solid insert, in which noise sources are concentrated at the trailing edge, those on the porous insert are distributed across the porous medium surface, and they promote phase interference effect that causes noise attenuation.

^{[25]}To verify the accuracy of the conditions, macroscopic solutions are compared to feature-resolving simulations and excellent agreement is demonstrated, even for values of the Reynolds number larger than those for which the theory is formally applicable and for a large value of the porosity which results in significant infiltration of the fluid into the porous medium.

^{[26]}Normally this liquid will not flow until its saturation exceeds the critical condensate saturation (Scc) due to the capillary pressure and relative permeability of the porous medium.

^{[27]}The porous medium anisotropy induces a lower effective resistance when the pillars are flow-opposing oriented, suppressing front roughening and capillary fingering.

^{[28]}As an application, we show the short time existence of solutions to the porous medium equation.

^{[29]}This analysis is considered through a porous medium.

^{[30]}Clogging (including physical, chemical and biological clogging) of the porous medium not only directly reduces the hydraulic load (treatment efficiency), but also reduces the service life.

^{[31]}Solving local problems allows us to find accurate local distributions of velocities, pressure and viscosity inside a separate pore, and also to evaluate the permeability of the porous medium and the effective viscosity of the fluid when only the geometry of the pore is known.

^{[32]}Throughout the paper, we explored answers to the following questions: (1) What conditions (asphaltene content) allow asphaltenes to stabilize W/O emulsions? (2) What are optimal aqueous conditions (pH, salinity) for stable emulsions? (3) Can such emulsions stabilized by asphaltenes be formed in-situ in a porous medium and thus improve recovery? These questions were answered through glass vial tests under controlled experimental conditions and sandpack flooding experiments.

^{[33]}In the present work we establish sharp regularity estimates for the solutions of the porous medium equation, along their zero level-sets.

^{[34]}Vegetation is modeled as a porous medium for the flow of moist air and a leaf energy balance model is used to determine the heat fluxes and transpiration at leaf surfaces.

^{[35]}Motivated by this geophysical problem, we analyze the influence of the porous medium properties on the evolution of a buoyant current that is weakly soluble with the ambient fluid.

^{[36]}There are known a lot of software for calculating the motion of salt solutions in the porous medium.

^{[37]}The cylinder is embedded in a porous medium and is considered to move vertically with nonlinear velocity.

^{[38]}The effective permeability of a composite composed of staggered slats embedded in a porous medium is determined.

^{[39]}, a solid particle moving in a liquid and fluid flowing through a porous medium, this method is applied to a high shear mixing problem of two types of solid particles mixed in a viscous liquid by a four-bladed mixer.

^{[40]}The main governing parameters of the problem are the Rayleigh, the Hartmann, the Soret, the Dufour and the Lewis numbers, the buoyancy ratio, the enclosure aspect ratio, and the normalized porosity of the porous medium.

^{[41]}In this paper, we derive several differential Harnack estimates (also known as Li-Yau-Hamilton type estimates) for nonnegative solutions and positive solutions of the porous medium equation f t = f f x x + f x 2 + f 2 , which is a nonlinear parabolic equation.

^{[42]}In this study, a mathematical model modelling the flowback behavior of salt in a multi-porous medium consisting of hydraulic fractures (HF), induced natural fractures (IF) and shale matrix (MP) under multi-phase flow conditions is proposed.

^{[43]}In the strong-deformability regime, we find that diffusion is initially described by the porous medium equation, thus yielding subdiffusive behavior of an initially localized cloud of particles.

^{[44]}The actual behavior of reservoir systems is determined by complexity of moving fluids rheology and the morphological structure of the porous medium, as well as the variety of interaction processes between the fluid and the porous medium.

^{[45]}The aim of study is to investigate the mass and heat transfer phenomena in hybrid hydro-nanofluidic system involving Al2O3–Cu–H2O over the rotating disk in porous medium with viscous dissolution and Joule heating through the stochastic solver by way of Levenberg-Marquardt backpropagation neural networks.

^{[46]}The dependence of the oil displacement front by the solvent on the dispersion parameter of the porous medium is considered.

^{[47]}A distinctive feature of this model is taking into account the porous medium strain in the deformation zone.

^{[48]}However, the scale of the research is bigger than porous medium layer of the particle that can’t meet the actual mechanism of the simulation.

^{[49]}Based on Darcy–Forchheimer model was implemented to illustrate the flow during the porous medium.

^{[50]}

## local thermal non

A general analytical solution procedure for the Brinkman–Forchheimer-extended Darcy model has been proposed to obtain local thermal non-equilibrium solutions for the fully developed forced convection in a channel filled with a porous medium subject to constant wall heat flux.^{[1]}The onset of thermal convection of a fluid saturated anisotropic bidisperse porous medium under the condition of local thermal non-equilibrium is investigated.

^{[2]}This work presents a local thermal non-equilibrium model for fully developed flow in a channel filled with a porous medium where Bi itself varies across the channel.

^{[3]}Global and linear stability analyses of Darcy–Brinkman–Benard convection in a liquid-saturated porous medium with a non-uniform gravity field using the local thermal non-equilibrium (LTNE) model are investigated.

^{[4]}The present study investigates the onset of Darcy–Bénard convection in a liquid-saturated anisotropic porous medium when phases are in local thermal non-equilibrium (LTNE) analytically.

^{[5]}The problem further involves mixed convection, entropy generation, local thermal non-equilibrium and non-linear thermal radiation within the porous medium.

^{[6]}Rayleigh-Benard-Taylor convection in a Newtonian, nanoliquid-saturated high porous medium using the local thermal non-equilibrium model (LTNE) is studied analytically using the single term Galerkin technique.

^{[7]}An analytical and numerical study was made on thermally developing forced convective flow in a channel filled with a fluid-saturated porous medium, subject to constant heat flux, under local thermal non-equilibrium.

^{[8]}In the paper, we make a linear stability analysis in a Newtonian, liquid-saturated high porous medium in the presence of a heat source under the assumption of local thermal non-equilibrium(LTNE).

^{[9]}For the porous medium, the Local Thermal Non-Equilibrium (LTNE) model is used.

^{[10]}

## mixed convection flow

This study reflects the combined impact of double dispersion and injection/suction on mixed convection flow over a vertical cone in an incompressible viscous fluid-saturated porous medium.^{[1]}This paper reports the instability mechanism of parallel mixed convection flow in a differentially heated vertical channel filled with a highly permeable porous medium.

^{[2]}This article analyses the thermal decomposition in an unsteady MHD mixed convection flow of a reactive, electrically conducting Casson fluid within a vertical channel filled with a saturated porous medium and the influence of the temperature dependent properties on the flow.

^{[3]}Therefore, the aim of this study is to analyze the hydrodynamic and thermal behaviors of unsteady mixed convection flow of nanofluid in a micro-channel filed with a saturated porous medium.

^{[4]}In the present study, the silent characteristics of magnetized mixed convection flow in porous medium suspended with both microorganisms and nanoparticles induced by two expanded or contracted disks are investigated.

^{[5]}The steady mixed convection flow towards an isothermal permeable vertical cylinder nested in a fluid-saturated porous medium is studied.

^{[6]}An efficient overlapping multi-domain spectral method is presented and used in the analysis of a two-dimensional steady laminar MHD mixed convection flow, heat and mass transfer over a vertical flat plate embedded in a porous medium.

^{[7]}

## partial differential equation

method to solve one of the nonlinear partial differential equation, which is the two-dimensional porous medium equation.^{[1]}This article models a system of partial differential equations (PDEs) for the thermal and solute characteristics under gradients (concentration and temperature) in the magnetohydrodynamic flow of Casson liquid in a Darcy porous medium.

^{[2]}A mathematical model that describes a partial differential equation with boundary, internal and initial conditions was developed in the article, to study the gas-dynamic parameters of the gas filtration process in a porous medium under isothermal conditions.

^{[3]}In this work, we examine a nonlinear partial differential equation of fluid mechanics, namely, the generalized nonlinear advection–diffusion equation, which portrays the motion of buoyancy driven plume in a bent-on porous medium.

^{[4]}A porous medium equation is a nonlinear parabolic partial differential equation that presents many physical occurrences.

^{[5]}The present study is a valid contribution to the existing literature, by developing a nonstandard line method for the partial differential equation, in order to obtain a numerical solution of unsteady flow of gas through nano porous medium.

^{[6]}A realistic nonlinear system of partial differential equations is developed for coupled water and solute transport through a drying porous medium when the solute has a mobile state (monomers) as well as an immobile state (micelles).

^{[7]}

## inclined magnetic field

The present investigation is focused on the study of a micropolar flow in a porous medium subject to inclined magnetic field, mass transpiration and internal radiation.^{[1]}In this analysis, a theoretical model is proposed to examine the collective effect of slip velocity, magnetic field, and inclination angles on an unsteady non-Newtonian particulate suspension flow in an inclined diseased tapered tube with a porous medium by applying an external inclined magnetic field.

^{[2]}The present study deal with the effective heat transfer with the activity of connected inclined magnetic field of the asymmetric channel through the porous medium.

^{[3]}The impact of the induced magnetic description on magnetohydrodynamic porous medium flow becomes significant by decisive importance to an inclined magnetic field.

^{[4]}A theoretical study on the flow of Casson nanofluid past a linear stretching sheet in a non-Darcian porous medium under the influence of inclined magnetic field is performed.

^{[5]}

## natural convection heat

Steady conjugate natural convection heat transfer in a two-dimensional enclosure filled with fluid saturated porous medium is studied numerically.^{[1]}Natural convection heat transfer in a rectangular cavity filled with a saturated porous medium with variable permeability is investigated analytically and numerically.

^{[2]}This paper represents an experimental study of natural convection heat transfer in a square enclosure filled with saturated porous medium and partially heated from below.

^{[3]}A numerical study is performed to investigate the magnetic field effects on steady laminar natural convection heat transfer between two concentric horizontal cylinders filled with a porous medium.

^{[4]}This study uses the heat transfer equation and the Navier-Stroke equation to describe heat transfer phenomena in porous mediums and fluid flow phenomena within the 3D human eye model, based on considering the natural convection heat transfer of aqueous humor and vitreous humor, under inconstant solar irradiation.

^{[5]}

## steady two dimensional

The steady two-dimensional stagnant flow of Sutterby nanofluid inside the boundary layer over a stretching wedge placed in a porous medium is investigated.^{[1]}Moreover, a similarity solution is given for steady two-dimensional flow subjected to Buongiorno’s theory to investigate the nature of magnetohydrodynamics (MHD) in a porous medium, utilizing the local thermal non-equilibrium conditions (LTNE).

^{[2]}Entropy generation analysis in steady two-dimensional, viscous, incompressible forced convective Falkner–Skan flow of Maxwell nanofluid over a static wedge embedded in a porous medium with temperature-dependent viscosity is examined.

^{[3]}This work examines the steady two-dimensional mixed convection boundary layer flow of non-Newtonian Carreau fluid embedded in a porous medium.

^{[4]}The steady two-dimensional, laminar, viscous, incompressible boundary layer flow of Cu/Ag-H2O nanofluid in a diverging channel formed by two non-parallel walls in a Darcian porous medium is numerically studied in the presence of mass suction/injection of equal magnitude on both the walls.

^{[5]}

## three dimensional flow

The aim of this paper is to determine the time-dependent MHD fractionalized three-dimensional flow of viscoelastic fluid in porous medium with heat transfer by traveling wave solution.^{[1]}Purpose The purpose of this study is to find the multiple branches of the three-dimensional flow of Cu-Al2 O3/water rotating hybrid nanofluid perfusing a porous medium over the stretching/shrinking surface.

^{[2]}The present study delivers the mathematical model and theoretical analysis of a three-dimensional flow in a free convection for an electrically conducting incompressible second-grade fluid through a very high porous medium circumscribed by an infinite vertical porous plate subject to a constant suction.

^{[3]}In this study, we model and theoretically analyze a free convective three-dimensional flow of an incompressible second-grade fluid through a highly porous medium bounded by an infinite vertical porous plate subjected to a constant suction.

^{[4]}

## vertical magnetic field

Purpose The effect of magnetic field dependent (MFD) viscosity on the onset of convection in a ferromagnetic fluid layer heated from below saturating rotating porous medium in the presence of vertical magnetic field is investigated theoretically by using Darcy model.^{[1]}This paper deals with the convection of micropolar fluids heated and soluted from below in the presence of suspended particles (fine dust) and uniform vertical rotation and uniform vertical magnetic field in a porous medium.

^{[2]}This study features a model for double-diffusive convection in a bidisperse porous medium where a vertical magnetic field chemical reaction’s effects are present.

^{[3]}The thermal convection of a plasma in porous medium is investigated to include simultaneously the effect of rotation and the finiteness of the ion Larmor radius (FLR) in the presence of a vertical magnetic field.

^{[4]}

## electrically conducting fluid

A mixed convective momentum and heat transfer in a composite system, consisting of electrically conducting fluid and saturated porous medium in a vertical channel, have been studied.^{[1]}In the current article, we have thoroughly investigated the collective impact of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid (Cattaneo–Friedrich Maxwell-CFM model) over a permeable surface embedded in a porous medium.

^{[2]}This paper investigates the unsteady flow and heat transfer of a viscous, incompressible, and electrically conducting fluid through a porous medium in a horizontal channel.

^{[3]}An attempt has been made to study laminar convective heat and mass transfer flow of an incompressible, viscous and electrically conducting fluid over an impulsively started vertical plate with conduction-radiation embedded in a porous medium in presence of transverse magnetic field.

^{[4]}

## enhanced oil recovery

Mathematical solutions for multiphase flow in porous medium are key for determining relative permeability curves from laboratory data, to check numerical reservoir simulation results and for screening an enhanced oil recovery technique.^{[1]}The presence of microorganisms could alter the porous medium permeability, which is vital for several applications, including aquifer storage and recovery (ASR), enhanced oil recovery (EOR) and underground hydrogen storage.

^{[2]}The efficacy of surfactant flooding as a Chemical Enhanced Oil Recovery (EOR) method depends on the amount of the surfactant loss in the porous medium.

^{[3]}The efficiency of emulsion injection as an enhanced oil recovery (EOR) method is investigated in a synthetic porous medium consisting of sintered bi-disperse glass beads, which emulates the porosity and permeability of real oil-bearing rocks.

^{[4]}

## heat source sink

Bejan number and entropy production are analyzed through the existence of porous medium, viscous dissipation, magnetic field, thermal radiation and heat source/sink.^{[1]}The presence of magnetic field and non-uniform heat source/sink is considered along with the flow over a stretching sheet enclosed in a porous medium.

^{[2]}It is assumed that the cylinder is embedded in a porous medium and, external magnetic field, heat source/sink are also taken into account.

^{[3]}With emphasis to the inherent aforementioned concepts together with heat source/sink and thermal radiation, this paper presents insight into the dynamics of unsteady Ethelene glycol conveying graphene nanoparticles through porous medium.

^{[4]}

## finite element method

This paper aims to develop a moving-mesh type Finite Element Method for the computation of the transient unconfined seepage flow through the porous medium.^{[1]}In this paper, transient coupled heat and moisture transfer in the cylinders and cylindrical panels of the porous medium are analyzed with three-dimensional finite element methods.

^{[2]}In present work Cahn–Hilliard phase field coupled with Navier-Stokes equations were solved using finite element method to simulate polymer flooding in a dual-permeability porous medium, the effect of injection rate, oil viscosity and oil density on polymer displacement oil process was quanlitatively and quantitatively investigated.

^{[3]}A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium by employing a mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics (CEMFEM) for the concentration equation.

^{[4]}

## stagnation point flow

The mixed convection on a stagnation-point flow of a thermo micropolar hybrid nanofluid through a vertical surface in a saturated porous medium having inertial and microstructure characteristics is investigated using Darcy–Brinkman model.^{[1]}In this study, a two-dimensional magnetohydrodynamic stagnation point flow of magnetite ferrofluid past a stretching/shrinking sheet through a Darcy–Forchheimer porous medium is investigated in the occurrence of viscous dissipation, suction/injection, and convective heating.

^{[2]}This paper analyzes the collective effects of buoyancy force, thermal radiation, convective heating, and magnetic field on stagnation point flow of an electrically conducting nanofluid past a permeable stretching/shrinking sheet in a porous medium.

^{[3]}This study communicates stagnation-point flow in magneto-Williamson nanofluid along a convectively heated nonlinear stretchable material in a porous medium.

^{[4]}

## power law fluid

On this basis, we constructed a physical model of blood coagulation in a porous medium integrated with the power-law fluid model to study the proposed poly(vinyl alcohol)-chitosan (PVA-CS) composite hemostatic material.^{[1]}This study deals the influence of power law fluid and inclined magnetic field on a porous medium in staggered cavity.

^{[2]}In this paper, we study the stationary incompressible power law fluid flow in a thin porous medium.

^{[3]}This problem deals with the power-law fluid flow with thermally stable stratification in a non-Darcy porous medium over a convectively heated truncated cone and this work is very useful in actual and applied circumstances due to presence of non-linear Boussinesq approximation.

^{[4]}

## two dimensional mixed

The current assessment deals with thermal and solutal stratifications in two-dimensional mixed convection tangent hyperbolic fluid past a stretched surface in porous medium under the effects of magnetic field.^{[1]}This paper studies the local thermal nonequilibrium (LTNE) model for two-dimensional mixed convection boundary-layer flow over a wedge, which is embedded in a porous medium in the presence of radiation and viscous dissipation. <