Anisotropic Porous Media Research Articles

Removability conditions for anisotropic parabolic equations in a computational validation

The article investigates removability conditions for singularities of anisotropic parabolic equations and in particular for the anisotropic porous medium equation and it aims in the numerical validation of the analytical results. The preconditions on the strength of the anisotropy are analyzed, and the analytical estimates for the growth behavior of the solutions near the singularities are compared with the observed growth in numerical simulations. Despite classical estimates used in the proof, we find that the analytical estimates are surprisingly close to the numerically observed solution behavior.

Multiscale modeling and simulation of turbulent flows in porous media

Numerical simulation is an important tool for understanding the physics of flows in porous media and for making predictions. The state of the art of multiscale modeling and simulation of turbulent flows in porous media is reviewed in this paper. Numerical simulations of flows in porous media can be classified as microscopic simulations, in which both macroscopic and pore-scale flows are directly resolved, and macroscopic simulations, in which the pore-scale motions are modeled while the volume-averaged equations are solved. Studies in the past few years have shown that microscopic simulations improve the understanding of turbulent flows in porous media considerably; this motivates the development of more efficient and more accurate turbulence models for macroscopic simulations. On the basis of this review, we believe that simulation of flows with higher Reynolds numbers, understanding the transport of macroscopic turbulence, modeling of turbulent flows in inhomogeneous and anisotropic porous media, simulation of compressible and multiphase turbulent flows in porous media, and fluid–structure interaction in deformable porous matrices are important topics to be studied in the future.

Transient dispersion of a reactive solute in an oscillatory Couette flow through an anisotropic porous medium

This research discusses the significance of reactive solute dispersion relevant to ecological, biological, and geological contexts. It examines solute movement under oscillatory Couette flow through anisotropic porous media between parallel plates under the effect of heterogeneous boundary reactions. The flow is driven by the combined effect of upper plate oscillation in its plane and time-dependent pressure gradient. The lower plate is assumed rough, which introduces slip velocity. A semi-analytical approach, with the method of moments and finite difference scheme, is utilized to explore the transient dispersion in steady and oscillatory flows with or without a non-zero mean. Using the Hermite polynomial, the mean concentration for purely oscillatory and combined flow is obtained, highlighting notable variations based on flow factors. The results suggest that increment in anisotropic angle ϕ reduces dispersion and enhances mean concentration for permeability ratio K < 1, but this reverses for K > 1. Three dispersion phases emerge: diffusive, anomalous, and Taylor's regimes. Gaussian cloud distribution occurs at small and large times, with intermediate stages displaying anomalous dispersion and asymmetric longitudinal distribution. Effects of boundary absorption stabilize over time. The research focuses on the practical significance of different permeabilities of porous media, emphasizing applications of anisotropic porous media in fields such as chemical engineering and industrial processes.

IMPACT OF AN ANISOTROPIC POROUS MEDIA ON THERMOBIOCONVECTION INSTABILITY IN THE PRESENCE OF GYROTACTICMICROORGANISMS AND HEATING FROM BELOW

The onset of thermobioconvection in a horizontal fluid layer saturated by gyrotactic microorganisms into an anisotropic porous medium is examined. The modeling of the governing equations considers heating from below, Darcy flow, and Boussinesq approximations along with the presence of gyrotactic microorganisms. The system of ordinary differential equations is obtained using linear stability analysis and the normal mode technique. The single-term Galerkin method casts the analytical solutions while the higher-order Galerkin technique is employed to compute the numerical solutions. The influence of the mechanical and thermal anisotropy parameters along with all bioconvection parameters on the onset of thermobioconvection are analytically as well as numerically discussed. It is perceived that bioconvection Rayleigh-Darcy number <i>Rb</i>, the gyrotactic number<i> G</i>, and Péclet number <i>Q</i> are to fast-forward the onset of convective motion. On the other hand, thermal anisotropy produces a stable system and acts to postpone the bioconvection pattern formulation. Mechanical anisotropy is found to have a destabilizing impact on the stability of the suspension and helps in the development of bioconvection. The enhancement of mechanical and thermal anisotropy is found to increase the size of bioconvective cells whereas the augmentation of the concentration of gyrotactic specie reduces the convective cell size.

Anisotropic Darcy–Brinkman Magnetic Fluid Convection under the Influence of a Time-Dependent Sinusoidal Magnetic Field

The impact of the sinusoidal mode of a magnetic field involving time fluctuations on the threshold of the ferromagnetic smart liquid convection in a saturated permeable medium is investigated using the regular perturbation technique. The Darcy–Brinkman model with anisotropic permeability is used to describe the flow through porous media. The thermal anisotropy is implemented in the energy equation. The problem might be useful in thermal engineering applications such as dynamic loudspeakers and computer hard discs and in medical applications like the treatment of tumor cells and the cell separation, to name a few. The regular perturbation technique is based on the minimum amplitude of a magnetic field modulation, and the onset criterion is dealt with in terms of a correction in the critical Rayleigh number and wavenumber. The thermal Rayleigh number correction depends on the magnetic field modulation frequency, magnetic force, anisotropies, porosity, and Prandtl number. At moderate values of the magnetic field modulation frequency, the impact of various physical factors is perceived to be noteworthy. The influences of the magnetic mechanism, Prandtl number, porosity parameter, and Brinkman number are shown to augment the destabilizing effect of the magnetic field modulation for moderate values of the frequency of a modulation. However, the destabilizing effect of the magnetic field modulation is diminished due to an increase in the values of the mechanical anisotropy parameter and thermal anisotropy parameter. The study reveals that the effect of the magnetic field modulation could be exploited to control the convective instability in an anisotropic porous medium saturated by a ferromagnetic fluid.

Effects of anisotropic permeability on EMHD nanofluid flow and heat transfer in porous microchannel with wavy rough walls

This theoretical study thoroughly examines the complexities of fluid flow and heat transfer within microfluidic devices by employing anisotropic porous media with wavy walls. The effect of anisotropic permeability and electromagnetohydrodynamics (EMHD) on nanofluid transport is examined in a wavy microchannel under a constant pressure gradient. The nonlinear coupled governing equations for electric double-layer potential, velocity, temperature, and nanoparticle volume fraction are solved numerically using shooting techniques with the Runge–Kutta method. The numerical results are validated with the asymptotic analytical solutions. We explore the impact of parameters like the Hartmann number, Darcy number, and joule heating on nanofluidic flow. The interplay between anisotropic permeability ratio and angle significantly influences flow, temperature profiles, and nanoparticle volume fraction. Reduction in nanoparticle concentration is attributed to constrained entry into the porous medium due to elevated anisotropic permeability. Frictional heating, influenced by the Forchheimer inertial effect, enhances temperature and induces slug flow at low Darcy numbers. The Nusselt number peaks at low Darcy numbers with an anisotropic angle of φ=π/2, experiencing a minimum when φ=0. These findings deepen our understanding of the role of anisotropic permeability in shaping fluid flow and heat transfer in microfluidic systems influenced by EMHD effects.

Regimes of Fluid Displacement from an Anisotropic Formation in the Gravity Field

The problem of two-phase flow of incompressible fluids through anisotropic porous media in the gravity field is considered. Similarity criteria characterizing the flow directions of the displacing and displaced fluids are determined. The regimes of displacement from an anisotropic formation are classified within the framework of study of numerical solutions to the profile problem of flow through an anisotropic porous medium. It is shown that there are four regimes corresponding to qualitatively different flows. Their effectiveness is compared in terms of the coefficient of fluid extraction from the formation and the displacement sweep factor. The influence of the capillary pressure on the displacement efficiency in various flow regimes is studied. It is shown that in some cases increase in the influence of the capillary pressure leads to increase in the displacement coefficient, while in other regimes, on the contrary, to its decrease.

Anisotropy distorts the spreading of a fixed volume porous gravity current

We consider the release and subsequent gravity-driven spreading of a dense finite volume of fluid in an anisotropic porous medium bounded by an impermeable substrate. When the permeability in the vertical direction is much smaller than in the horizontal direction, as is the case in many real geological reservoirs, this restricts the spread of the current to a very thin layer near the impermeable base. Using a combination of asymptotic analysis and finite difference computations of Darcy flow, we show that there exist two distinct flow regimes. At early times, the bulk of the current descends slowly and uniformly, injecting fluid into thin finger-like regions near the base. At much later times, the current transitions to the classical gravity-driven solution and continues to spread with a self-similar shape. One interesting consequence is that the swept volume of the current grows differently depending on the anisotropy of the medium. This has important consequences for managing contaminant spills, where it is important to minimize the contacted volume of the aquifer, or during geological CO 2 sequestration where a larger contacted volume results in more CO 2 being stored.

Nano‐bioconvective anisotropic slip flow in anisotropic porous medium with Coriolis force effects

AbstractThe present paper examines the impact of the Coriolis force and anisotropic Darcian porous medium on the nonsteady convective flow of bio‐nanofluid along a three‐dimensional stretchable surface. The problem has applications in extrusion processes and the geographical movement of tectonic plates under a moveable ocean. Emphasis is placed on the combined effects of anisotropic porous medium, anisotropic momentum slip and temperature, and microorganism slips at the wall. The pertaining equations are condensed to a set of similarity equations before being simulated using the Keller‐box method, an efficient numerical technique. The study reveals that the friction factor along the y‐axis decreases as the rotation parameter increases. Moreover, the heat and nanoparticle volume fraction movement gradient decrease along the y‐axis when the Darcy number and stretching rate increase. However, these rates increase with a higher velocity slip. In addition, the density of microorganisms in the bio‐nanofluid decreases when there are higher levels of unsteadiness, rotation, and Darcy numbers, as well as with increased bioconvection parameters.

A monotone finite volume scheme for single phase flow with reactive transport in anisotropic porous media

This paper deals with development and analysis of a finite volume scheme for a nonlinear parabolic equation modeling a single phase flow with reactive solute transport, including equilibrium sorption, in heterogeneous porous media. We present a second-order, finite volume scheme by combining a nonlinear two point flux approximation for the diffusion term, and an upwind method for the advective term on unstructured meshes. It is shown that, under a CFL condition, this scheme ensures the non-negativity of the discrete solution, taking into account the anisotropy and heterogeneity of the domain. The existence of a solution for the discrete nonlinear system is studied, and the convergence of a fixed point method is proved. Our theoretical results are illustrated by numerical simulations in a bidimensional domain, in particular we show the second-order convergence rate for the numerical solution and confirm the monotonicity of the scheme.

Thermogravitational Convection in a Controlled Rotating Darcy-Brinkman Nanofluids Layer Saturated in an Anisotropic Porous Medium Subjected to Internal Heat Source

Thermogravitational convection in a controlled rotating Darcy-Brinkman nanofluids layer saturated in an anisotropic porous medium heated from below is Thermogravitational convection in a controlled rotating Darcy-Brinkman nanofluids layer saturated in an anisotropic porous medium heated from below is investigated. The presence of a uniformly distributed internal heat source and considers the Brinkman model for different boundary conditions: rigid-rigid, free-free, and lower-rigid and upper-free are considered. The effect of a control strategy involving sensors located at the top plate and actuators positioned at the bottom plate of the nanofluids layer is analysed. Linear stability analysis based on normal mode technique is employed. The resulting eigenvalue problem is solved numerically using the Galerkin method implemented with Maple software. The model used for the nanofluids associates with the mechanisms of Brownian motion and thermophoresis. The influences of the internal heat source strength, mechanical anisotropy parameter, modified diffusivity ratio, nanoparticles concentration Darcy-Rayleigh number and nanofluids Lewis number are found to advance the onset of convection. Conversely, the Darcy number, thermal anisotropy parameter, porosity, rotation, and controller effects are observed to slow down the process of convective instability.investigated. The presence of a uniformly distributed internal heat source and considers the Brinkman model for different boundary conditions: rigid-rigid, free-free, and lower-rigid and upper-free are considered. The effect of a control strategy involving sensors located at the top plate and actuators positioned at the bottom plate of the nanofluids layer is analysed. Linear stability analysis based on normal mode technique is employed. The resulting eigenvalue problem is solved numerically using the Galerkin method implemented with Maple software. The model used for the nanofluids associates with the mechanisms of Brownian motion and thermophoresis. The influences of the internal heat source strength, mechanical anisotropy parameter, modified diffusivity ratio, nanoparticles concentration Darcy-Rayleigh number and nanofluids Lewis number are found to advance the onset of convection. Conversely, the Darcy number, thermal anisotropy parameter, porosity, rotation, and controller effects are observed to slow down the process of convective instability.

Physics of generalized couette flow of immiscible fluids in anisotropic porous medium

This study presents an investigation on the generalized Couette flow of two immiscible Newtonian fluids in an anisotropic porous medium. The flow occurs between two parallel plates, driven by the combined effects of an axial pressure gradient and the movement of the upper plate. The region between the plates is filled with an anisotropic porous medium characterized by directional permeability. An exact solution is derived for the proposed problem by considering appropriate boundary and interface conditions. The slip condition is employed at the lower plate of the channel which has a significant role in the flow mechanics of fluids flowing through porous medium. The effects of directional permeability on the flow of immiscible fluids are explored, and specific cases are presented. This study reveals a significant influence of directional permeability on the flow velocity of both fluids. Furthermore, quantitative conclusions are drawn regarding the shear stress at the lower and upper wall of the channel, with implications for the anisotropic nature of blood arteries. The findings contribute to a deeper understanding of fluid flow in anisotropic porous media and have potential applications in various engineering domains.

Generalized finite difference method-based numerical modeling of oil–water two-phase flow in anisotropic porous media

This paper applies generalized finite difference method (GFDM) to a compressible two-phase flow in anisotropic porous media with the aim of further extending the wider application of this class of meshless methods. We develop an implicit Euler scheme in time and a GFDM discretization in space based on two treatments of the anisotropic permeability tensor in continuous function expression and discrete distribution. The effectiveness and generality of GFDM for two-phase flow problems in anisotropic porous media are verified by three examples with rectangular, irregular, and complex boundaries. Also, the computational performance of the method is verified according to the error calculation with L2 absolute error functions in different node collocation schemes. In addition, the sensitivity analysis of the radius of the influence domain to the transient pressure equation (parabolic equation) and the saturation equation (hyperbolic equation) is considered. It generally holds that the larger the radius of the influence domain, the lower the calculation accuracy in the case of Cartesian collocation. This may be a preliminary rule for the radius choice of the influence domain for GFDM. In sum, this work provides an efficient and accurate meshless solver to handle two-phase flow problems in anisotropic porous media under the GFDM framework, which reveals the great application potential of GFDM in reservoir numerical simulation.

Global stability of Bénard–Marangoni convection in an anisotropic porous medium

Surface tension is essential in many industrial applications, especially where the liquid surface is in contact with the environment, such as crystal growth, semiconductor manufacturing, and welding. The present article reports a numerical analysis of convection induced by the combined effects of buoyancy force and surface tension in an infinitely extended horizontal fluid-saturated anisotropic porous layer with high permeability. We assume that the bottom boundary is rigid and the top is exposed to the air. Biot numbers establish general thermal conditions at both ends instead of commonly used adiabatic and isothermal boundary conditions. The anisotropy of the porous structure results in thermal and mechanical anisotropy parameters while examining layer's stability. The Chebyshev Tau technique yields the critical Marangoni number, MaLc and MaEc, representing linear and energy stability boundaries. We compare the constraints obtained from linear and energy analyses and conclude that the energy bounds for the current problem are less than linear bounds, indicating subcritical instabilities may exist. It is also observed that thermal anisotropy and Biot numbers stabilize the system. In contrast, mechanical anisotropy and the Darcy number advance the onset of convection. The existing results of limiting cases of the present problem are recovered with remarkable accuracy.

Three-dimensional unsteady state solution of multilayered saturated anisotropic finite media subjected to point source

The three-dimensional (3D) unsteady state solution of a saturated anisotropic finite porous medium due to a point source is presented in this paper. First, Laplace transform is introduced to address the governing equations of anisotropic poroelasticity. Next, the eigen equation method and the modified pseudo-Stroh formalism are used to obtain the transient solution for homogeneous saturated finite media. Then, the propagation matrix method is used to address the corresponding multilayered porous elastic media, and the 3D unsteady state solution of the layered porous medium subjected to a point source is obtained. The solution for a finite porous foundation is presented to demonstrate the use of the closed form solution. The numerical results of a homogeneous foundation due to an internal time-dependent point source are given. Two different point sources are used to show the transient responses of the proposed method. By comparing with those obtained by the finite element method, the accuracy of the proposed method has been proved. Meanwhile, the influence of the depth of the point source on the hydromechanical responses of the homogeneous foundation is discussed. Finally, the transient response of an anisotropic sandwich foundation due to a point source at different depths is investigated. These numerical examples prove the feasibility of the transient 3D solution. The closed form solution presented in this paper can provide help for 3D transient fluid-deforming coupling analysis of laminated porous media.

Onset of triply diffusive thermo-bio-convection in the presence of gyrotactic microorganisms and internal heating into an anisotropic porous medium: Oscillatory convection

This paper accentuates different aspects of an important and novel triply-diffusive thermo-bio-convection instability problem in the presence of inconsistent internal heating and the gyrotactic microorganism species into an anisotropic porous fluid layer. This instability study may be important from the application point of view in microfluidic devices, enhanced oil recovery, antibiotics, and many other areas. The normal mode analysis and linear stability analysis along with the weighted Galerkin method are used to establish the instability criteria in terms of the thermal Rayleigh number for both the oscillatory and non-oscillatory modes of convection as a secular equation among prominent parameters used in this study. It is perceived that the presence of internal heating and gyrotactic microorganism species destabilize the suspension and fast forward the triply-diffusive thermo-bio-convection. It is also established that the increment of mechanical anisotropy hastens the convection and advances the heat transfer in the porous medium while mounting thermal anisotropy and the concentration of solutes constitute a stable system that postpones the onset of triply-diffusive thermo-bio-convection.

Magnetohydrodynamic forced convection and heat transfer of a Casson fluid flow in an anisotropic porous channel with Isoflux boundaries

Theoretical analysis of magnetohydrodynamic fully developed forced convective flow and heat transfer characteristics of a non-Newtonian electrically conducting Casson fluid through a porous channel filled with anisotropic porous material bounded by two impermeable horizontal walls subject to constant heat flux applied to the outer walls is investigated. The extended Darcy-Brinkmann model govern the flow inside the porous channel and accounts for the presence of the inertia term, which allows for the no-slip boundary conditions at the walls. The principal axis of anisotropic permeability is oriented from to radians. The equations governing the system are solved using the inbuilt “DSolve” in Mathematica 11.1 for the velocity and temperature profiles as well as the Nusselt number (Nu), which characterizes the rate of heat transfer in the system. The effects of various parameters on the velocity, temperature and heat transfer profiles are displayed through graphs and discussed quantitatively. From the results we observed that the velocity profile reduced as the Casson parameter (ß), anisotropic ratio (Kr), orientation angle (), and Hartmann number (Ha) increased, whereas the velocity profile increased when the Darcy number (Da) and apparent viscosity () increased. Similarly, these variables also had similar effect on the temperature profile of the fluid. On the other hand, the heat transfer profile, as measured by Nusselt number, increased when the Casson parameter (ß), anisotropic ratio (Kr) and Hartmann number (Ha) increased, while increasing the orientation angle () and apparent viscosity ( ) decreased the Heat Transfer Profile. Upon further investigation, the heat transfer profile was shown to be maximum along the path of least permeability in the anisotropic porous media under investigation.

A numerical investigation of wave-induced fluid flows in anisotropic fractured porous media

A numerical investigation of wave-induced fluid flows in anisotropic fractured porous media

Three-Temperature Boundary Element Modeling of Ultrasound Wave Propagation in Anisotropic Viscoelastic Porous Media

The main goal of this work is to develop a novel boundary element method (BEM) model for analyzing ultrasonic wave propagation in three-temperature anisotropic viscoelastic porous media. Due to the problems of the strong nonlinearity of ultrasonic wave propagation in three-temperature porous media, the analytical or numerical solutions to the problems under consideration are always challenging, necessitating the development of new computational techniques. As a result, we use a new BEM model to solve such problems. A time-stepping procedure based on the linear multistep method is obtained after solving the discretized boundary integral equation with the quadrature rule. The calculation of a double integral is required to obtain fundamental solutions, but this increases the total BEM computation time. Our proposed BEM technique is used to solve the current problem and improve the formulation efficiency. The numerical results are graphed to demonstrate the effects of viscosity and anisotropy on the nonlinear ultrasonic stress waves in three-temperature porous media. The validity, accuracy, and efficiency of the proposed methodology are demonstrated by comparing the obtained results to a corresponding solution obtained from the finite difference method (FDM).

Anisotropic fast diffusion equations

We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The combination of self-similarity and anisotropy is essential in our analysis and creates serious mathematical difficulties that are addressed by means of novel methods.