Horizontal Porous Layer Research Articles

Gravity modulation effect on ferromagnetic convection in a Darcy-Brinkman layer of porous medium

The influence of a timedependent body force on the threshold of convective instability in a magnetic fluid filled horizontal porous layer is investigated. The gravity modulation effect is treated by employing a perturbation method. The correction Rayleigh number is computedas a function of the modulation frequency, porous and magnetic parameters. It is expounded that, for small and reasonable values of the modulation frequency, gravity modulation and magnetic mechanism have opposing influence on the stability. The study further explicates that, when the gravity modulation frequency increases beyond all bounds, manifestation of the disappearance of the magnetic and porous medium effects on the stability is highly likely.

Double-diffusive Soret convection phenomenon in porous media: effect of Vadasz inertia term

The onset of double-diffusive convection in horizontal porous layers for the thermo-diffusive Soret phenomenon in the case of a generalized Darcy model including inertia term is investigated. Via a linearization principle recently introduced in Rionero (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur 28:21–47, 2017) the coincidence between linear and nonlinear (global) stability thresholds of the thermo-solutal conduction solution, is proved. Necessary and sufficient conditions guaranteeing the onset of steady or oscillatory convection in a closed algebraic form are obtained.

Stability of homogeneous pulsating vertical flow of mixture through horizontal porous layer with solute immobilization

The paper is devoted to the investigation of the linear stability of homogeneous pulsating vertical seepage of mixture through horizontal porous layer with solute immobilization. The mixture flow through porous medium is studied within the standard Darcy-Boussinesq approach taking into account the solutal convection. The immobilization of solute particles by solid matrix of medium is described by the mobile/immobile media (MIM) model with second-order kinetics. It was shown earlier that the immobilization leads to oscillatory mode of convective instability due to transition of solute from the mobile phase to immobile one. This means that immobilization effect on the flow pulsations should be non-trivial. The equations for the convection with immobilization are derived, the regime of homogeneous filtration is obtained and its stability is investigated. The neutral curves in the parameter space are plotted. The effect of homogeneous filtration stabilization under intensification of external seepage and dynamics interphase exchange is studied. It is found that the effect of pulsations is significant in the case when frequency of pulsations is close to the natural frequency of instability. The parametric excitation of convection is not observed.

Instability of the Benard–Marangoni Convection in a Porous Layer Affected by a Non-Vertical Magnetic Field

The onset of the Benard–Marangoni convection in a horizontal porous layer permeated by a magnetohydrodynamic fluid with a nonlinear magnetic permeability is examined. The porous layer is assumed to be governed by the Brinkman model; it is bounded by a rigid surface from below and by a non-deformable free surface from above and subjected to a non-vertical magnetic field. The critical effective Marangoni number and the critical Rayleigh number are obtained for different values of the effective Darcy number, Biot number, Chandrasekhar number, nonlinear magnetic parameter, and angle from the vertical axis for the cases of stationary convection and overstability. The related eigenvalue problem is solved by using the first-order Chebyshev polynomial method.

Effect of vertically varying permeability on the onset of convection in a porous medium

Considering the vertically varying permeability of a porous medium, we conducted theoretical and numerical analyses on the onset of buoyancy-driven instability in an initially quiescent, fluid-saturated, horizontal porous layer. Darcy’s law was employed to explain the fluid flow through a porous medium and linear and nonlinear analyses are conducted. In the semi-infinite domain, the growth of disturbance and the onset of convection were analyzed with and without the quasi-steady state approximation. The present analysis of initial growth rate shows that the system is initially unconditionally stable regardless of a vertical heterogeneity parameter. The onset conditions of buoyancy-driven instabilities were investigated as a function of the Darcy-Rayleigh number and the heterogeneity parameter. To find the effect of a vertical heterogeneity on the flow after the onset of convection, nonlinear numerical simulations also were conducted using the result of the linear analysis as a starting point. Nonlinear numerical simulations show that the finger-like instability motion is not readily observable at a critical time and it becomes visible approximately when a mass transfer rate substantially increases.

Influence of Vertical Vibrations on the Stability of a Binary Mixture in a Horizontal Porous Layer Subjected to a Vertical Heat Flux

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid mixture and subjected to vertical high-frequency and small-amplitude vibrations. Two configurations have been considered and compared: an infinite horizontal layer and a bounded domain with a large aspect ratio. In both cases, the initial temperature gradient is produced by a constant uniform heat flux applied on the horizontal boundaries. A formulation using time-averaged equations is used. The linear stability of the equilibrium solution is carried out for various Soret separation ratios $$\varphi $$ , vibrational Rayleigh numbers Rv, Lewis numbers Le and normalized porosity. For an infinite horizontal layer, the critical Rayleigh number $$\mathrm{Ra}_c$$ is determined analytically. For a steady bifurcation to a one-cell solution (the critical wavenumber is zero), we obtain $$\mathrm{Ra}_{c}={12}/{({\varphi }(\mathrm{Le}+1)+1)}$$ for all Rv. When the bifurcation is a Hopf bifurcation or when the critical wavenumber is not zero, we use a Galerkin method to compute the critical values. Our study is completed by a nonlinear analysis of the bifurcation to one-cell solutions in an infinite horizontal layer that is compared to numerical simulations in bounded horizontal domains with large aspect ratio.

Linear Stability of the Double-Diffusive Convection in a Horizontal Porous Layer with Open Top: Soret and Viscous Dissipation Effects

The linear stability of the double-diffusive convection in a horizontal porous layer is studied considering the upper boundary to be open. A horizontal temperature gradient is applied along the upper boundary. It is assumed that the viscous dissipation and Soret effect are significant in the medium. The governing parameters are horizontal Rayleigh number ( $$Ra_\mathrm{H}$$ ), solutal Rayleigh number ( $$Ra_\mathrm{S}$$ ), Lewis number (Le), Gebhart number (Ge) and Soret parameter (Sr). The Rayleigh number (Ra) corresponding to the applied heat flux at the bottom boundary is considered as the eigenvalue. The influence of the solutal gradient caused due to the thermal diffusion on the double-diffusive instability is investigated by varying the Soret parameter. A horizontal basic flow is induced by the applied horizontal temperature gradient. The stability of this basic flow is analyzed by calculating the critical Rayleigh number ( $$Ra_\mathrm{cr}$$ ) using the Runge–Kutta scheme accompanied by the Shooting method. The longitudinal rolls are more unstable except for some special cases. The Soret parameter has a significant effect on the stability of the flow when the upper boundary is at constant pressure. The critical Rayleigh number is decreasing in the presence of viscous dissipation except for some positive values of the Soret parameter. How a change in Soret parameter is attributing to the convective rolls is presented.

Inclined porous medium convection at large Rayleigh number

High-Rayleigh-number ($Ra$) convection in an inclined two-dimensional porous layer is investigated using direct numerical simulations (DNS) and stability and variational upper-bound analyses. When the inclination angle $\unicode[STIX]{x1D719}$ of the layer satisfies $0^{\circ }<\unicode[STIX]{x1D719}\lesssim 25^{\circ }$, DNS confirm that the flow exhibits a three-region wall-normal asymptotic structure in accord with the strictly horizontal ($\unicode[STIX]{x1D719}=0^{\circ }$) case, except that as $\unicode[STIX]{x1D719}$ is increased the time-mean spacing between neighbouring interior plumes also increases substantially. Both DNS and upper-bound analysis indicate that the heat transport enhancement factor (i.e. the Nusselt number) $Nu\sim CRa$ with a $\unicode[STIX]{x1D719}$-dependent prefactor $C$. When $\unicode[STIX]{x1D719}>\unicode[STIX]{x1D719}_{t}$, however, where $30^{\circ }<\unicode[STIX]{x1D719}_{t}<32^{\circ }$ independently of $Ra$, the columnar flow structure is completely broken down: the flow transitions to a large-scale travelling-wave convective roll state, and the heat transport is significantly reduced. To better understand the physics of inclined porous medium convection at large $Ra$ and modest inclination angles, a spatial Floquet analysis is performed, yielding predictions of the linear stability of numerically computed, fully nonlinear steady convective states. The results show that there exist two types of instability when $\unicode[STIX]{x1D719}\neq 0^{\circ }$: a bulk-mode instability and a wall-mode instability, consistent with previous findings for $\unicode[STIX]{x1D719}=0^{\circ }$ (Wen et al., J. Fluid Mech., vol. 772, 2015, pp. 197–224). The background flow induced by the inclination of the layer intensifies the bulk-mode instability during its subsequent nonlinear evolution, thereby favouring increased spacing between the interior plumes relative to that observed in convection in a horizontal porous layer.

Dynamics and Stability of Air Bubbles in a Porous Medium

A numerical method is developed for reliably computing the evolution of the boundary of a multiply connected water-saturated domain with air bubbles in the case when the pressure inside them depends on the bubble volume. It is assumed that the distance between the gas bubbles is comparable with their size. Gas bubbles can be near an extended phase transition boundary separating a porous medium flow and a domain saturated with a mixture of air and water vapor. The numerical method is verified by comparing the numerical solution of a test problem with its analytical solution. Caused by finite-amplitude perturbations of the phase interface, the deformation of an air bubble in an extended horizontal water-saturated porous layer with a constant pressure gradient is studied. It is shown that the instability of the bubble boundary with respect to finite perturbations leads to the splitting of the bubble. An analysis of the numerical solution shows that, although all circular bubbles move at the same velocity irrespective of their size, nevertheless, due to instability, a portion of the bubble boundary where the air displaces the fluid moves faster than an opposite portion where the fluid displaces the air. As a result, nearby bubbles are capable of merging before splitting.

Effect of Constant Magnetic Field on Convective Heat Transfer through Anisotropic River Beds

An analytical investigation is conducted to study the effect of magnetic field on convection heat transfer through packed porous beds which consists of a horizontal fluid layer (river bed) and a porous zone with anisotropic permeability and underlined by a surface heated by a constant temperature T1. The free surface of the fluid layer overlying the horizontal porous layer receives solar rays to length of day and is then considered heated isothermally at temperature T2 such as T1 T2. Flow in porous medium is assumed to be governed by the generalized Brinkman-extended Darcy law and in the fluid layer by the Navier-Stokes model. The Beavers-Joseph condition is applied at the interface between the two layers. The influence of Hartmann number and hydrodynamic anisotropy on the convective phenomenon is investigated analytically. It is found that the magnetic field, the anisotropic permeability and the thickness of the porous lining, ε, have a strong influence of the geothermal convective flow and the heat transfer rate.

Onset of convection in a non-Newtonian viscous flow through a horizontal porous channel

An impermeable horizontal fluid saturated porous layer is studied with respect to the onset of thermal convection. The saturating fluid is a Ostwald-de Waele type of non-Newtonian fluid. The viscous dissipation is the only heat source present inside the layer. A basic throughflow whose direction is arbitrary in the horizontal plane occurs. The basic flow produces, by means of viscous dissipation, a basic non-uniform vertical temperature gradient. The l​inear stability analysis of the basic state is carried out. The critical values are obtained as functions of the magnitude and of the inclination of the basic throughflow and as functions of the power-law index. The identification of the most unstable mode is performed: while for pseudoplastic fluids the most unstable modes are transverse rolls, for dilatant fluids the most unstable modes can be either longitudinal or transverse rolls.

Linear and Nonlinear Stability of Double-Diffusive Convection with the Soret Effect

The onset of double-diffusive convection in a horizontal fluid-saturated porous layer is examined by taking the Soret effect into consideration. The linear and nonlinear stability analyses are derived, and the corresponding eigenvalue problems are solved. The nonlinear stability analysis is achieved by using the energy method. In both the cases of linear and nonlinear stability theories, the onset criterion for all possible modes is derived analytically. For numerical computations of the eigenvalue problem, the Chebyshev tau method is employed. It is observed that the effect of stabilization or destabilization caused by the Soret parameter is significant for the Soret parameters which are less than \(Sr = 2\). In the absence of the Soret effect, the linear and nonlinear stability thresholds coincide.

Effect of solute immobilization on the stability problem within the fractional model in the solute analog of the Horton-Rogers-Lapwood problem.

The paper is devoted to the linear stability analysis within the solute analogue of the Horton-Rogers-Lapwood (HRL) problem. The solid nanoparticles are treated as solute within the continuous approach. Therefore, we consider the infinite horizontal porous layer saturated with a mixture (carrier fluid and solute). Solute transport in porous media is very often complicated by solute immobilization on a solid matrix of porous media. Solute immobilization (solute sorption) is taken into account within the fractal model of the MIM approach. According to this model a solute in porous media immobilizes within random time intervals and the distribution of such random variable does not have a finite mean value, which has a good agreement with some experiments. The solute concentration difference between the layer boundaries is assumed as constant. We consider two cases of horizontal external filtration flux: constant and time-modulated. For the constant flux the system of equations that determines the frequency of neutral oscillations and the critical value of the Rayleigh-Darcy number is derived. Neutral curves of the critical parameters on the governing parameters are plotted. Stability maps are obtained numerically in a wide range of parameters of the system. We have found that taking immobilization into account leads to an increase in the critical value of the Rayleigh-Darcy number with an increase in the intensity of the external filtration flux. The case of weak time-dependent external flux is investigated analytically. We have shown that the modulated external flux leads to an increase in the critical value of the Rayleigh-Darcy number and a decrease in the critical wave number. For moderate time-dependent filtration flux the differential equation with Caputo fractional derivatives has been obtained for the description of the behavior near the convection instability threshold. This equation is analyzed numerically by the Floquet method; the parametric excitation of convection is observed.

Numerical Investigations of Haline-Convective Flows of Saline Groundwater

Numerical investigations of saline water flows in a horizontal porous layer resulted from water evaporation at the upper boundary and salt accumulation nearby are carried out. The mathematical model includes the continuity, Darcy’s and contaminant transport equations. Over the threshold of saline profile stability, different regimes of haline convection in low- and high-permeable layers are simulated. As found, steady regular convection occupying a whole domain is induced in the case of weak evaporation. Stochastic convection with a small-scale structure of salt “drops” and salt precipitation are developed in the case of intensive evaporation. The maps of regimes are plotted and boundaries between regimes are determined.

Effect of uniform inclined magnetic field on natural convection and entropy generation in an open cavity having a horizontal porous layer saturated with a ferrofluid

ABSTRACTNumerical analysis of natural convection combined with entropy generation in a square open cavity partially filled with a porous medium has been performed for a ferrofluid under the effect of inclined uniform magnetic field. Governing equations with corresponding boundary conditions formulated in dimensionless stream function and vorticity using Brinkman–extended Darcy model for porous layer have been solved numerically using finite difference method. An influence of key parameters on ferrofluid flow and heat transfer has been analyzed. It has been found that an inclusion of spherical ferric oxide nanoparticles can lead to a diminution of entropy generation in the case of similar flow and heat transfer structures.

Dispersion-driven instability of mixed convective flow in porous media

This paper investigates the impact of hydrodynamic dispersion on the stability of free convection in a saturated horizontal porous layer subject to a transient vertical concentration gradient and a steady horizontal background flow. A linear stability analysis (LSA) was conducted using the quasi-steady-state approximation to obtain neutral stability curves, critical times, and the corresponding wavenumbers as a function of dispersivity ratio (α) and longitudinal dispersion strength (β). The LSA results showed that the dispersive boundary layer becomes less unstable as longitudinal and transverse dispersivity increase. In addition, for the isotropic dispersive system with α = 1, the critical time and its corresponding wavenumber follow τc = 167.6/(1 − β) and κc = 0.0696 (1 − β), respectively. The nonlinear dynamics of the system were studied by examining the interaction of convective fingers, dissolution flux, and the time-dependent Sherwood number. Finally, the results were applied to 24 deep saline aquifers in the Alberta Basin.

Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux

We analyze the thermal convection thresholds and linear characteristics of the primary and secondary instabilities for viscoelastic fluids saturating a porous horizontal layer heated from below by a constant flux. The Galerkin method is used to solve the eigenvalue problem by taking into account the elasticity of the fluid, the ratio between the viscosity of the solvent and the total viscosity of the fluid and the lateral confinement of the medium. For the primary instability, we found out that depending on the rheological parameters, two types of convective structures may appear when the basic conductive solution loses its stability: stationary long wavelength instability as for Newtonian fluids and oscillatory convection. The effect of the lateral confinement of the porous medium by adiabatic walls is to stabilize the oblique and longitudinal rolls and therefore selects transverse rolls at the onset of convection. In the range of the rheological parameters where stationary long wave instability develops first, we use a parallel flow approximation to determine analytically the velocity and temperature fields associated with the monocellular convective flow. The linear stability analysis of the monocellular flow is performed, and the critical conditions above which the flow becomes unstable are determined. The combined influence of the viscoelastic parameters and the lateral confinement on the characteristics of the secondary instability is quantified. The major new findings concerning the secondary instabilities may be summarized as follows: (i) For concentrated viscoelastic fluids, computations showed that the most amplified mode of convection corresponds to oscillatory transverse rolls, which appears via a Hopf bifurcation. This pattern selection is independent of both the fluid elasticity and the lateral confinement of the porous medium. (ii) For diluted viscoelastic fluids, the preferred mode of convection is found to be oscillatory transverse rolls for a very laterally-confined medium. Otherwise, stationary or oscillatory longitudinal rolls may develop depending on the fluid elasticity. Results also showed the destabilizing effect of the relaxation fluid elasticity and the stabilizing effect of the viscosity ratio for the onset of both primary and secondary instabilities.

Effect of uniform inclined magnetic field on mixed convection in a lid-driven cavity having a horizontal porous layer saturated with a ferrofluid

MHD mixed convection in a lid-driven cavity with partially filled with a porous medium saturated with a ferrofluid has been analyzed numerically. The domain of interest consists of a bottom porous layer and a nanofluid layer over the porous one with a heated motionless bottom wall and cooled upper moved wall. The governing partial differential equations formulated on the basis of a single-phase model for nanofluid, Brinkman-extended Darcy model for porous layer and Boussinesq approximation for buoyancy force have been solved by finite difference method of the second-order accuracy. Analysis has been carried out for a wide range of Hartmann number, magnetic field inclination angle, Darcy number, porous layer height, and nanoparticles volume fraction. It has been revealed that average Nusselt number is a non-monotonic function of Darcy number and porous layer height, while a growth of Hartmann number illustrates the heat transfer rate reduction.

Thermal Instability of a Power-Law Fluid Flowing in a Horizontal Porous Layer with an Open Boundary: A Two-Dimensional Analysis

A two-dimensional analysis of the onset of thermal convective instability in a horizontal porous layer with open upper boundary is carried out. The saturating fluid is non-Newtonian of power-law behaviour, and its flow is represented through a suitable extension of Darcy’s law. A model of temperature-dependent viscosity is employed where the consistency index is considered as variable, while the power-law index is assumed to be constant. Numerical data for the neutral stability and for the critical values of a modified Darcy–Rayleigh number have been obtained. The feasibility of an experimental validation of the theoretical results predicted by the stability analysis is discussed in detail. An experimental set-up based on a Hele-Shaw cell is described, and preliminary results relative to the onset of convective cells are described. Observed hysteretic effects and deviations from the rheological model are identified as potential sources of uncertainty.

Linear Stability Analysis of Penetrative Convection via Internal Heating in a Ferrofluid Saturated Porous Layer

Penetrative convection due to purely internal heating in a horizontal ferrofluid-saturated porous layer is examined by performing linear stability analysis. Four different types of heat supply functions are considered. The Darcy model is used to incorporate the effect of the porous medium. Numerical solutions are obtained by using the Chebyshev pseudospectral method, and the results are discussed for all three boundary conditions: when both boundaries are impermeable and conducting; when both boundaries are conducting with lower boundary impermeable and free upper boundary; and when both boundaries are impermeable with lower boundary conducting and upper with constant heat flux. The effect of the Langevin parameter, width of ferrofluid layer, permeability parameter, and nonlinearity of the fluid magnetization has been observed at the onset of penetrative convection for water- and ester-based ferrofluids. It is seen that the Langevin parameter, width of ferrofluid layer, and permeability parameter have stabilizing effects on the onset of convection, while the nonlinearity of the fluid magnetization advances the onset of convection.