Horizontal Porous Layer Research Articles

Laterally Penetrative Onset of Convection in a Horizontal Porous Layer

The onset of Darcy–Bénard convection in an unlimited horizontal porous layer is studied theoretically. The thermomechanical boundary conditions of Dirichlet or Neumann type at the lower and upper plane are switched from one type to another, at certain values of the horizontal x-coordinate. A semi-infinite portion of the lower boundary is defined as thermally conducting and impermeable, while the remaining boundary is open and with given heat flux. At the upper boundary, the same thermomechanical conditions are applied, but with a relative spatial displacement L and in the opposite spatial order. A domain of local destabilization around the origin is generated between the lines of discontinuity x = pm ,L/2. The marginal state of convection is triggered centrally, while it is penetrative in the domains exterior to the central domain. The onset problem is solved numerically, with a general 3D mode of disturbance, but 2D disturbances are preferred in most cases. The critical Rayleigh number is given as a function of the dimensionless gap width L and the wavenumber k in the y direction along the lines of discontinuity in the boundary conditions. An asymptotic formula for 2D penetrative eigenfunctions is shown to be in agreement with the numerical results.

Soret phenomenon in porous Magneto-Hydrodynamics

The onset of double-diffusive convection in horizontal porous layers filled by an electrically conducting fluid embedded in a magnetic field, is investigated on taking into account of Soret phenomenon. Sufficient conditions guaranteeing the onset of steady or oscillatory convection in a closed algebraic form are obtained. Conditions guaranteeing that Hopf convection in the absence of Soret phenomenon, turns into steady convection by the Soret parameter, have been found.

Stationary convection in a binary mixture of ferrofluids in a porous medium

The Brinkman model is used to study the criterion of appearance of stationary convection in a porous horizontal layer saturated by a binary mixture of ferrofluids and heated from below. The analytical expression of the Rayleigh number is determined by dimensionless numbers for the combined effects of buoyancy and magnetic force on the one hand and magnetic force alone on the other. The effect of Darcy’s number, ratio of viscosities, each of the magnetic parameters and the binary parameter is illustrated on the appearance and size of the convection cells. The increasing of the number of Darcy Da, each of the magnetic parameters M1, M3, ψm and the separation ratio ψ accelerates the appearance of stationary convection. On the other hand, when the ratio of the viscosity ∧ or the magnetic parameter M2 is increased, the beginning of the convection is delayed and the system is stable. We have also noticed that the size of the convection structures is contracted with each of the parameters M1, M2, ψ, ψm and for low values of the parameter M3. But by increasing the number of Darcy, the ratio of viscosities or for very high values of M3 the size of the convection cells is enlarged. Finally, it should be noted that convection sets in with great delay when the magnetic mechanisms alone are highlighted.

Analytical and Numerical Study of Soret and Dufour Effects on Thermosolutal Convection in a Horizontal Brinkman Porous Layer with a Stress-Free Upper Boundary

In this paper, thermo-diffusion (Soret effect) and diffusion-thermo (Dufour effect) effects on double-diffusive natural convection induced in a horizontal Brinkman porous layer with a stress-free upper boundary are investigated. The cavity is filled with a binary fluid and subjected to uniform fluxes of heat and mass on its long sides. An analytical solution based on the parallel flow approximation is developed for the problem considered in order to allow prompt determination of the thresholds of stationary and finite amplitude solutions and also heat and mass transfer characteristics. The analytical solution is validated numerically by using a finite difference method. The combined effects of the Soret and Dufour parameters, the thermal Rayleigh number, the buoyancy ratio, and the Darcy number on the flow intensity and heat and mass transfer are illustrated graphically, and some particular behaviors observed are discussed. The analytical solution proves the existence of different regions in the buoyancy ratio-Dufour parameter plane, corresponding to different parallel flow behaviors. The number, the location, and the extent of these regions, which are impossible to predict numerically, depend strongly on Soret and Dufour parameters. The effect of thermo-diffusion and diffusion-thermo on flow intensity and heat and mass transfer is found to be important.

Buoyancy-driven convection in a horizontal porous layer saturated by a power-law fluid: The effect of an open boundary

An infinitely wide horizontal porous layer saturated by a power-law fluid is heated from below by an imposed heat flux. The basic stationary state is characterised by a horizontal throughflow and a constant temperature gradient along the vertical direction. A linear stability analysis is performed by employing the method of normal modes. The eigenvalue problem obtained is solved by employing two different approaches: a hybrid analytical/numerical technique and a fully numerical technique. The critical values defining the threshold configurations for the onset of thermal convection are presented. These threshold values display a non monotonic dependence on the basic flow rate while they are monotonically increasing functions of the power-law index.

Effect of Hydraulic Resistivity on a Weakly Nonlinear Thermal Flow in a Porous Layer

Heat and mass transfer through porous media has been a topic of research interest because of its importance in various applications. The flow system in porous media is modelled by a set of partial differential equations. The momentum equation which is derived from Darcy’s law contains a resistivity parameter. We investigate the effect of hydraulic resistivity on a weakly nonlinear thermal flow in a horizontal porous layer. The present study is a realistic study of nonlinear convection flow with variable resistivity whose rate of variation is arbitrary in general. This is a first step for considering more general problems in applications that involve variable resistivity that may include both variations in permeability and viscosity of the porous layer. Such problems are important for understanding properties of underground flow, migration of moisture in fibrous insulations, underground disposal of nuclear waste, welding process, petrochemical generation, drug delivery in vascular tumor, etc. Using weakly non-linear procedure, the linear and first-order systems are derived. The critical Rayleigh number and the critical wave number are obtained from the linear system using the normal mode approach for the two-dimensional case. The linear and first-order systems are solved numerically using the fourth-order Runge-Kutta and shooting methods. Numerical results for the temperature are presented in tabular and graphical forms for different resistivities. Through this study, it is observed that a stabilizing effect on the dependent variables occurs in the case of a positive vertical rate of change in resistivity, whereas a destabilizing effect is noticed in the case of a negative vertical rate of change in resistivity. The results obtained indicate that the convective flow due to the buoyancy force is more effective for weaker resistivity.

КОНВЕКЦИЯ ВОЗДУХА В СНЕЖНОМ ПОКРОВЕ МОРСКОГО ЛЬДА

For the first time, data on stability of stationary convective filtration within infinite horizontal layer of snow covering the flat surface of floating ice is presented in this article. An analytical solution of the linearized problem was obtained with the use of the Galerkin method, and the parametric analysis of the problem was performed. It was found that the stability criteria (Rayleigh filtration numbers) obtained with consideration for the heat exchange of snow cover with the atmosphere did not exceed the known value of 4π2 for a horizontal porous layer with impermeable isothermal boundaries. As expected, the interaction with the atmosphere has the most significant impact on the critical Rayleigh numbers, while influence of variations in snow density and ice thickness and the thickness of the underlying layer of ice are small. Based on data of ice and meteorological observations made in the winter of 2015/16 in the Western part of the Laptev Sea together with calculations of the fast ice evolution, the values and temporal variability of temperature gradients and the Rayleigh numbers in the snow cover were obtained using a thermodynamic model. It was found that both, the model and observed magnitudes, exceeded their critical values determined by solving the stability problem. The conclusion is made that the convective regime of the heat transfer does really exist in the snow cover, and thus its contribution to the thermal and mass balance of sea ice during winter period should be taken into account.

Instability of Vertical Throughflows in Porous Media under the Action of a Magnetic Field

The instability of a vertical fluid motion (throughflow) in a binary mixture saturating a horizontal porous layer, uniformly heated from below, uniformly salted from below by one salt and permeated by an imposed uniform magnetic field H , normal to the layer, is analyzed. By employing the order-1 Galerkin weighted residuals method, the critical Rayleigh numbers for the onset of steady or oscillatory instability, have been determined.

Thermosolutal convective instability of power-law fluid saturated porous layer with concentration based internal heat source and Soret effect

The onset of double-diffusive convective instability in a horizontal porous layer saturated by a power-law fluid that is subjected to concentration based internal heat source and Soret effect is investigated. Both these physical processes have a significant influence on the concentration field in porous media. Here the fluid flow is induced by constant but different temperature and concentration maintained across the boundary planes. The basic flow considered is a uniform throughflow which is inclined with respect to the horizontal x -axis. Considering the heat source ($ \gamma$) and Soret effect (Sr) results in a non-linear basic temperature and concentration profiles. The resulting eigenvalue system of coupled ordinary differential equations along with the boundary conditions is solved to explore the combined effect of internal heat source and Soret parameter on the instability of the power-law fluid. The instability of the base flow appears in the form of oblique rolls, which can be reduced to longitudinal rolls or transverse rolls. Stationary instability is seen for the longitudinal rolls. For transverse rolls, the onset of stationary and oscillatory convective instability depends on Lewis number, Peclet number, Soret parameter and presence of the concentration based internal heat source. It is also evident that $ \gamma$ = 0 and Sr = 0 are singularities for the basic temperature and concentration profiles. The limiting behavior of these parameters is also investigated for their linear stability. Both stationary and oscillatory convective instability corresponding to the pseudoplastic and dilatant fluids is discussed for the case of $ \gamma\rightarrow 0$.

Convection in a Horizontal Porous Layer with Vertical Pressure Gradient Saturated by a Power-Law Fluid

The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such a way that a vertical throughflow is described. The threshold for a buoyancy-driven cellular flow is investigated by means of a linear stability analysis. This study consists in introducing disturbances with small amplitude. The disturbances are plane waves, i.e. a normal modes stability analysis of the basic stationary solution is performed. The resulting problem is an ordinary differential equation eigenvalue problem which is solved numerically by coupling the Runge–Kutta method with the shooting method. Results are presented in the form of marginal stability curves and their critical points representing the values of the control parameters such that the growth rate of the disturbances is zero. It is found that, among roll disturbances, the most unstable modes are stationary and uniform with infinite wavelength. For this reason, an asymptotic analysis for vanishing wave numbers is carried out. The results of this asymptotic analysis are obtained analytically displaying a very good agreement with the numerical solution. It is found that vertical throughflow plays a destabilising role for pseudoplastic fluids and a stabilising role for dilatant fluids.

Robin Boundary Effects in the Darcy–Rayleigh Problem with Local Thermal Non-equilibrium Model

The contribution of Robin boundaries on the onset of convection in a horizontal saturated porous layer covered by a free surface on the top is investigated here. The saturated solid matrix is assumed in a regime where the temperature profile of the solid phase differs from a fluid one. Two energy equations are adopted as a consequence of the local thermal non-equilibrium model (LTNE), and four Biot numbers are arising out of the third kind of boundaries imposed on both surfaces. The dimensionless parameters H and $$\gamma $$ which rule the transition from local thermal equilibrium (LTE) to non-equilibrium one or vice versa are taken into account. The cases of equal and different Biot numbers have been considered beside the asymptotic limits of LTE and LTNE one. A linear stability analysis of the basic motionless state has been performed. The perturbation terms of the main steady flows are evaluated in the form of plane waves. The eigenvalue problem is solved either analytically or numerically depending on the temperature gradient of the fluid phase. The analytical solution is handled through a dispersion relation, while the numerical one is computed by the Runge–Kutta solver combined with the shooting method. The variation in Darcy–Rayleigh number and wave number is obtained with respect to Biot numbers for all resulting cases.

On the stability of carbon sequestration in an anisotropic horizontal porous layer with a first-order chemical reaction.

Carbon dioxide (CO2) sequestration in deep saline aquifers is considered to be one of the most promising solutions to reduce the amount of greenhouse gases in the atmosphere. As the concentration of dissolved CO2 increases in unsaturated brine, the density increases and the system may ultimately become unstable, and it may initiate convection. In this article, we study the stability of convection in an anisotropic horizontal porous layer, where the solute is assumed to decay via a first-order chemical reaction. We perform linear and nonlinear stability analyses based on the steady-state concentration field to assess neutral stability curves as a function of the anisotropy ratio, Damköhler number and Rayleigh number. We show that anisotropy in permeability and solutal diffusivity play an important role in convective instability. It is shown that when solutal horizontal diffusivity is larger than the vertical diffusivity, varying the ratio of vertical to horizontal permeabilities does not significantly affect the behaviour of instability. It is also noted that, when horizontal permeability is higher than the vertical permeability, varying the ratio of vertical to horizontal solutal diffusivity does have a substantial effect on the instability of the system when the reaction rate is dominated by the diffusion rate. We used the Chebyshev-tau method coupled with the QZ algorithm to solve the eigenvalue problem obtained from both the linear and nonlinear stability theories.

Stability of the Horizontal Throughflow in a Power-Law Fluid Saturated Porous Layer

Double-diffusive convective instability of horizontal throughflow in a power-law fluid saturated porous layer is investigated. The boundaries of this horizontal porous layer are impermeable, isothermal and isosolutal. The generalized Darcy’s equation for power-law fluid is employed in the momentum balance, along with Oberbeck–Boussinesq approximation for buoyancy effect in the system. It is considered that the viscous dissipation is non-negligible. The basic temperature and concentration profiles for the considered horizontal throughflow are determined analytically, and the linear stability analysis is performed on this basic solution. The eigenvalue problem is solved numerically using bvp4c in MATLAB. The conditions for the onset of instability of base flow in terms of critical Rayleigh number and the corresponding wave number are obtained. It is found that critical values are affected by the flow governing parameters such as the Peclet number (Pe), power-law index (n), viscous dissipation (Ge), angle of inclination $$(\alpha )$$ , buoyancy ratio (N) and Lewis number (Le). For the considered range of these parameter values, the instability of the basic flow is discussed for the convective rolls. It is observed that instability of any general convective roll lies between that of the longitudinal and transverse rolls. Also, the effect of viscous dissipation on the onset of instability for both the longitudinal and transverse rolls is discussed in detail. This source of heating inside the porous layer increases the thermal instability of base flow. Further, the asymptotic analysis for vanishingly small Peclet number is presented as $$Pe = 0$$ is a singular point. The boundary between the stationary and the oscillatory instabilities is given using the buoyancy ratio parameter and the Lewis number in this limiting case of $$Pe\rightarrow 0$$ .

Thermo-convective carbon sequestration in horizontal porous layers

This article explores the linear and nonlinear stability of double-diffusive density-driven convection in the context of carbon sequestration in deep saline aquifers. The anisotropy, due to the thermal and solutal diffusivities, is considered in a porous layer where the permeability is assumed to be layer dependent. Solute concentration is assumed to decay via a first-order chemical reaction. It is observed that varying the ratio of vertical to horizontal solutal and thermal diffusitivies does not significantly affect the behaviour of linear instability. This is in contrast to the nonlinear stability results. It is also observed that when the solute and thermal diffusion rates dominate the solute and thermal reaction rates, a change in the permeability has no significant effect on the onset of convection. However, when the solute reaction rate dominates the diffusion rate, a change in permeability has a notable effect on the instability of the system. It is observed that the effects of geothermal gradient on the onset of convection are negligible as compared to the solutal effects induced by the diffusion and dissolution of CO2 in deep saline aquifers.

Mono‐diffusive Hadley‐Prats flow in a horizontal porous layer subject to variable gravity and internal heat generation

AbstractAnalysis of internal heated and gravity effect on the onset of Hadley‐Prats flow in a horizontal porous layer with inclined temperature gradients is investigated using the linear and nonlinear instability analysis. The transformed eigenvalue problem is evaluated numerically to find the eigenvalue, which is treated as a vertical thermal Rayleigh number (Rz). It is evaluated by applying shooting and Runge‐Kutta method. Also, the critical Rz is investigated for different parameters governing the flow. A theoretical study is made to understand the influence of gravity field on the mechanism of mono‐diffusive instability of Hadley‐Prats convection in a fluid saturated horizontal porous layer. Nonlinear stability is evaluated by using energy functional. The comparison between linear and nonlinear instability results are presented and it is noted that linear theory of instability may not be useful to capture the complete picture of stability and instabilities may arise before one attains the linear stability threshold. This subcritical instability region is identified between the linear and energy thresholds in the parameter space of the problem considered.

The Onset of Convection in a Two-Layered Porous Medium with Anisotropic Permeability

We consider convection in a horizontal porous layer of uniform thickness which is heated from below and which is composed of two anisotropic sublayers with principal axes lying in the three coordinate directions. The aim is to determine criteria for the onset of convection by finding the critical Rayleigh number, wavenumber and roll orientation relative to the coordinate axes. The full set of nondimensional parameters has at least six members even when the sublayers are considered to be thermally isotropic, and therefore, we select some special cases in order to illuminate the type of qualitative behaviour which may be expected. One such case is where the anisotropic sublayers are identical except that one sublayer is rotated by an angle of 90^circ to the other. In this situation, the most unstable roll is found to lie at an angle of pm ,45^circ to the principal axes. It is also found that fluid particles exhibit a mean longitudinal flow as they circulate about the vortex axis. This drift along the vortex is balanced by an equal and opposite drift in the two neighbouring vortices. Convection in each sublayer is shown to be two dimensional even though the full flow field is three dimensional.

Instability of Vertical Constant Through Flows in Binary Mixtures in Porous Media with Large Pores

A binary mixture saturating a horizontal porous layer, with large pores and uniformly heated from below, is considered. The instability of a vertical fluid motion (throughflow) when the layer is salted by one salt (either from above or from below) is analyzed. Ultimately boundedness of solutions is proved, via the existence of positively invariant and attractive sets (i.e. absorbing sets). The critical Rayleigh numbers at which steady or oscillatory instability occurs are recovered. Sufficient conditions guaranteeing that a secondary steady motion or a secondary oscillatory motion can be observed after the loss of stability are found. When the layer is salted from above, a condition guaranteeing the occurrence of “cold” instability is determined. Finally, the influence of the velocity module on the increasing/decreasing of the instability thresholds is investigated.

On the Stability of Hadley-Flow in a Horizontal Porous Layer with Non-Uniform Thermal Gradient and Internal Heat Source

The present study deals with the effects of non-uniform inclined thermal gradient and internal heat source effect on the stability of buoyant flows in a fluid-saturated horizontal porous layer. Both linear and non-linear stability analyses have been performed. The non-linear stability analysis has been carried out by using the energy functional. The eigenvalue problems in both cases are solved numerically by shooting and Runga-Kutta methods. It is observed that the preferred mode at the onset of convection is longitudinal stationary mode. Comparison between linear and energy thresholds is given and found that the flow is stabilized at higher horizontal Rayleigh number for linear and non-linear cases irrespective of heat source. For fixed horizontal thermal Rayleigh number, increasing the value of internal heat source parameter destabilizes the system and favors the convection to commence.

Effect of Exponentially Temperature-Dependent Viscosity on the Onset of Penetrative Ferro-Thermal-Convection in a Saturated Porous Layer via Internal Heating

The effect of viscosity depending exponentially on temperature on the onset of penetrative ferro-thermal-convection (FTC) in a saturated horizontal porous layer in the presence of vertical magnetic field is investigated. The bounding surface of the ferrofluid layer is considered to be rigid-rigid and insulated to temperature perturbations. The resulting eigenvalue problem is solved numerically using the Galerkin technique and also analytically by a regular perturbation technique with wave number as a perturbation parameter. The analytical and numerical results are found to be concurrence. The characteristics of stability of the system are strongly dependent on the viscosity parameter B. The effect of B on the onset of ferroconvection in a porous layer is dual in nature depending on the choices of physical parameters and a sublayer starts to form at higher values of B. Whereas, increase in magnetic number M1 and the Darcy number Da is to advance the onset of ferroconvection in a porous layer. The nonlinearity of fluid magnetization M3 is found to have no influence on the onset of ferroconvection.

Linear and nonlinear stability of Soret‐Dufour Lapwood convection near double codimension‐2 points

AbstractIn this study, the Dufour and Soret effects on natural double‐diffusion convection in a horizontal porous layer was studied numerically using FORTRAN 90 programming and analytically near various convection onset thresholds. The porous layer was subject to a uniform heat and mass fluxes on the horizontal walls while the vertical walls were impermeable and adiabatic. The Darcy model along with the Boussinesq approximation was assumed in the problem formulation. The governing parameters of the problem are the thermal Rayleigh number, RT, the buoyancy ratio, N, the Lewis number, Le, the aspect ratio of the cavity, A, and the Dufour, Du, and Soret, Sr, numbers. For a shallow enclosure, the analytical solution was derived assuming zero convection wave number, which is valid near and above criticality. The onset of subcritical, supercritical and oscillatory convection was investigated. Two linear and nonlinear codimension‐2 points were found to exist. Whether the system was subject to constant fluxes and heat and solute, regardless of the aspect ratio of the layer, the subcritical convection behavior remained the same with similarity in the thresholds expressions for subcritical bifurcation.