The Onset of Prandtl–Darcy–Prats Convection in a Horizontal Porous Layer

We consider the effect of finite Prandtl–Darcy numbers of the onset of convection in a porous layer heated isothermally from below and which is subject to a horizontal pressure gradient. A dispersion relation is found which relates the critical Darcy–Rayleigh number and the induced phase speed of the cells to the wavenumber and the imposed Péclet and Prandtl–Darcy numbers. Exact numerical solutions are given and these are supplemented by asymptotic solutions for both large and small values of the governing nondimensional parameters. The classical value of the critical Darcy–Rayleigh number is $4\pi ^2$ 4 π 2 , and we show that this value increases whenever the Péclet number is nonzero and the Prandtl–Darcy number is finite simultaneously. The corresponding wavenumber is always less than $\pi $ π and the phase speed of the convection cells is always smaller than the background flux velocity.     

Instability of Hadley–Prats Flow with Viscous Heating in a Horizontal Porous Layer

The onset of convective rolls instability in a horizontal porous layer subject to a basic temperature gradient inclined with respect to gravity is investigated. The basic velocity has a linear profile with a non-vanishing mass flow rate, i.e., it is the superposition of a Hadley-type flow and a uniform flow. The influence of the viscous heating contribution on the critical conditions for the onset of the instability is assessed. There are four governing parameters: a horizontal Rayleigh number and a vertical Rayleigh number defining the intensity of the inclined temperature gradient, a Péclet number associated with the basic horizontal flow rate, and a Gebhart number associated with the viscous dissipation effect. The critical wave number and the critical vertical Rayleigh number are evaluated for assigned values of the horizontal Rayleigh number, of the Péclet number, and of the Gebhart number. The linear stability analysis is performed with reference either to transverse or to longitudinal roll disturbances. It is shown that generally the longitudinal rolls represent the preferred mode of instability.     

Onset of Double-Diffusive Reaction–Convection in an Anisotropic Rotating Porous Layer

The linear and non-linear stability of a rotating double-diffusive reaction–convection in a horizontal anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated considering a Darcy model that includes the Coriolis term. The effect of Taylor number, mechanical, and thermal anisotropy parameters, reaction rate, solute Rayleigh number, Lewis number, and normalized porosity on the stability of the system is investigated. We find that the Taylor number has a stabilizing effect, chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite-amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.     

Linear Instability of the Isoflux Darcy–Bénard Problem in an Inclined Porous Layer

The linear stability for convection in an inclined porous layer is considered for the case where the plane bounding surfaces are subjected to constant heat flux boundary conditions. A combined analytical and numerical study is undertaken to uncover the detailed thermoconvective instability characteristics for this configuration. Neutral curves and decrement spectra are shown. It is found that there are three distinct regimes between which the critical wavenumber changes discontinuously. The first is the zero-wavenumber steady regime which is well known for horizontal layers. The disappearance of this regime is found using a small-wavenumber asymptotic analysis. The second consists of unsteady modes with a nonzero wavenumber, while the third consists of a steady mode. Linear stability corresponds to inclinations which are greater than 32.544793° from the horizontal.     

Effects of a.c. Electric Field and Rotation on Bénard–Marangoni Convection

The coupled buoyancy and thermocapillary instability, the Bénard–Marangoniproblem, in an electrically conducting fluid layer whose upper surface is deformed and subject to a temperature gradient is studied. Both influences of an a.c. electric field and rotation are investigated. Special attention is directed at the occurrence of convection both in the form of stationary motion and oscillatory convection. The linear stability problem is solved for different values of the relevant dimensionless numbers, namely the a.c. electric Rayleigh number, the Taylor, Rayleigh, Biot, Crispation and Prandtl numbers. For steady convection, it is found that by increasing the angular velocity, one reinforces the stability of the fluid layer whatever the values of the surface deformation and the applied a.c. electric field. We have also determined the regions of oscillatory instability and discussed the competition between both stationary and oscillatory convections. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Onset of Darcy–Brinkman Electroconvection in a Dielectric Fluid Saturated Porous Layer

The combined effect of a vertical AC electric field and the boundaries on the onset of Darcy–Brinkman convection in a dielectric fluid saturated porous layer heated either from below or above is investigated using linear stability theory. The isothermal bounding surfaces of the porous layer are considered to be either rigid or free. It is established that the principle of exchange of stability is valid irrespective of the nature of velocity boundary conditions. The eigenvalue problem is solved exactly for free–free (F/F) boundaries and numerically using the Galerkin technique for rigid–rigid (R/R) and lower-rigid and upper-free (F/R) boundaries. It is observed that all the boundaries exhibit qualitatively similar results. The presence of electric field is emphasized on the stability of the system and it is shown that increasing the AC electric Rayleigh number R _ea is to facilitate the transfer of heat more effectively and to hasten the onset of Darcy–Brinkman convection. Whereas, increase in the ratio of viscosities Λ and the inverse Darcy number Da ⁻¹ is to delay the onset of Darcy–Brinkman electroconvection. Besides, increasing R _ea and Da ⁻¹ as well as decreasing Λ are to reduce the size of convection cells.     

Thermal Convection in a Cattaneo–Fox Porous Material with Guyer–Krumhansl Effects

The Guyer–Krumhansl equation is coupled with the Cattaneo–Fox law for the temperature and heat flux fields to study thermal convection in a fluid-saturated Darcy porous material. In particular the effects of the Guyer–Krumhansl terms on oscillatory convection are studied. It is found that for a certain range of the Guyer–Krumhansl coefficient stationary convection occurs while changing the range results in oscillatory convection. Numerical results quantify this effect.     

Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.     

Double-Diffusive Natural Convection Flows with Thermosolutal Symmetry in Porous Media in the Presence of the Soret–Dufour Effects

The double-diffusive natural convection past a vertical plate embedded in a fluid-saturated porous medium is considered in the boundary-layer and Boussinesq approximations. It is assumed that the Soret–Dufour cross-diffusion effects are significant. The heat and mass fluxes on the plate are prescribed as functions of the surface coordinate x. The general similarity reduction of the problem for power-law and exponential variation of the wall fluxes is given. In the case of thermosolutal symmetry, when the similar temperature and concentration fields become coincident, exact analytical as well as numerical solutions are reported and discussed in some detail. For the flows without thermosolutal symmetry, the final similarity equations have been solved numerically, by paying attention to the influence of the Soret and Dufour numbers on the departure from thermosolutal symmetry. The reported results focus on the wall temperatures and concentrations, whose reciprocals are Nusselt and Sherwood numbers, respectively.     

A Semilinear Schrödinger Equation in the Presence of a Magnetic Field

We consider the semilinear stationary Schrödinger equation in a magnetic field: (–i+A)² u+V(x)u=g(x,|u|)u in ^N , where V is the scalar (or electric) potential and A is the vector (or magnetic) potential. We study the existence of nontrivial solutions both in the critical and in the subcritical case (respectively g(x,|u|)=|u|^{2 * –2} and |g(x,|u|)|c(1+|u| ^{p –2}), where 2<p<2^*). The results are obtained by variational methods. For g critical we use constrained minimization and for subcritical g we employ a minimax-type argument. In the latter case we also study the existence of infinitely many geometrically distinct solutions.     

“Inviscid Finger” Instability in Regular Models of a Porous Medium

Displacement of a fluid from a porous medium is considered. The flow is assumed to be fast enough, i.e., the Reynolds number based on the characteristic pore size is large. If he driving fluid is less dense (for example, a gas), the interface is unstable. This instability is similar to the well–known viscous finger instability but the governing parameter is density instead of viscosity. The instability is demonstrated experimentally using two–dimensional models. In square lattices of perpendicular channels, noticeable branching of fingers is not observed, which is attributed to the anisotropy of such an artificial porous medium. A more ordinary pattern with finger branching is obtained in a two–dimensional layer of spheres, which appears to be more isotropic. A simple model describing flow in a square lattice is proposed. The initial stage of growth is considered, and the instability increment is estimated. A qualitative analysis of the nonlinear stage is performed.     

Entropy Generation in Double-Diffusive Convection in a Square Porous Cavity using Darcy–Brinkman Formulation

The article reports a numerical study of entropy generation in double-diffusive convection through a square porous cavity saturated with a binary perfect gas mixture and submitted to horizontal thermal and concentration gradients. The analysis is performed using Darcy–Brinkman formulation with the Boussinesq approximation. The set of coupled equations of mass, momentum, energy and species conservation are solved using the control volume finite-element method. Effects of the Darcy number, the porosity and the thermal porous Rayleigh number on entropy generation are studied. It was found that entropy generation considerably depends on the Darcy number. Porosity induces the increase of entropy generation, especially at higher values of thermal porous Rayleigh number.     

Diffusion–Convection in a Porous Medium with Impervious Inclusions at Low Flow Rates

The aim of this paper is to develop a macroscopic model for the transport of a passive solute, by diffusion and convection, in a heterogeneous medium consisting of impervious solids periodically distributed in a porous matrix. In the porous part, the flow is described by Darcy's law. Attempt is made to derive the macroscopic equation governing the average concentration field in the equivalent macroscopic medium and the macroscopic transport parameters. The analysis is conducted in the case when convection and diffusion are of the same order of magnitude at the macroscopic level, that is, when the Péclet number is of order 1. The proposed macroscopic model is obtained using the homogenization method for periodic structures with a double scale asymptotic expansion, in which the small parameter is the ratio between the two characteristic lengths l (the period scale of the impervious bodies distribution) and L (the scale of the macroscopic sample). The macroscopic parameters, which characterize the multiporous medium, depend solely on the transport parameters in the porous matrix and on the geometry of the impervious inclusions without any phenomenological assumption. Numerical computations are performed using a finite element method for several geometries of the solid inclusions, in two- and three-dimensional cases.     

Mixed Convection in a Darcy–Brinkman Porous Medium with a Constant Convective Thermal Boundary Condition

The characteristics of the boundary layer flow past a plane surface adjacent to a saturated Darcy–Brinkman porous medium are investigated in this paper. The flow is driven by an external free stream moving with constant velocity. The surface is heated with a convective boundary condition with constant heat transfer coefficient. The problem is non-similar and is investigated numerically by a finite difference method. The problem is governed by four non-dimensional parameters, that is, the convective Darcy number, the convective Grashof number, the Prandtl number, and the axial distance along the plate. The influence of these parameters on the results is investigated, and the results are presented in tables and figures. The Darcy term and the Grashof term in the momentum equation contradict each other and this contradiction makes the problem complicated. However, the wall shear stress and the wall temperature increase continuously along the plate and the wall temperature always tends to 1.     

Rayleigh–Bénard Instability of an Ellis Fluid Saturating a Porous Medium

Transport in Porous Media - The Ellis model describes the apparent viscosity of a shear–thinning fluid with no singularity in the limit of a vanishingly small shear stress. In particular,...     

Thermal Dispersion–Radiation Effects on Non-Darcy Natural Convection in a Fluid Saturated Porous Medium

The effects of thermal dispersion and thermal radiation on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium are studied. Forchheimer extension is considered in the flow equations. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. Rosseland approximation is used to describe the radiative heat flux in the energy equation. Similarity solution for the transformed governing equations is obtained. Numerical results for the details of the velocity and temperature profiles which are shown on graphs have been presented. The combined effect of thermal dispersion and thermal radiation, for the two cases Darcy and non-Darcy porous medium, on the heat transfer rate which are entered in tables is discussed.     

Onset of Darcy–Brinkman Ferroconvection in a Rotating Porous Layer Using a Thermal Non-Equilibrium Model: A Nonlinear Stability Analysis

This article presents a nonlinear stability analysis of a rotating thermoconvective magnetized ferrofluid layer confined between stress-free boundaries using a thermal non-equilibrium model by the energy method. The effect of interface heat transfer coefficient ( H^￠){( {{\mathcal H}^{\prime}})}, magnetic parameter (M ₃), Darcy–Brinkman number ( [^(D)]a){( {\hat{{\rm D}}{\rm a}})}, and porosity modified conductivity ratio (γ′) on the onset of convection in the presence of rotation (T_A1){({T_{{\rm A}_1}})} have been analyzed. The critical Rayleigh numbers predicted by energy method are smaller than those calculated by linear stability analysis and thus indicate the possibility of existence of subcritical instability region for ferrofluids. However, for non-ferrofluids stability and instability boundaries coincide. Asymptotic analysis for both small and large values of interface heat transfer coefficient (H^￠){({{\mathcal H}^{\prime}})} is also presented. A good agreement is found between the exact solutions and asymptotic solutions.     

Weakly Nonlinear Stability Analysis of Temperature/Gravity-Modulated Stationary Rayleigh–Bénard Convection in a Rotating Porous Medium

The effect of time-periodic temperature/gravity modulation on thermal instability in a fluid-saturated rotating porous layer has been investigated by performing a weakly nonlinear stability analysis. The disturbances are expanded in terms of power series of amplitude of convection. The Ginzburg–Landau equation for the stationary mode of convection is obtained and consequently the individual effect of temperature/gravity modulation on heat transport has been investigated. Further, the effect of various parameters on heat transport has been analyzed and depicted graphically.     

Instability of Streaming Viscoelastic Fluids in the Presence of ��Effective Interfacial Tension�� Through Porous Medium

The ??effective interfacial tension?? effect on the instability of the plane interface between two uniform, superposed, and streaming Rivlin?CEricksen viscoelastic fluids through a porous medium is considered. The case of two uniform streaming Rivlin?CEricksen viscoelastic fluids separated by a horizontal boundary is studied. In the absence of ??effective interfacial tension??, stability/instability of the system as well as perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if the perturbations in the direction of streaming are ignored, whereas for perturbations in all other directions, there exists instability for a certain wave number range. ??Effective interfacial tension?? is able to suppress this Kelvin?CHelmholtz instability for small wavelength perturbations, the medium porosity reduces the stability range given in terms of a difference in streaming velocities.