The Onset of Convection in a Heterogeneous Porous Medium with Vertical Throughflow

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from vertical throughflow, is studied analytically using linear stability theory. It is found that, to first order, a linear variation of the reciprocal of permeability with depth has no effect on the critical value of the Rayleigh number Ra _c based on the harmonic mean of the permeability, but a quadratic variation increasing in the upwards direction leads to a reduction in Ra _c.     

The Onset of Convection in a Layered Porous Medium with Vertical Throughflow

An analytical investigation of the effect of vertical throughflow on the onset of convection in a composite porous medium consisting of two horizontal layers has been made. The cases of iso-flux and iso-temperature boundaries are both investigated. The critical Rayleigh number depends on a Péclet number $Q$ , a permeability ratio $K_{r}$ , a thermal conductivity ratio $k_{r}$ , and a depth ratio $\delta $ . For the case of small $Q$ an approximate solution is obtained, which shows that in general throughflow has a stabilizing effect whose magnitude may be increased or decreased by the heterogeneity. This solution is supplemented by an asymptotic solution valid for large $Q.$      

Onset of Convection in a Porous Medium with Strong Vertical Throughflow

The problem for determining the critical Rayleigh number for the onset of convection in a horizontal porous layer with vertical throughflow is re-examined with the aim of obtaining analytical formulas applicable in the cases of weak and strong throughflow. For the case of strong throughflow an asymptotic analysis is performed.     

The Onset of Double-Diffusive Convection in a Vertical Cylinder Occupied by a Heterogeneous Porous Medium with Vertical Throughflow

This paper investigates the onset of convection in a vertical cylinder occupied by a saturated porous medium of vertically heterogeneous permeability. The flow is induced by an applied vertical temperature gradient and an imposed solute concentration gradient. The main interest of this paper is studying the effect of vertical throughflow on the onset of instability in this system. The study is performed using linear stability theory. The problem is of considerable interest for hydrological and geophysical situations.     

The Effect of Vertical Throughflow on the Onset of Convection in a Porous Medium in a Rectangular Box

The effect of vertical throughflow on the onset of convection in a rectangular box occupied by a saturated porous medium uniformly heated from below, is studied using linear stability theory. It is found that, for small values of the throughflow, the stabilizing effect of the throughflow and the stabilizing effect of the confining lateral walls of the box are approximately independent of each other.     

The Effect of Vertical Throughflow on the Onset of Convection Induced by Internal Heating in a Layered Porous Medium

We present an analytical investigation of the effect of vertical throughflow on the onset of convection, induced by internal heating, in a composite porous medium consisting of two horizontal layers. If convection is induced by internal heating, the bulk of the convection occurs in the upper half of the layer where the buoyancy force is destabilizing. For the case of heterogeneous porous medium, if the permeability increases in the upward direction, or if the thermal conductivity decreases in the upward direction, instability is increased. It is also found that upward throughflow is stabilizing but a modest amount of downward throughflow is destabilizing.     

The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Non-Periodic Global Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been extended to deal with the case of vertical throughflow.     

Non-Darcian Effects on the Onset of Convection in a Porous Layer with Throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.     

The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Periodic and Localized Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated with the aid of a computer package. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. Previous work on the case of non-periodic global variation is now extended to the case of either periodic variation or localized variation.     

The Effect of Strong Heterogeneity on the Onset of Convection in a Porous Medium

The effect of strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is discussed. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been developed to implement an algorithm giving a criterion for instability. This package is applied to the case of a cube partitioned into octants and to the cases where the permeability and thermal conductivity vary continuously across a cube in either a linear or a quadratic manner.     

Local Thermal Non-Equilibrium and Heterogeneity Effects on the Onset of Convection in a Layered Porous Medium

The effect of local thermal non-equilibrium on the onset of convection in a porous medium consisting of two horizontal layers is studied analytically. Linear stability theory is applied. Variations of permeability, fluid conductivity, solid conductivity, interphase heat transfer coefficient and porosity are considered. It is found that heterogeneity of permeability and fluid conductivity have a major effect, heterogeneity of interphase heat transfer coefficient and porosity have a lesser effect, while heterogeneity of solid conductivity is relatively unimportant.     

Local Thermal Non-equilibrium and Heterogeneity Effects on the Onset of Convection in an Internally Heated Porous Medium

The effect of local thermal non-equilibrium on the onset of convection in a porous medium consisting of two horizontal layers, each internally heated, is studied analytically. Linear stability theory is applied. Variations of permeability, fluid thermal conductivity, solid thermal conductivity, source strength in the solid and fluid phases, interphase heat-transfer coefficient and porosity are considered. It is found that heterogeneity of permeability, fluid thermal conductivity and source strength in the fluid phase have a major effect; heterogeneity of interphase heat-transfer coefficient and porosity have a lesser effect, while heterogeneity of solid thermal conductivity and source strength in the solid phase are relatively unimportant.     

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part I. Normal Modes

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number R _h < 49, the instability is absolute implying that, as the vertical Rayleigh number, R _v, increases passing through its critical value, R _vc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as R _v, increases beyond the critical value, while the horizontal Rayleigh number, R _h, and the Péclet number, Q _v, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, R _v, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient.     

The Onset of Convection in a Suspension of Gyrotactic Microorganisms in Superimposed Fluid and Porous Layers: Effect of Vertical Throughflow

The purpose of this paper is to investigate the effect of vertical throughflow on the onset of bioconvection in a suspension of gyrotactic microorganisms. A dilute suspension of gyrotactic microorganisms in a shallow system that consists of superimposed fluid and porous layers is considered. A linear instability analysis of this problem is performed and the Galerkin method is utilized to solve the eigenvalue problem. The analysis leads to an equation for the critical Rayleigh number. It is shown that the vertical throughflow stabilizes the system.     

The Onset of Convection in a Heterogeneous Porous Medium with Transient Temperature Profile

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from transient heating or otherwise, is studied analytically using linear stability theory. Two particular situations, corresponding to instantaneous bottom heating and constant-rate bottom heating, are studied. Estimates of the timescale for the development of convection instability are obtained.     

Thermal Dispersion Effects on Combined Convection in Non-Newtonian Fluids Along a Nonisothermal Vertical Plate in a Porous Medium

The method of non-similarity solution is used to study the influence of thermal dispersion on combined convection from vertical surfaces in a porous medium saturated with a power-law type non-Newtonian fluid. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The transformed conservation laws are solved numerically for the case of variable surface heat flux conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented.     

The Effect of Pulsating Deformation on the Onset of Convection in a Porous Medium

A linear instability analysis has been performed for the onset of convection in a horizontal layer of a porous medium whose permeability pulsates with time as a result of vertical movement of one of the boundaries. It was found that, to a first-order approximation in the pulsation amplitude, the effect of deformation is destabilizing for disturbances whose period is of the order of the thermal diffusion time scale. The effects of the average porosity, pulsation amplitude, and pulsation frequency were investigated.