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.     

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 $Q.$      

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.     

The Onset of Convection in an Anisotropic Porous Layer Using a Thermal Non-Equilibrium Model

The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.     

Effect of Local Thermal Non-equilibrium on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is analytically studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Darcy model is employed. The effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases is investigated using a three-temperature model. The analysis reveals that in some circumstances the effect of LTNE can be significant, but for a typical dilute nanofluid (with large Lewis number and with small particle-to-fluid heat capacity ratio) the effect is small.     

The Effect of Heterogeneity on the Onset of Double-Diffusive Convection Induced by Internal Heating in a Porous Medium: A Layered Model

In this paper we have investigated the effects of both weak and strong heterogeneity on the onset of double-diffusive convection which is induced by combined effects of internal heating and solutal gradient. To make analytical progress, we considered a composite porous medium consisting of two horizontal layers. We investigated the effects of heterogeneities in permeability, thermal conductivity, volumetric heat source strength, and porosity. The solutal diffusivity, which becomes effective when a vertical salinity gradient is imposed, is affected by variation of porosity. We found that the effect of solutal diffusivity is stabilizing when the porosity increases upwards.     

The Effect of Strong Heterogeneity on the Onset of Convection Induced by Internal Heating in a Porous Medium: A Layered Model

An analytical investigation of the onset of convection, induced by internal heating, in a composite porous medium consisting of two horizontal layers has been made. The two-layer model that we adopted makes it possible to investigate and compare the effects of both weak and strong heterogeneity. Both cases of constant heat flux and constant wall temperature boundary conditions have been treated. In general, we established that anything that aids convection in the upper portion of the layer is destabilizing. In agreement with this rule, we found that conductivity increasing upward leads to a more stable situation, permeability increasing upward leads to a less stable situation, and source strength increasing upward generally leads to a less stable situation.     

The Effects of Combined Horizontal and Vertical Heterogeneity on the Onset of Convection in a Porous Medium with Vertical Throughflow

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, with weak vertical throughflow, are studied analytically for the case of weak heterogeneity. It is found that when the boundary conditions at the upper and lower boundaries are symmetric, the throughflow magnitude and the permeability and conductivity gradients enter the expression for the critical Rayleigh number at second order. The throughflow on its own is stabilizing but the combination of throughflow and heterogeneity may be either stabilizing or destabilizing.     

Interaction of Transverse Heterogeneity and Thermal Development of Forced Convection in a Porous Medium

The classical Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium whose permeability and thermal conductivity vary in the transverse direction. It was found that there is a significant interaction between heterogeneity and thermal development.     

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.     

Effect of Thermal Modulation on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid

The effect of time-periodic temperature modulation at the onset of convection in a Boussinesq porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. The perturbation method is applied for computing the critical Rayleigh and wave numbers for small amplitude temperature modulation. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation, concentration Rayleigh number, porosity, Lewis number, and thermal capacity ratio. It has been shown that it is possible to advance or delay the onset of convection by time-periodic modulation of the wall temperature. The nanofluid is found to have more stabilizing effect when compared to regular fluid. Low frequency is destabilizing, while high frequency is always stabilizing for symmetric modulation. Asymmetric modulation and only lower wall temperature modulation is stabilizing for all frequencies when concentration Rayleigh number is greater than one.     

Thermal Convection in a Non-darcy Hemispherical Porous Medium

A simple mathematical theory is proposed to investigate the development of the flow field which is the response of a fluid to the buoyancy force due to the existence of a temperature gradient in a hemispherical fluid-saturated porous medium, assuming the validity of the Brinkman model. The induced flow is assumed to be slow, and Stokes approximation is invoked. It is shown, at all times, the induced fluid motion occurs in the form of eddies on either side of the axis of symmetry. In the steady state, the behavior of the fluid motion on the free surface is similar to that of axial fluid flow.     

Effect of Thermal Modulation on the Onset of Convection in Walters B Viscoelastic Fluid-Saturated Porous Medium

The linear stability of Walters B viscoelastic fluid-saturated horizontal porous layer is examined theoretically when the walls of the porous layer are subjected to time-periodic temperature modulation. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and the corresponding wave number. The shift in critical Rayleigh number is calculated as a function of modulation frequency, viscoelastic parameter, and Prandtl number. The effect of all three types of modulations is found to be destabilizing as compared to the unmodulated system. This result is in contrast to the system with other types of fluids. Besides, the influence of physical parameters on the control of convective instability of the system is discussed.     

Thermal Non-Equilibrium Porous Convection with Heat Generation and Density Maximum

Linear stability criterion for the onset of natural convection in a fluid saturated porous medium with uniform internal heat generation and density maximum is determined. The porous medium is not in local thermal equilibrium (LTE) and we follow a two-field model for the energy equation. It is found that both the heat generation and density maximum have an additive effect in advancing the onset condition. In general the destabilising effect of density maximum increases for large values of the fluid heat generation parameter. This effect becomes prominent even for small values of the fluid heat generation parameter when the flow is of Darcy type and LTE is not valid.     

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 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.     

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.     

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.