The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium

We investigate Rayleigh–Benard convection in a porous layer subjected to gravitational and Coriolis body forces, when the fluid and solid phases are not in local thermodynamic equilibrium. The Darcy model (extended to include Coriolis effects and anisotropic permeability) is used to describe the flow, whilst the two-equation model is used for the energy equation (for the solid and fluid phases separately). The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection and the effect of both thermal and mechanical anisotropy on the critical Rayleigh number is discussed.     

Stationary Fingering Instability in a Non-equilibrium Porous Medium with Coupled Molecular Diffusion

Finger type double diffusive convective instability in a fluid-saturated porous medium is studied in the presence of coupled heat-solute diffusion. A local thermal non-equilibrium (LTNE) condition is invoked to model the Darcian porous medium which takes into account the energy transfer between the fluid and solid phases. Linear stability theory is implemented to compute the critical thermal Rayleigh number and the corresponding wavenumber exactly for the onset of stationary convection. The effects of Soret and Dufour cross-diffusion parameters, inter-phase heat transfer coefficient and porosity modified conductivity ratio on the instability of the system are investigated. The analysis shows that positive Soret mass flux triggers instability and positive Dufour energy flux enhances stability whereas their combined influence depends on the product of solutal Rayleigh number and Lewis number. It also reveals that cell width at the convection threshold gets affected only in the presence of both the cross-diffusion fluxes. Besides, asymptotic solutions for both small and large values of the inter-phase heat transfer coefficient and porosity modified conductivity ratio are found. An excellent agreement is found between the exact and asymptotic solutions.     

The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail.     

Onset of convection in a horizontal porous channel with uniform heat generation using a thermal nonequilibrium model

This paper considers the onset of free convection in a horizontal fluid-saturated porous layer with uniform heat generation. Attention is focused on cases where the fluid and solid phases are not in local thermal equilibrium, and where two energy equations describe the evolution of the temperature of each phase. Standard linearized stability theory is used to determine how the criterion for the onset of convection varies with the inter-phase heat transfer coefficient, H, and the porosity-modified thermal conductivity ratio, γ. We also present asymptotic solutions for small values of H. Excellent agreement is obtained between the asymptotic and the numerical results.     

Stability analysis of thermosolutal convection in a horizontal porous layer using a thermal non-equilibrium model

A stability analysis is carried out to investigate the onset of thermosolutal convection in a horizontal porous layer when the solid and fluid phases are not in a local thermal equilibrium, and the solubility of the dissolved component depends on temperature. To study how the reaction and thermal non-equilibrium affect the double-diffusive convection, the effects of scaled inter-phase heat transfer coefficient H and dimensionless reaction rate k on thermosolutal convection are discussed . The critical Rayleigh number and the corresponding wave number for the stability and overstability convections are obtained. Specially, asymptotic analysis for both small and large values of H and k is presented, and the corresponding asymptotic solutions are compared with numerical results. At last, a nonlinear stability analysis is presented to study how H and k affect the Nusselt number.     

Free Convection Heat Transfer over a Vertical Cylinder in a Saturated Porous Medium Using a Local Thermal Non-equilibrium Model

In this article, free convection heat transfer over a vertical cylinder with variable surface temperature distributions in a porous medium is analyzed. It is assumed that the fluid and solid phases are not in local thermal equilibrium and, therefore, a two-temperature model of heat transfer is applied. The coupled momentum and energy equations are presented and then they are transformed into ordinary differential equations. The similarity equations are solved numerically. The resulting velocity, streamlines, temperature distributions for fluid and solid phases are shown for different values of parameters entering into the problem. The calculated values of the local Nusselt numbers for both solid and fluid phases are also shown.     

Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.     

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.     

Effect of Thermal Non-Equilibrium on Convective Instability in a Ferromagnetic Fluid-Saturated Porous Medium

The effect of local thermal non-equilibrium (LTNE) on the onset of thermomagnetic convection in a ferromagnetic fluid-saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. A modified Forchheimer-extended Darcy equation is employed to describe the flow in the porous medium, and a two-field model is used for temperature representing the solid and fluid phases separately. It is found that both the critical Darcy–Rayleigh number and the corresponding wave number are modified by the LTNE effects. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H _t are presented and compared with those computed numerically. An excellent agreement is obtained between the asymptotic and the numerical results. Besides, the influence of magnetic parameters on the instability of the system is also discussed. The available results in the literature are recovered as particular cases from the present study.     

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.     

TEMPERATURE PROFILES OF LOCAL THERMAL NONEQUILIBRIUM FOR THERMAL DEVELOPING FORCED CONVECTION IN POROUS MEDIUM PARALLEL PLATE CHANNEL

Based on the two-energy equation model, taking into account viscous dissipation due to the interaction between solid skeleton and pore fluid flow, temperature expressions of the solid skeleton and pore fluid flow are obtained analytically for the thermally developing forced convection in a saturated porous medium parallel plate channel, with walls being at constant temperature. It is proved that the temperatures of the two phases for the local thermal nonequilibrium approach to the temperature derived from the one-energy equation model for the local thermal equilibrium when the heat exchange coefficient goes to infinite. The temperature profiles are shown in figures for different dimensionless parameters and the effects of the parameters on the local thermal nonequilibrium are revealed by parameter study.     

Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium

The stability analysis of the quiescent state in a Maxwell fluid-saturated densely packed porous medium subject to vertical concentration and temperature gradients is presented. A single phase model with local thermal equilibrium between the porous matrix and the Maxwell fluid is assumed. The critical Darcy–Rayleigh numbers and the corresponding wave numbers for the onset of stationary and oscillatory convection are determined. A Lorenz like system is obtained for weakly nonlinear stability analysis.     

Free Convection Heat Transfer From A Sphere In A Porous Medium Using A Thermal Non-equilibrium Model

This work studies the free convection heat transfer from a sphere with constant wall temperature embedded in a fluid-saturated porous medium using a thermal non-equilibrium model. The governing equations are transformed into boundary-layer partial differential equations by the coordinate transform, and the obtained governing equations are then solved by the cubic spline collocation method. The temperature distributions for fluid and solid phases are shown for different values of the porosity scaled thermal conductivity ratio, the interphase heat transfer parameter, and the streamwise coordinate. The effects of the porosity scaled thermal conductivity ratio and the interphase heat transfer parameter between solid and fluid phases on the local Nusselt numbers for fluid and solid phases are examined. Results show the local Nusset number for the porous medium can be increased by increasing the porosity scaled thermal conductivity ratio. Moreover, the thermal non-equilibrium effect is more significant for low values of the porosity scaled thermal conductivity ratio or the interphase heat transfer parameter.     

Linear and nonlinear double diffusive convection in a rotating sparsely packed porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and nonlinear stability analyses. The Brinkman model that includes the Coriolis term is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for the energy equation. The onset criterion for stationary, oscillatory, and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal diffusion, solute diffusion, and rotation that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number, and Taylor number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

A Local Thermal Non-Equilibrium Analysis of Fully Developed Forced Convective Flow in a Tube Filled with a Porous Medium

A local thermal non-equilibrium model has been considered for the case of thermally fully developed flow within a constant heat flux tube filled with a porous medium. Exact temperature profiles for the fluid and solid phases are found after combining the two individual energy equations and then transforming them into a single ordinary differential equation with respect to the temperature difference between the solid phase and the wall subject to constant heat flux. The exact solutions for the case of metal-foam and air combination reveal that the local thermal equilibrium assumption may fail for the case of constant heat flux wall. The Nusselt number is presented as a function of the Peclet number, which shows a significant increase due to both high stagnant thermal conductivity and thermal dispersion resulting from the presence of the metal-foam.     

Non-Linear Two Dimensional Double Diffusive Convection in a Rotating Porous Layer Saturated by a Viscoelastic Fluid

Double diffusive convection in a rotating anisotropic porous layer, saturated by a viscoelastic fluid, heated from below and cooled from above has been studied making linear and non-linear stability analyses. The fluid and solid phases are considered to be in equilibrium. In momentum equation, we have employed the Darcy equation which includes both time derivative and Coriolis terms. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary and oscillatory convection. A weak non-linear analysis based on minimal representation of truncated Fourier series analysis containing only two terms has been used to find the Nusselt number and Sherwood number as functions of time. We have solved the finite amplitude equations using a numerical scheme. The results obtained, during the above analyses, have been presented graphically and the effects of various parameters on heat and mass transfer have been discussed. Finally, we have drawn the steady and unsteady streamlines, isotherms, and isohalines for various parameters.     

Local Thermal Non-equilibrium Analysis of the Instability in a Vertical Porous Slab with Permeable Sidewalls

Buoyant flow in a fluid-saturated porous vertical slab with isothermal and permeable boundaries is performed. Two reservoirs, maintained at different uniform temperatures, confine the slab. The permeable plane boundaries of the slab are modelled by imposing a condition of hydrostatic pressure. Darcy’s law and the Oberbeck–Boussinesq approximation are employed. The hypothesis of local thermal equilibrium between the fluid and the solid phase is relaxed. A two-temperature model is adopted, so that two local energy balance equations govern the heat transfer in the porous slab. The basic stationary buoyant flow consists of a single convective cell of infinite height. The time evolution of normal mode perturbations superposed onto the basic state is investigated in order to determine the onset conditions for thermal instability. A pressure–temperature formulation is employed. Major asymptotic cases are investigated. It is shown that departure from local thermal equilibrium implies in general a destabilisation of the basic stationary flow.     

Ferromagnetic Convection in a Heterogeneous Darcy Porous Medium Using a Local Thermal Non-equilibrium (LTNE) Model

The combined effects of vertical heterogeneity of permeability and local thermal non-equilibrium (LTNE) on the onset of ferromagnetic convection in a ferrofluid saturated Darcy porous medium in the presence of a uniform vertical magnetic field are investigated. A two-field model for temperature representing the solid and fluid phases separately is used. The eigenvalue problem is solved numerically using the Galerkin method for different forms of permeability heterogeneity function Γ(z) and their effect on the stability characteristics of the system has been analyzed in detail. It is observed that the general quadratic variation of Γ(z) with depth has more destabilizing effect on the system when compared to the homogeneous porous medium case. Besides, the influence of LTNE and magnetic parameters on the criterion for the onset of ferromagnetic convection is also assessed.     

多孔介质平板通道强迫对流中热局部非平衡时的热应力

根据微观不可压饱和多孔介质热-力-流相互作用的一般理论,在固相骨架小变形的假定下,考虑固相和流相相互作用的粘性耗散,研究了多孔介质平板通道强迫对流热局部非平衡的热应力问题.建立了问题的热-力数学模型,根据饱和多孔介质的平衡方程,在固相骨架只存在横向位移的假定下,求解了固相骨架的位移和相应的热应力,数值考察了各种物性参数对热应力分布的影响,讨论了热局部平衡模型的适用性.     

Convective Transport in a Nanofluid Saturated Porous Layer With Thermal Non Equilibrium Model

The effect of local thermal non-equilibrium on linear and non-linear thermal instability in a horizontal porous medium saturated by a nanofluid has been investigated analytically. The Brinkman Model has been used for porous medium, while nanofluid incorporates the effect of Brownian motion along with thermophoresis. A three-temperature model has been used for the effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases. The linear stability is based on normal mode technique, while for nonlinear analysis, a minimal representation of the truncated Fourier series analysis involving only two terms has been used. The critical conditions for the onset of convection and the heat and mass transfer across the porous layer have been obtained numerically.