Effect of thermal dispersion on free convection in a fluid saturated porous medium

The present article considers a numerical study of thermal dispersion effect on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. 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. The non-dimensional governing equations are solved by the finite element method (FEM) with a Newton–Raphson solver. Numerical results for the details of the stream function, velocity and temperature contours and profiles as well as heat transfer rates in terms of Nusselt number are obtained. The study shows that the increase in thermal dispersion coefficient of the porous medium results in more heat energy to disperse away in the normal direction to the wall. This induces more fluid to flow along the wall, enhancing the heat transfer coefficient particularly near the wall.     

The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium

The onset of buoyancy-driven convection in an initially quiescent ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. The Brinkman-Lapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-paramagnetic, while the upper paramagnetic boundary is considered to be either rigid or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective–radiative exchange at the upper boundary, which encompasses fixed temperature and fixed heat flux as particular cases. The resulting eigenvalue problem is solved numerically using the Galerkin technique. It is found that increase in the Biot number Bi, porous parameter σ, viscosity ratio Λ, magnetic susceptibility χ, and decrease in the magnetic number M ₁ and non-linearity of magnetization M ₃ is to delay the onset of ferroconvection in a porous medium. Further, increase in M ₁, M ₃, and decrease in χ, Λ, σ and Bi is to decrease the size of convection cells.     

Linear and non-linear analyses of convection in a micropolar fluid occupying a porous medium

Linear and weakly non-linear analyses of convection in a micropolar fluid occupying a high-porosity medium are performed. The Brinkman–Eringen momentum equation is considered. The linear and non-linear analyses are, respectively, based on the normal mode technique and truncated representation of Fourier series. The linear theory for a two-phase system reiterates that the preferred mode of convection is stationary as in the case of a single-phase system. An autonomous system of differential equations representing cellular convection arising in the study is considered to analyse the critical points. The Nusselt number is obtained as a function of micropolar and porous medium parameters.     

Radiation effect on mixed convection from a horizontal surface in a porous medium

The effect of thermal radiation on the non-Darcy mixed convection flow over a non-isothermal horizontal surface immersed in a saturated porous medium has been studied. The wall temperature is assumed to have a power-law variation with the distance measured from the leading edge of the plate. The non-linear coupled parabolic partial differential equations governing the flow have been solved numerically using a finite-difference scheme. For some particular cases, the self-similar solution has also been obtained. The heat transfer is found to be strongly influenced by the radiative flux number, buoyancy parameter, variation of wall temperature, non-Darcy parameter and the nature of the free stream velocity.     

Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a porous medium

Stability analysis of double-diffusive convection for viscoelastic fluid with Soret effect in a porous medium is investigated using a modified-Maxwell-Darcy model. We use the linear stability analysis to investigate how the Soret parameter and the relaxation time of viscoelastic fluid effect the onset of convection and the selection of an unstable wavenumber. It is found that the Soret effect is to destabilize the system for oscillatory convection. The relaxation time also enhances the instability of the system. The effects of Soret coefficient and relaxation time on the heat transfer rate in a porous medium are studied using the nonlinear stability analysis, the variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Some previous results can be reduced as the special cases of the present paper.     

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.     

The inertial effect on the natural convection flow within a fluid-saturated porous medium

The general momentum equation for fluid flow within a porous medium is supposedly valid for any fluid-porous medium configuration. One of the main concerns of using the general equations refers to the inclusion of both inertia terms, namely, the convective inertia term and the Forchheimer term. In this study, we go beyond the important discussion about the correctness of including both terms in the general momentum equations by focusing upon the effect of the convective inertia term on the heat transfer results. The fluid-porous medium system considered here is a cavity bounded by solid surfaces with vertical walls maintained at constant but different temperatures. The natural convection problem is solved numerically, and the results are compared with a general theory developed by using the method of scale analysis. It is demonstrated that the convective inertia term effect upon the heat transfer results is minor for 0.01 ≤ Pr ≤ 1, 10 ≤ Ra_D ≤ 10⁴, 10⁻⁸ ≤ Da ≤ 10⁻², and porosities 0.4 and 0.8. It is also shown that, contrary to the general belief, the convective inertial effect upon the heat transfer within the cavity is minimized when the Prandtl number is reduced.     

Radiation effect on free convection of a non-Newtonian fluid over a vertical cone embedded in a porous medium with heat generation

The effects of thermal radiation on free convection in a non-Newtonian fluid over a vertical cone embedded in a porous medium in the presence of heat generation are studied. By using similarity transformations, the governing equations describing the problem are transformed to a system of nonlinear ordinary differential equations, which are solved numerically. The results are presented in the graphical form. The effects of various physical parameters and of the local Nusselt number on the velocity and the temperature profiles are discussed.     

Radiation effect on Darcy free convection flow along an inclined surface placed in porous media

The paper investigates the effect of radiation on Darcy's buoyancy induced flow of an optically dense viscous incompressible fluid along a heated inclined flat surface maintained at uniform temperature placed in a saturated porous medium with Rosseland diffusion approximation employing the implicit finite difference method together with Keller box elimination technique. Both the streamwise and normal components of the buoyancy force are retained in the momentum equations. The numerical results show that as the buoyancy parameter, ξ, increases the local Nusselt number increases. The results for the locally nonsimilar solutions are compared with the locally similar solutions for small angle of inclination and approximate similar solutions along vertical surface. The effect of the conduction-radiation parameter, R _d , and the surface temperature excess ration, θ _w , on the local Nusselt number, the tangential velocity distribution and the temperature distribution are also shown graphically.     

Horizontal thermal convection in a porous medium

Linearised instability and nonlinear stability bounds for thermal convection in a fluid-filled porous finite box are derived. A nonuniform temperature field in the steady state is generated by maintaining the vertical walls at different temperatures. The linearised instability threshold is shown to be well above the global stability boundary, which is strongly suggested by the lack of symmetry in the perturbed system. The numerical results are evaluated utilising a newly developed Legendre-polynomial-based spectral method.     

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.     

Soret and Dufour effect on double diffusion mixed convection from a vertical surface in a porous medium saturated with a non-Newtonian fluid

A non-similar boundary layer analysis is presented to study the flow, heat and mass transfer characteristics of non-Darcian mixed convection of a non-Newtonian fluid from a vertical isothermal plate embedded in a homogeneous porous medium with the effect of Soret and Dufour and in the presence of either surface injection or suction. The value of the mixed-convection parameter lies between 0 and 1. In addition, the power-law model is used for non-Newtonian fluids with exponent n < 1 for pseudoplastics n = 1 for Newtonian fluids and n > 1 for dilatant fluids. Furthermore, the coordinates and dependent variables are transformed to yield computationally efficient numerical solutions that are valid over the entire range of mixed convection, from the pure forced-convection limit to the pure free-convection limit, and the whole domain of non-Newtonian fluids, from pseudoplastics to dilatant fluids. The numerical solution of the problem is derived using a Runge–Kutta integration scheme with Newton–Raphson shooting technique. Distributions for velocity, temperature and concentration, as well as for the rate of wall heat and mass transfer, have been obtained and discussed for various physical parametric values.     

Thermal dispersion and viscous dissipation effects on non-darcy mixed convection in a fluid saturated porous medium

Mixed convection flow and heat transfer about an isothermal vertical wall embedded in a fluid saturated porous medium with uniform free stream velocity is considered and the effects of thermal dispersion and viscous dissipation in both aiding and opposing flows are analysed. Similarity solution is not possible due to the inclusion of the viscous dissipation term, series solution is obtained, first and second order effects of dissipation revealed that viscous dissipation lowers the heat transfer rate. Observations also revealed that the thermal dispersion effect enhances the heat transfer rate and the effect of viscous dissipation is observed to increase with increasing values of the dispersion parameter. Received on 21 March 1997     

Analytical investigation of steady convection of a near-critical Van der Waals gas in a porous thin annular cylinder embedded in a heat-conducting medium

Natural convection of a near-critical Van der Waals gas in a horizontal porous thin annular cylinder embedded in a heat-conducting space with a temperature gradient given at infinity is considered. The two-dimensional problem in a plane orthogonal to the cylinder axis is investigated using analytical methods, accurate to quantities of the order of the annulus thickness. A criterion of the onset of an annulus-section-average flow is obtained. The critical Rayleigh-Darcy number is determined for the general case in which the physical properties of the gas can considerably vary throughout the medium. Several limiting cases are considered and the ranges of their applicability are discussed. It is shown that, as the thermodynamic critical point is approached, the asymptotics of the critical Rayleigh-Darcy number depend on the relation between non-Boussinesq parameters, such as hydrostatic compressibility criteria, the temperature difference, and the nearness to the critical point. In the case of steady convection, an analytical solution is also derived in the case in which the above-mentioned stability threshold is exceeded and the physical properties of the gas vary throughout space only slightly. A comparison with the case of a perfect gas is made.     

Linear stability of triple-diffusive convection in micropolar ferromagnetic fluid saturating porous medium

The triple-diffusive convection in a micropolar ferromagnetic fluid layer heated and soluted from below is considered in the presence of a transverse uniform magnetic field. An exact solution is obtained for a flat fluid layer contained between two free boundaries. A linear stability analysis and a normal mode analysis method are carried out to study the onset convection. For stationary convection, various parameters such as the medium permeability, the solute gradients, the non-buoyancy magnetization, and the micropolar parameters (i.e., the coupling parameter, the spin diffusion parameter, and the micropolar heat conduction parameter) are analyzed. The critical magnetic thermal Rayleigh number for the onset of instability is determined numerically for a sufficiently large value of the buoyancy magnetization parameter M ₁. The principle of exchange of stabilities is found to be true for the micropolar fluid heated from below in the absence of the micropolar viscous effect, the microinertia, and the solute gradients. The micropolar viscous effect, the microinertia, and the solute gradient introduce oscillatory modes, which are non-existent in their absence. Sufficient conditions for the non-existence of overstability are also obtained.     

Transient non-Darcy free convection between parallel vertical plates in a fluid saturated porous medium

Transient non-Darcy free convection between two parallel vertical plates in a fluid saturated porous medium is investigated using the generalized momentum equation proposed by Vafai and Tien. The effects of porous inertia and solid boundary are considered in addition to the Darcy flow resistance. Exact solutions are found for the asymptotic states at small and large times. The large time solutions reveal that the velocity profiles are rather sensitive to the Darcy number Da when Da<1. It has also been found that boundary friction alters the velocity distribution near the wall, considerably. Finite difference calculations have also been carried out to investigate the transient behaviour at the intermediate times in which no similarity solutions are possible. This analytical and numerical study reveals that the transient free convection between the parallel plates may well be described by matching the two distinct asymptotic solutions obtained at small and large times.Nomenclature C empirical constant for the Forchheimer term - f velocity function for the small time solution - F velocity function for the large time solution - g acceleration due to gravity - Gr* micro-scale Grashof number - H a half distance between two infinite plates - K permeability - Nu Nusselt number - Pr Prandtl number - t time - T temperature - u, v Darcian velocity components - x, y Cartesian coordinates - effective thermal diffusivity - coefficient of thermal expansion - porosity - dimensionless time - similarity variable - dimensionless temperature - viscosity - kinematic viscosity - density - the ratio of heat capacities     

Mixed convection of an incompressible viscous fluid in a porous medium past a hot vertical plate

This paper analyses steady two-dimensional mixed convection of an imcompressible viscous fluid in a porous medium past a hot vertical plate. Assuming Darcy-Brinkman model for the flow in a porous medium, the boundary layer equations are integrated numerically to obtain the non-similar solution for the velocity and temperature distribution for several values of the permeability and viscous dissipation parameters. It is shown that for a fixed value of Prandtl number Pr and dissipation parameter E, the skin-friction at the plate decreases with increase in the permeability parameter K₁. However for the same value or Pr and E, the heat transfer rate at the plate increases with increasing K₁. The dimensionlcss velocity and temperature functions in the flow are plotted for several values of E and K₁ with Pr = 0.73. It is also shown that for fixed values of K₁, and KPr, the skin-friction increases with increase in the dissipation parameter E.     

Measurements of thermally-induced convection in a model porous medium

The velocity field generated by thermal convection in a model porous medium is experimentally determined by means of both PIV and LDA techniques. Details of matching refraction index under non isothermal conditions are given. Fields are measured in the empty parallelepipedic cell and in a model medium made of parallel circular bundles. Results are in good agreement. Moreover, by an averaging technique, we are able to measure seeping velocity profiles.