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     

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     

Effect of variable viscosity on non-Darcy free or mixed convection flow on a vertical surface in a fluid saturated porous medium

This paper analyzes the variable viscosity effects on non-Darcy free or mixed convection flow on a vertical surface in a fluid saturated porous medium. The viscosity of the fluid is assumed to be a inverse linear function of temperature. Velocity and heat transfer are found to be significantly affected by the variable viscosity parameter, Ergun number, Peclet number or Rayleigh number.     

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.     

Variable porosity and thermal dispersion effects on vortex instability of a horizontal natural convection flow in a saturated porous medium

A numerical analysis is made to analyze the variable porosity and thermal dispersion effects on the vortex mode of instability of a horizontal natural convection boundary layer flow in a saturated porous medium. The porosity of the medium is assumed to vary exponentially with distance from the wall. In the base flow, the governing equations are solved by using a suitable variable transformation and employing an implicit finite difference Keller Box method. The stability analysis is based on the linear stability theory and the resulting eigenvalue problem is solved by the local similarity approximations. The results indicate that both effects increase the heat transfer rate. In addition, the thermal dispersion effect stabilizes the flow to the vortex mode of disturbance, while the variable porosity effect destabilizes it.

     

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.     

Non-darcy natural convection on a vertical cylinder in a saturated porous medium

The steady free convection boundary layer flow of non-Darcy fluid along an isothermal vertical cylinder embedded in a saturated porous medium using the Ergun model has been studied. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme developed by Keller. It is found that the heat transfer is strongly affected by the modified Grashof number which characterizes the non-Darcy fluid, and the curvature parameter. Also the heat transfer is found to be more than that of the flat plate.     

Effects of thermal dispersion and stratification on combined convection on a vertical surface embedded in a porous medium

The effects of thermal dispersion and thermal stratification on mixed convection about a vertical surface in a porous medium are studied. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations are solved on the basis of the local similarity approximation. The results indicate that both dispersion and stratification effects have considerable influence on the heat transfer rate.     

Plane waves in a semi-infinite fluid saturated porous medium

The field equations governing the propagation of waves in an incompressible liquid-saturated porous medium are investigated and a general solution is presented. It has been revealed that coupled longitudinal and transverse waves propagate in the porous medium. The propagation of transverse waves in the fluid phase is completely due to the interaction between the solid and fluid phases. The dispersion relationship and attenuation features are discussed. Unlike other investigations, all explicit forms of the arguments are derived. The reflection of the plane harmonic waves at the plane, traction-free boundary, which shows the influence of the dissipation on the velocity, and the attenuation coefficients of the reflected waves is studied. It is of interest that pore pressure is produced in the process of reflection, even in the case of the incidence of transverse waves.     

Effects of thermal dispersion and stratification on non-darcy mixed convection from a vertical plate in a porous medium

A boundary layer analysis is presented for the mixed convection from a vertical plate embedded in a porous medium. The effects of thermal dispersion and stratification on the flow and temperature fields are investigated. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations were solved on the basis of the local similarity approach. Received on 12 February 1998     

Free convection in a saturated porous medium beyond the similarity solution

The non-similar problems associated with a non-isothermal vertical flat plate embedded in a fluid-saturated porous medium were considered to assess the performances of the two distinctive boundary layer solution methods, namely, the local similarity solution and the integral method. The results generated from these two approximate solution methods are compared against the results from a two-point finite difference and those based on a Merk-type series expansion. Comparison of the results reveals that both integral and local similarity methods perform excellently. Especially, the accuracy acquired by the local similarity solution is so high that the difference between the results from the local similarity solution and those from the two-point finite difference and local non-similarity solution methods is hardly discernible for the case of monotonic increasing wall temperature.     

Convective cooling/heating induced thermal stresses in a fluid saturated porous medium undergoing local thermal non-equilibrium

Local thermal non-equilibrium (LTNE) may have profound effects on the pore pressure and thermal stresses in fluid saturated porous media under transient thermal loads. This work investigates the temperature, pore pressure, and thermal stress distributions in a porous medium subjected to convective cooling/heating on its boundary. The LTNE thermo-poroelasticity equations are solved by means of Laplace transform for two fundamental problems in petroleum engineering and nuclear waste storage applications, i.e., an infinite porous medium containing a cylindrical hole or a spherical cavity subjected to symmetrical thermo-mechanical loads on the cavity boundary. Numerical examples are presented to examine the effects of LTNE under convective cooling/heating conditions on the temperature, pore pressure and thermal stresses around the cavities. The results show that the LTNE effects become more pronounced when the convective heat transfer boundary conditions are employed. For the cylindrical hole problem of a sandstone formation, the thermally induced pore pressure and the magnitude of thermal stresses are significantly higher than the corresponding values in the classical poroelasticity, which is particularly true under convective cooling with moderate Biot numbers. For the spherical cavity problem of a clay medium, the LTNE effect may become significant depending on the boundary conditions employed in the classical theory.     

Horizontal boundary-layer natural convection in a porous medium saturated with a gas

Boundary-layer analysis is performed for free convection flow over a hot horizontal surface embedded in a porous medium saturated with a gas of variable properties. The variable gas properties are accounted for via the assumption that thermal conductivity and dynamic viscosity are proportional to temperature. A similarity solution is shown to exist for the case of constant surface temperature. Numerical results for the stream function, horizontal velocity, and temperature profiles within the boundary layer as well as for the mass of entrained gas, surface slip velocity, and heat transfer rate at different values of the wall-temperature parameter are presented. Asymptotic solutions for large heating are also available to support the numerical work.     

Free convection above a horizontal circular disk in a saturated porous medium

The boundary-layer flow above a horizontal impermeable circular disk embedded in a saturated porous medium is considered in the cases both when the disk is held at a constant temperature above ambient and when heat is supplied to the convective fluid by the disk at a constant rate. Series solutions are obtained based on the flat plate solution, which holds at the edge of the disk, as the leading order terms. These series solutions can then be used to describe the flow nearly all the way across the disk. A simple approximate solution, based on an integrated form of the energy equation, is also obtained and is shown, for the constant wall temperature case, to be useful in indicating how the solution behaves near the centre of the disk. The solutions asr0, wherer measures distance from the centre is discussed in both cases, and it is shown that the solution develops a singularity with the boundary layer having a thickness of 0[(–logr)^1/2].

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Freie Konvektion über einer waagerechten Kreisscheibe in einem gesättigten porösen Medium

Zusammenfassung Es wird die Grenzschicht über einer waagerechten undurchlässigen Kreisscheibe, die in einem gesättigten porösen Medium eingebettet ist, untersucht. Zwei Möglichkeiten werden betrachtet: bei der einen ist die Oberflächentemperatur der Scheibe konstant, aber größer als die der Umgebung, bei der anderen ist die Scheibe gleichförmig beheizt. Es werden Lösungen in Form von Reihenentwicklungen erzielt, wobei die ersten Glieder den Lösungen einer ebenen Platte entsprechen, die an dem Rand der Scheibe gültig sind. Danach werden diese Lösungen zur Beschreibung der Bewegung der Flüssigkeit über einen großen Bereich der Scheibe benutzt. Eine einfache Lösung für den Fall der konstanten Wandtemperatur wird durch Integration der Energiegleichung erhalten und zur Beschreibung des Verhaltens der Lösung in der Nähe der Scheibenmitte verwendet. Man untersucht die Lösung fürr0 in beiden betrachteten Fällen, wobeir der Abstand von der Scheibenmitte ist. Bei einer Dicke von 0[(–logr)^1/2] der Grenzschicht weist die Lösung eine Singularität auf.

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Nomenclature a radius of disk - g acceleration due to gravity - K permeability of the porous medium - k thermal conductivity - q prescribed wall heat flux - Q (non-dimensional) heat transfer coefficient - r, coordinate measuring distance from the centre of the disk - R _a,R _a ^* Rayleigh number - T temperature of the convective fluid - T ₀ ambient temperature - T ₁ prescribed wall temperature - u Darcy's law velocity in ther-direction - V _w (non-dimensional) wall velocity - w Darcy's law velocity in thez-direction - x coordinate measuring distance from the edge of the disk - z, coordinate measuring distance normal to the disk - equivalent thermal diffusivity - coefficient of thermal expansion - non-dimensional temperature - _w (non-dimensional) wall temperature - viscosity of the convective fluid - stream-function - stream function at the edge of the boundary layer     

On the similarity solutions for free convection in a saturated porous medium adjacent to impermeable horizontal surfaces

The similarity solutions for free convection about a horizontal surface in a saturated porous medium when the wall temperature is prescribed as a power law of distance from the origin, i.e. proportional tox ^m, are considered. Solutions valid form1 are obtained. Numerical solutions are obtained which reveal that there is a valuem ₀ such that a solution of the similarity equations is possible only form>m ₀. It is shown thatm ₀=–2/5 and the nature of the singularity is discussed and an asymptotic expansion is obtained.

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Ähnlichkeitslösungen für eine freie Konvektion in einem gesättigten porösen Medium, das an eine undurchlässige, horizontale Oberfläche grenzt

Zusammenfassung Die Ähnlichkeitslösungen sind für eine freie Konvektion über eine horizontale Oberfläche in einem gesättigten porösen Medium in Betracht gezogen worden, wenn die Wandtemperatur beschrieben ist als eine Potenzfunktion des Abstandes vom Ursprung, z. B. proportional zux ^m.Gültige Lösungen sind fürm1 bestimmt. Numerische Lösungen sind bestimmt worden, die offenbaren, daß es einen Wertm ₀ geben muß, so daß eine Lösung von ähnlichen Lösungen nur fürm>m ₀ möglich ist. Es wird gezeigt, daßm ₀=–2/5 und daß die Beschaffenheit der Singularität besprochen und die asymptotische Ausdehnung bestimmt worden ist.

     

Investigation of convection in a horizontal cylindrical layer of a saturated porous medium

The structures of the convective motions and the nature of the heat transfer in a horizontal cylindrical layer are studied numerically for the Forchheimer model of a porous medium in the Boussinesq approximation. New asymmetric solutions of the equations of convection flow through a porous medium are found. Their development, domains of existence, and stability are investigated. One consists of a multivortex structure with asymmetric vortices in the near-polar region. Another asymmetric solution is realized at large Grashof numbers in the form of a convective plume deflected from the vertical. The threshold Grashof number of formation of the asymmetric motions depends on the Prandtl number and the cylindrical layer thickness.     

Pressure waves in a porous medium saturated with a gassy fluid

Experimental data on the evolution of pressure waves in a consolidated porous medium saturated with a gassy fluid are obtained. These data are generalized on the basis of a theoretical analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 85–92, July–August, 1987.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Influence of variable permeability on combined free and forced convection flow past a semi-infinite vertical plate in a saturated porous medium

Combined free and forced convection flow of viscous incompressible fluid past a semi-infinite vertical plate embedded in a porous medium incorporating the variation of permeability and thermal conductivity are studied. Similarity solutions are obtained, for two cases namely uniform permeability (UP) and variable permeability (VP). Velocity and temperature profiles are shown graphically and the numerical values of the skin friction and the rate of heat transfer are entered in tabular form. The effects of the parameters Gr/Re ² (Gr – Grashof number, Re – the Reynolds number), (coefficient of viscositys of the fluid and porous medium), (the Darcy number), * (ratio of thermal conductivity of the solid to the liquid), Pr (the Prandtl number) and E (the Eckert number) on the flow field are discussed.