Non-darcy double-diffusive mixed convection from heated vertical and horizontal plates in saturated porous media

The non-darcy mixed convection flows from heated vertical and horizontal plates in saturated porous media have been considered using boundary layer approximations. The flows are considered to be driven by multiple buoyancy forces. The similarity solutions for both vertical and horizontal plates have been obtained. The governing equations have been solved numerically using a shooting method. The heat transfer, mass transfer and skin friction are reduced due to inertial forces. Also, they increase with the buoyancy parameter for aiding flow and decrease for the opposing flow. For aiding flow, the heat and mass transfer coefficients are found to approach asymptotically the forced or free convection values as the buoyancy parameter approaches zero or infinity.     

Natural convection from a vertical cylinder in a thermally stratified porous medium

The problem of non-Darcy natural convection adjacent to a vertical cylinder embedded in a thermally stratified porous medium has been analyzed. Nonsimilarity solutions are obtained for the case that the ambient temperature increases linearly with height of the cylinder. A generalized flow model was used in the present study to include the effects of the macroscopic viscous term and the microscopic inertial force. Also, the thermal dispersion effect is considered in the energy equation. Thus, the main aim of this work is to examine the effects of thermal stratification and non-Darcy flow phenomena on the free convection flow and heat transfer characteristics. It was found that the present problem depends on six parameters, namely, the local thermal stratification parameter ξ, the boundary effect parameter Bp, the modified Grashof number Gr^*, wall temperature exponent m, the curvature parameter ω, and the modified Rayleigh number based on pore diameter Ra _d . The impacts of these governing parameters on the local heat transfer parameter are discussed in great detail. Also, representative velocity and temperature profiles are presented at selected values of the thermal stratification parameter. In general, the local heat transfer parameter is increased with increasing the values of m, ω, and Ra _d ; while it is decreased with increasing the values of ξ, Bp, and Gr^*. Received on 19 May 1998     

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.     

Darcian mixed convection plumes along vertical adiabatic surfaces in a saturated porous medium

The mixed convection flow due to a line thermal source embedded at the leading edge of an adiabatic vertical plane surface immersed in a saturated porous medium has been studied. Both weakly and strongly buoyant plume regimes have been considered. The cases of buoyancy assisting and buoyancy opposing flow conditions have been incorporated in the analysis. The results are presented for the entire range of buoyancy parameter from the pure forced convection (ξ=0) to the pure free convection (ξ → ∞@#@) regimes. For buoyancy-assisting flow, the wall temperature and the velocity at the wall increase as the plume strength increases. However, they all decrease as the free-stream velocity increases. For buoyancyopposing flow, the temperature at the wall increases as the strength of the plume increases but velocity at the wall decreases.     

Non-Darcy natural convection from a vertical wavy surface in a porous medium

We examine the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection induced by a vertical heated surface embedded in a fluid-saturated porous medium. We consider the boundary-layer regime where the Darcy-Rayleigh number, Ra, is very large, and assume that the surface waves have O(1) amplitude and wavelength. The resulting boundary-layer equations are found to be nonsimilar only when the surface is nonuniform and inertia effects are present; self-similarity results when either or both effects are absent. Detailed results for the local and global rates of heat transfer are presented for a range of values of the inertia parameter and the surface wave amplitude.     

Free convection in a saturated porous medium adjacent to a non-isothermal vertical impermeable wall

Boundary layer analysis is performed for free convection in a saturated porous medium adjacent to non-isothermal vertical impermeable surfaces. The impermeable surface temperature is assumed to be an arbitrary function of the distance along the surface. The solutions are obtained in the form of perturbations to the isothermal case. Using the differentials of the wall temperature, which are functions of distance along the surface, as perturbation elements, universal functions are derived. These universal functions can be used to estimate the heat transfer to any type of wall temperature variation. Solutions for some specialized wall temperature variations are derived using these universal functions and are compared with the solutions that are available in the literature. The agreement is found to be good. The case of uniform wall heat flux turns out to be a special case of non-isothermal wall solution.Es wurde eine Untersuchung der Grenzschicht bei freier Konvektion in einem gesättigten porösen Medium, das an eine nicht isotherme undurchlässige Oberfläche angrenzt, durchgeführt. Die Temperatur der undurchlässigen Oberfläche wurde als beliebige Ortsfunktion der Oberfläche angenommen. Lösungen werden in Form von Störungen des isothermen Falles erhalten. Unter Benutzung der Differentiale der Wandtemperatur, welche Ortsfunktionen der Oberfläche sind, werden als Störfunktion universelle Funktionen erhalten. Diese universellen Funktionen können benutzt werden, um die Wärmeübertragung für beliebige Variationen der Wandtemperatur zu bestimmen. Für einige spezielle Wandtemperaturen werden Lösungen unter Benutzung dieser universellen Funktionen abgeleitet und mit Lösungen aus der Literatur verglichen; gute Übereinstimmung kann festgestellt werden. Der Fall der einheitlichen Wandwärmestromdichte stellt sich als Spezialfall der nicht isothermen Lösung heraus.     

Conjugate free convection over a vertical slender hollow cylinder embedded in a porous medium

A numerical study of the steady conjugate free convection over a vertical slender, hollow circular cylinder with the inner surface at a constant temperature and embedded in a porous medium is reported. The governing boundary layer equations for the fluid-saturated porous medium over the cylinder along with the one-dimensional heat conduction equation for the cylinder are cast into dimensionless form, by using a non-similarity transformation. The resulting non-similarity equations with their corresponding boundary conditions are solved by using the Keller box method. Emphasis is placed on the effects caused by the wall conduction parameter, p, and calculations have covered a wide range of this parameter. Heat transfer results including the temperature profiles, the interface temperature profiles and the local Nusselt number are presented. Received on 17 November 1997     

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     

Freezing around a vertical cylinder immersed in porous media incorporating the natural convection effect

The present paper analyses the freezing heat transfer on a cold cylinder immersed in a superheated liquid-saturated porous medium. The main emphasis of this paper is to clarify the effect of natural convection in the liquid phase on the freezing process. A mathematical model was developed, based on several reasonable assumptions for fluid flow in a liquid phase. The governing equations were numerically solved. Numerical results were reported for the thickness distribution of the frozen layer and the heat transfer coefficients at the cold wall and interface. The influences of various parameters such as Rayleigh number, liquid superheat, Stefan number and the dimensionless radius of a cylinder were demonstrated.Die vorliegende Arbeit untersucht den Wärmeübergang bei Erstarrung an einem kalten Zylinder, der in ein überhitztes, flüssigkeitsgesättigtes und poröses Medium getaucht wurde. Das Hauptanliegen dieser Arbeit ist es, den Einfluß der freien Konvektion, in einer flüssigen Phase, auf den Erstarrungsvorgang zu klären. Es wurde ein mathematisches Verfahren entwickelt, das auf mehreren vernünftigen Annahmen für die Fluid-Strömung in einer flüssigen Phase basiert. Die bestehenden Gleichungen sind numerisch gelöst worden. Für die Verteilung der gefrorenen Schichtdicke und für die Wärmeübergangskoeffizienten der kalten Wand und Grenzfläche sind numerische Ergebnisse ermittelt worden. Die Einflüsse der unterschiedlichen Parameter wie die Raleighzahl, die Flüssigkeitsüberhitzung, die Stefanzahl und den dimensionslosen Radius des Zylinders werden dargestellt.     

Stability of a fluid-saturated porous medium contained in a vertical cylinder heated from below by forced convection

A linear stability analysis determining the critical Rayleigh number R _c for onset of convection in a bounded vertical cylinder containing a fluid-saturated porous medium is performed for insulated sidewalls, isothermal top surface, and bottom surface heated by forced convection. This Newtonian heating of the bottom surface involves a Biot number Bi that allows consideration of the continuum of boundary conditions ranging from constant heat flux, with global minimum R _min=27.096 found as Bi→0, to isothermal, with global minimum R _min=4π² found as Bi→ ∞. In both cases and for most cylinder aspect ratios, incipient convection sets in as an asymmetric mode, though islands of aspect ratio exist where the onset mode is symmetric. Sample three-dimensional renderings of disturbance temperature distributions showing preferred modes at onset of convection for fixed Bi are provided and an analytical fit to R _min as a function of Bi is given.     

On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid

The stability of the conduction regime of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and Oldroyd-B fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for power-law fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter \(\Lambda _2 \) portrays stabilizing influence on the system while increasing stress relaxation parameter \(\Lambda _1\) displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to Oldroyd-B fluid.     

Nonlinear hydro-magnetic convection at a permeable cylinder in a porous medium

We consider the longitudinal steady nonlinear hydromagnetic convection flow over a permeable vertical cylinder in a porous medium. We assume that both the mainstream velocity at the outer edge of the boundary layer and the surface temperature of the cylinder vary linearly with axial distance from the leading edge, and extend the existing literature by including the nonlinear density temperature variation, magnetic field, and heat source/sink.     

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.     

Natural convection in a vertical strip immersed in a porous medium

In this work, the conjugated heat transfer characteristics of a thin vertical strip of finite length, placed in a porous medium has been studied using numerical and asymptotic techniques. The nondimensional temperature distribution in the strip and the reduced Nusselt number at the top of the strip are obtained as a function of the thermal penetration parameter s, which measures the thermal region where the temperature of the strip decays to the ambient temperature of the surrounding fluid. The numerical values of this nondimensional parameter permits to classify the different physical regimes, showing different solutions: a thermally long behaviour, an intermediate transition and a short strip limit.     

Improved perturbation solutions for laminar natural convection on a vertical cylinder

The method of extended perturbation series is applied to solve for laminar natural convection from an isothermal, thin vertical cylinder. The series in terms of the transverse curvature parameterξ extended to five terms and is subsequently improved by applying the Shanks transformation twice. The validity of the solution is extended up toξ=10 and possibly even beyond. Up toξ=10, the results for wall shear as well as the local and average Nusselt numbers agree very closely with those of local nonsimilarity and finite difference solutions. The ease of computation coupled with high accuracy makes the present approach far more attractive than the currently popular local nonsimilarity and finite difference methods. Its success with the present problem should motivate applications to a host of nonsimilar boundary layer flows.     

Free convection in a saturated porous medium adjacent to a vertical impermeable wall subjected to a non-uniform heat flux

Boundary layer analysis is performed for free convection in a fluid-saturated porous medium adjacent to a vertical impermeable wall subjected to a non-uniform heat flux. The wall heat flux is assumed to be an arbitrary function of the distance along the surface. The solutions are obtained in the form of perturbations to the uniform heat flux case. Using the differentials of the wall heat flux, which are functions of the distance along the surface, as perturbation elements, universal functions are obtained. These universal functions can be used to estimate the heat transfer for any type of wall heat flux variation. For the wall heat flux variation as a power function of the distance from the origin, the solutions obtained by using these universal functions have been compared with those obtained by similarity analysis and the agreement is found to be good. Further, solutions are presented for wall heat flux varying exponentially and sinusoidally but comparison could not be drawn due to non-availability of solutions in the literature.     

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     

Effect of weak rotation on natural convection in a horizontal porous cylinder

Natural convection in a fluid saturated porous medium confined in a horizontal circular cylinder and rotating about its axis, with isothermal boundary conditions and uniform internal heat sink, is studied by both numerical and perturbation methods. No symmetry with respect to the vertical diameter is expected for the flow and temperature fields and the whole region must be involved in the computation. Only the weak rotation regime, for which the centrifugal force is negligible compared to gravity, is considered. Governing equations for the two-dimensional flow field are solved in both rotating and non-rotating coordinate systems. Results indicate that rotation significantly decreases the radial amplitude of fluid particle trajectories in the radial direction and thus reduces the overall heat transfer.     

Mixed convection about a non-isothermal vertical surface in a porous medium

The problem of mixed convection about non-isothermal vertical surfaces in a saturated porous medium is analysed using boundary layer approximations. The analysis is made assuming that the surface temperature varies as an arbitrary function of the distance from the origin. A perturbation technique has been applied to obtain the solutions. Using the differentials of the wall temperature, which are functions of distance along the surface, as perturbation elements, universal functions are derived for various values of the governing parameter Gr/Re. Both aiding and opposing flows are considered. The universal functions obtained can be used to estimate the heat transfer and fluid velocity inside the boundary layer for any type of wall temperature variation. As a demonstration of the method, heat transfer results have been presented for the case of the wall temperature varying as a power function of the distance from the origin. The results have been studied for various combinations of the parameters Gr/Re and the power index m, taking both aiding and opposing flows into consideration. On comparing these results with those obtained by a similarity analysis, the agreement is found to be good.