Combined convection in non-Newtonian fluids along a nonisothermal vertical plate in a porous medium with lateral mass flux

 A boundary layer analysis has been presented for interaction of combined convection from vertical surface in a porous medium saturated with a power-low type non-Newtonian fluid with presence suction and injection. The transformed conservation laws are solved numerically for the case of variable surface heat flux conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented. The effect of fluid suction/injection on the heat transfer rate is discussed. Received on 21 December 1999     

Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium

An analysis is presented for the calculation of heat transfer due to free convective flow along a vertical plate embedded in a porous medium with an arbitrarily varying surface heat flux. By applying the appropriate coordinate transformations and the Merk series, the governing energy equation is expressed as a set of ordinary differential equations. Numerical solutions are presented for these equations which represent universal functions and several computational examples are provided.     

Effects of anisotropy on the free convection from a vertical plate embedded in a porous medium

The effects of anisotropy on the steady laminar boundary-layer free convection over a vertical impermeable surface are analysed by using the method of integral relations. If the permeability in the direction orthogonal to the plate is greater than the permeability along the plate, then there is an increase in the temperature field.     

Non-Darcy free convection in a thermally stratified porous medium along a vertical plate with variable heat flux

Free convection along an impermeable vertical plate embedded in a thermally stratified, fluid-saturated porous medium is analyzed. The wall heat flux is varied in a power-law form. The non-Darcian effects, such as solid-boundary viscous resistances, high-flow-rate inertia forces, near wall nonuniform porosity distribution and thermal dispersion, have been considered in the present study. Due to the variation of porosity in the near wall region, the stagnant thermal conductivity also varies accordingly. The nonsimilar system of transformed equations is solved with Keller's Box method. It is shown that the thermal stratification effect and the higher value of the exponent m can increase the local Nusselt number. Also the non-Darcian and thermal dispersion effects significantly influence the velocity and temperature profiles and local Nusselt number.     

Non-Darcy free convective flow along a vertical cylinder embedded in a porous medium with surface mass flux

The nonsimilar non-Darcian free-convection flow about a vertical cylinder with impermeable surface embedded in a saturated porous medium, where surface temperature of the cylinder varies as x^m, a power function of distance from the leading edge, has been studied by employing the implicit finite-difference method together with the Newton's quasilinearization technique. In the present investigation, effects of the surface mass flux together with the inertial effects on the rate of heat transfer at the surface, on the velocity distribution, and on the temperature distribution are shown graphically.     

Nonsimilar solutions for free convection from a vertical plate in porous media

A nonsimilar boundary layer analysis has been presented for the free convection along a vertical plate embedded in a fluid-saturated porous medium in the presence of surface mass transfer and internal heat generation. The transformed conservation laws are solved numerically for the cases of variable wall temperature and variable wall heat flux boundary conditions. Results are presented for the details of the velocity and temperature fields as well as Nusselt number. Received on 13 December 1996     

Steady laminar flow past a heated horizontal plate embedded in a saturated porous medium

In this paper we study the boundary layer equations for steady laminar flow past a heated horizontal plate embedded in a saturated porous medium by adopting the formulation of Chandrasekhara [3], Kolar and Sastri [7]. The velocity distribution and temperature distribution are determined by using the implicit Crank-Nicolson-Predictor-Corrector method of finite difference scheme [7] and [1]. With the help of a compute the distributions are estimated at both (i+1/2)th and (i+1)th levels and they are presented in tabular form. The curves for these distributions are plotted. We calculate the shear stress and skin friction at the wall and observe that the skin-friction directly depends upon the dimensions of the plate and inversely depends upon the Reynolds numberRe. The heat flux and the Nusselt number are evaluated. Further we observe that the Nusselt number depends upon the length of the porous plate.

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Stetige laminare Strömung über eine in einem gesättigten porösen Medium eingebettete horizontale Platte

Zusammenfassung Diese Untersuchung befaßt sich mit den Grenzschichtgleichungen für eine stetige laminare Strömung über eine beheizte Platte, die in ein gesättigtes poröses Medium eingebettet ist, mittels des Formalismus von Chandrasekhara [3], Kolar und Sastri [7]. Die Geschwindigkeits- und Temperaturverteilung wurden unter Benutzung der impliziten Crank-Nicolson-Korrekturmethode des Finiten-Elemente-Schemas bestimmt. Die (i+1/2). und die (i+1). Ebene der Verteilungen wurden mit Computer-Hilfe berechnet und in Tabellenform dargestellt. Die Graphen der Temperatur- und Geschwindigkeitsverteilung wurden ausgeplottet. Die Schubspannungen und die Oberflächenreibung an der Wand wurden berechnet und es konnte festgestellt werden, daß die Oberflächenreibung direkt von der Größe der Platte abhängt und umgekehrt proportional der Reynolds-ZahlRe ist. Der Wärmestrom und die Nusselt-Zahl wurden bestimmt. Weiterhin konnte festgestellt werden, daß die Nusselt-Zahl von der Länge der porösen Platte abhängt.

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Nomenclature C _p specific heat of the convective fluid - D skin friction - k permeability of the porous medium - k _f thermal conductivity of the fluid - k _m the coefficient of thermal conductivity of the porous medium - k _s the conductivity of the solid matrix - N(x) Nusselt number - q(x) specific heat flux - Re local Reynolds number - T temperature - T ₀ temperature of the free stream - T _w temperature of the plate - u velocity in thex-direction - u ₀ velocity of the free stream - V velocity iny-direction - x coordinate axis along the plate - y coordinate axis normal to the plate Greek symbols thermal diffusivity - thickness of the velocity boundary layers in thex direction - thickness of the velocity boundary layer in they-direction - the porosity of the medium - dimensionless variable - kinematic viscosity of the fluid - density of the fluid - shear stress     

Natural convection of nanofluid over vertical plate embedded in porous medium: prescribed surface heat flux

The aim of the present paper is to analyze the natural convection heat and mass transfer of nanofluids over a vertical plate embedded in a saturated Darcy porous medium subjected to surface heat and nanoparticle fluxes. To carry out the numerical solution, two steps are performed. The governing partial differential equations are firstly simplified into a set of highly coupled nonlinear ordinary differential equations by appropriate similarity variables, and then numerically solved by the finite difference method. The obtained similarity solution depends on four non-dimensional parameters, i.e., the Brownian motion parameter (N _b), the Buoyancy ratio (N _r), the thermophoresis parameter (N _t), and the Lewis number (Le). The variations of the reduced Nusselt number and the reduced Sherwood number with N _b and N _t for various values of Le and N _r are discussed in detail. Simulation results depict that the increase in N _b, N _t, or N _r decreases the reduced Nusselt number. An increase in the Lewis number increases both of the reduced Nusselt number and the Sherwood number. The results also reveal that the nanoparticle concentration boundary layer thickness is much thinner than those of the thermal and hydrodynamic boundary layers.     

Non-Darcy mixed convection from a vertical plate in saturated porous media-variable surface heat flux

Nonsimilarity solutions for non-Darcy mixed convection from a vertical impermeable surface embedded in a saturated porous medium are presented for variable surface heat flux (VHF) of the power-law form. The entire mixed convection region is divided into two regimes. One region covers the forced convection dominated regime and the other one covers the natural convection dominated regime. The governing equations are first transformed into a dimensionless form by the nonsimilar transformation and then solved by a finite-difference scheme. Computations are based on Keller Box method and a tolerance of iteration of 10⁻⁵ as a criterion for convergence. Three physical aspects are introduced. One measures the strength of mixed convection where the dimensionless parameter Ra^* _x /Pe^3/2 _x characterizes the effect of buoyancy forces on the forced convection; while the parameter Pe _x /Ra^*2/3 _x characterizes the effect of forced flow on the natural convection. The second aspect represents the effect of the inertial resistance where the parameter K′U _∞/ν is found to characterize the effect of inertial force in the forced convection dominated regime, while the parameter (K′U _∞/ν)(Ra^*2/3 _x /Pe _x ) characterizes the effect of inertial force in the natural convection dominated regime. The third aspect is the effect of the heating condition at the wall on the mixed convection, which is presented by m, the power index of the power-law form heating condition. Numerical results for both heating conditions are carried out. Distributions of dimensionless temperature and velocity profiles for both Darcy and non-Darcy models are presented. Received on 26 May 1997     

Mixed convection of non-newtonian fluids from a vertical plate embedded in a porous medium

This paper investigates mixed free and forced convection of non-Newtonian fluids from a vertical isothermal plate embedded in a homogenous porous medium. A mathematical model is developed based on the modified Darcy's law and boundary-layer approximations, and the exact similarity solution is obtained as well as an integral solution. These two solutions agree within 3% for aiding flows and 10% for opposing flows. It is found that, non-Newtonian characteristics of fluids have appreciable influences on velocity profiles, temperature distributions and flow regimes.     

An integral treatment for non-Darcy free convection over a vertical flat plate and cone embedded in a fluid-saturated porous medium

An integral treatment was proposed for analysis of non-Darcy free convection over a vertical flat plate and cone within a fluid-saturated porous medium. A flexible one-parameter family of third order polynomials was employed to cope with vast changes in the velocity and temperature profiles encountered in the Darcy flow limit through to the Forchheimer flow limit. Zero curvature requirement for the temperature profile at the wall was exploited as an auxiliary relation to determine the shape parameter. Comparison of the approximate results with the exact solution reveals a high performance of the present integral procedure for heat transfer rat prediction.     

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     

Asymptotic and numerical transient analysis of the free convection cooling of a vertical plate embedded in a porous medium

This paper analyzes the cooling process of a vertical thin plate originated by a fluid-saturated porous medium, taking into account the effects of both longitudinal and transversal heat conduction in the plate. Due to the finite thermal conductivity of the plate, a longitudinal temperature gradient arises within it, which prevents any similarity solution in the boundary layer, changing the mathematical character of the problem from parabolic to elliptic, for large values of the suitable Rayleigh number. The energy balance equations are reduced to a system of two differential equations with two parameters: the nondimensional plate thermal conductivity and the aspect ratio of the plate . In order to obtain the evolution of the temperature of the plate as a function of time and position, the coupled balance equations are integrated numerically for several values of the parameters, including the cases of very good and poor conducting plates. The results obtained, are compared with an asymptotic analysis based on the multiple scales technique carried out for the case of a very good conducting plate. There is at the beginning a fast transient in nondimensional time scale of order followed by a slow nondimensional time scale of order unity, which gives the evolution of the cooling process. Good agreement is achieved even for values of the conduction parameter of order unity. The asymptotic solution allows us to give closed form analytical solution for the plate temperature evolution in time and space. The overall thermal energy of the plate decreases faster for smaller values of .     

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     

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.     

An integral method for free convection from a vertical heated surface in a thermally stratified porous medium

The Kárman-Pohlhausen integral procedure was exploited to solve the problem of free convection from a vertical heated surface imbedded in a thermally stratified porous medium. Upon considering the Darcy law, the energy integral relation, and the auxiliary relation associated with the curvature of the temperature profile at the wall, a closed-form expression for the Nusselt number was reduced for the case where the distributions of the wall temperature and ambient temperature vary according to power-functions of distance. Results on an isothermal plate are found to be in close agreement with those from the exact solution. Effects of the thermal stratification on local heat transfer rates are also discussed.

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Freie Konvektion von einer beheizten, vertikalen Fläche in der thermischen Schichtung eines porösen Mediums

Zusammenfassung Die Kárman-Pohlhausen-Integration wurde angewandt, um das Problem der freien Konvektion von einer vertikalen Fläche, die in ein thermisch geschichtetes poröses Medium eingebettet ist, zu lösen. Unter Hinzuziehung des Darcyschen Gesetzes der integralen Energiebeziehung und einer Hilfsbeziehung, die mit der Krümmung des Temperaturprofiles an der Wand zusammenhing, wurde eine geschlossene Form für die Nusselt-Zahl gefunden, jeweils bezogen auf den Fall, wenn die Verteilung der Wandtemperatur und der Umgebungstemperatur entsprechend eines Exponentialansatzes mit dem Abstand abnimmt. Ergebnisse, die für eine isotherme Platte mit dieser Methode berechnet wurden, sind in guter Übereinstimmung mit denen der exakten Lösung. Es werden auch Einflüsse der thermischen Schichtung auf den örtlichen Wärmeübergang diskutiert.

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Nomenclature A, B, C, D shape factors - f profile function - g acceleration due to gravity - K permeability - m parameter associated with thermal stratification - n power-law exponent of the wall temperature - Nux local Nusselt number - Pr Prandtl number - Rax local Rayleigh number - T temperature - u, v velocity components inx andy directions - x, y boundary layer coordinates Greek symbols shape parameter - \ coefficient of thermal expansion - boundary layer thickness - similarity variable     

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.     

Subcooled forced convection film boiling over a vertical flat plate embedded in a fluid-saturated porous medium

The problem of subcooled forced convection film boiling on a vertical flat plate embedded in a porous medium was attacked exploiting similarity transformations on the governing equations and boundary conditions in both vapor and liquid layers. Similarity solutions were obtained to investigate the effects of the vapor super-heating and liquid subcooling. The heat transfer groupingNu _x /Ra _x ^1/2 is expressed in terms of a function of three parameters associated with the degree of liquid subcooling (Sub), the degree of vapor superheating (Sup) and the vapor buoyancy effect relative to the liquid forced convection effect (R). It is found that the level ofNu _x /Ra _x ^1/2 increases asSup orR decreases and asSub increases. Furthermore, asymptotic expressions were reduced considering the physical limiting conditions, namely, thin and thick vapor films.     

Mixed convection in non-Newtonian fluids along a vertical plate with variable surface heat flux in a porous medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law type non-Newtonian fluids along a vertical plate with power-law surface heat flux distribution. The mixed convection regime is divided into two regions, namely, the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate. Received on 13 October 1998