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.     

Free convection from a truncated cone subject to constant wall heat flux in a micropolar fluid

The paper studies the problem of free convection about a vertical frustum of a cone in a micropolar fluid. It is assumed that the flow is laminar, steady and the wall is subjected to a constant heat flux and the angle of the frustum of the cone is large enough so that the transverse curvature effects are negligible. Under these assumptions, the governing boundary layer equations subject to appropriate boundary conditions are transformed into a set of equations of parabolic type, that are solved using the local non-similarity method. The space of parameters contains the Prandtl number Pr, the micropolar parameter Δ and the microrotation parameter n. Numerical solutions are obtained by varying Pr from 6.7 to 100, Δ from 0 (Newtonian fluid) to 2 and considering two values of n with physical significance (0 and 0.5). Flow and heat transfer characteristics are determined and are shown in graphs. The results are discussed and compared at some extent with those reported by the present author in a previous study (Postelnicu in Int. J. Eng. Sci. 44:672–682, 2006) on the isothermal case.     

Free convection on an arbitrarily inclined plate with uniform surface heat flux

This paper proposed a proper inclination parameter and transformation variables for the analysis of free convection from an inclined plate with uniform surface heat flux to fluids of any Prandtl number. Very accurate numerical results and a simple correlation equation are obtained for arbitrary inclination from the horizontal to the vertical and for 0.001 Pr. Maximum deviation between the correlated and calculated data is less than 1.2%.

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Freie Konvektion an einer beliebig geneigten Platte mit erheblicher Wärmestromdichte an der Oberfläche

Zusammenfassung Für die Berechnung von freier Konvektion von Fluiden mit beliebiger Prandtl-Zahl an einer geneigten Platte mit einheitlicher Wärmestromdichte an der Oberfläche werden ein zweckmäßiger Neigungsparameter und Transformationsvariablen eingeführt. Sehr genaue numerische Ergebnisse und eine einfache Korrelationsgleichung wurden für beliebige Neigungen zwischen der Horizontalen und der Vertikalen und für 0.001Pr erhalten. Die größte Abweichung zwischen Korrelations- und berechneten Daten liegt bei weniger als 1.2%.

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Nomenclature f reduced stream function - g gravitational acceleration - h local heat transfer coefficient - k thermal conductivity - Nu local Nusselt number - p static pressure difference - Pr Prandtl number - q _w wall heat flux - Ra* modified local Rayleigh number,g(q _w x/k)x ³/ - T fluid temperature - T temperature of ambient fluid - u velocity component inx-direction - v velocity component iny-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - (Ra* |sin|)^1/5/( Ra* cos)^1/6 - ( Ra* cos)^1/6+( Ra*|sin|)^1/5 - (y/x) - dimensionless temperature, (T–T )/(q _w x/k) - kinematic viscosity - [1+( Ra* cos)^1/6/( Ra*|sin|)^1/5]^–1 - density of fluid - Pr/(1+Pr) - _w wall shear stress - angle of inclination measured from the horizontal - stream function - dimensionless static pressure difference, p x ²/ ⁴     

Laminar flow of micropolar fluid in rectangular microchannels

Compared with the classic flow on macroscale, flows in microchannels have some new phenomena such as the friction increase and the flow rate reduction. Papautsky and co-workers explained these phenomena by using a micropolar fluid model where the effects of micro-rotation of fluid molecules were taken into account. But both the curl of velocity vector and the curl of micro-rotation gyration vector were given incorrectly in the Cartesian coordinates and then the micro-rotation gyration vector had only one component in the z-direction. Besides, the gradient term of the divergence of micro-rotation gyration vector was missed improperly in the angular moment equation. In this paper, the governing equations for laminar flows of micropolar fluid in rectangular microchannels are reconstructed. The numerical results of velocity profiles and micro-rotation gyrations are obtained by a procedure based on the Chebyshev collocation method. The micropolar effects on velocity and micro-rotation gyration are discussed in detail.The project was supported by the National Natural Science Foundation of China (10472054). The English text was polished by Boyi Wang.     

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.     

Combined heat and mass transfer by laminar natural convection from a vertical plate with uniform heat flux and concentration

This paper studies combined heat and mass transfer by laminar natural convection from a vertical plate maintained with uniform surface heat flux and species concentration. Very accurate finite-difference solutions of a set of nonsimilarity equations have been obtained for most practical gaseous solutions (Pr?=?0.7, 0.21 ≤ Sc ≤ 2.1) and aqueous solutions (Pr?=?7, 140 ≤ Sc ≤ 1400). Variations of heat and mass transfer rates with buoyancy ratio and Lewis number are presented. Precise correlations have been developed for predicting heat and mass transfer rates of natural convection arising from single (solutal or thermal) buoyancy force and dual buoyancy forces.     

Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet

The mixed convection two-dimensional boundary layer flow of a micropolar fluid near the stagnation point on a stretching vertical sheet is investigated. The stretching velocity and the surface temperature are assumed to vary linearly with the distance from the stagnation point. The transformed ordinary differential equations are solved numerically for some values of the parameters involved using a finite-difference scheme known as the Keller-box method. The features of the flow and heat transfer characteristics are analyzed and discussed. Both assisting and opposing flows are considered. Results are presented in terms of the skin friction coefficient and the local Nusselt number with selections of velocity, microrotation and temperature profiles. Dual solutions are found to exist for the opposing flow.     

Combined natural and forced convection in a laminar wall jet along a vertical plate with uniform surface heat flux

The steady state flow and heat transfer characteristics of the combined natural and forced convection in a two dimensional, laminar, incompressible wall jet over a vertical wall are obtained for constant wall heat flux boundary condition. The velocity and temperature distribution are assumed to be power series, where the zeroth term corresponds to that for a plane wall jet in the absence of buoyancy effects. Numerical results for the momentum and thermal series functions are presented for a Prandtl number of 0.73. Wall values of the momentum and thermal series functions are presented for Prandtl numbers ranging from 0.01 to 1000.Nomenclature Gr* modified Grashof number - k thermal conductivity - Nu Nusselt number - Pr Prandtl number - q _w heat flux at the wall - Re Reynolds number - T temperature - u velocity component in x-direction - v velocity component in y-direction - x co-ordinate along the plane wall - y co-ordinate normal to the wall - () gamma function - non-dimensional co-ordinate defined in (6) - non-dimensional temperature - dynamic viscosity - kinematic viscosity - non-dimensional co-ordinate defined in (6) - density - w values at the wall - values at large distances away from the wall     

Flow of a micropolar fluid due to a porous stretching sheet and heat transfer

The present work investigates the micropolar fluid flow due to a permeable stretching sheet and the resulting heat transfer. Unlike the existing numerical works on the flow phenomenon in the literature, the prime interest here is to analytically work out shape of the solutions and identify whether they are unique. Indeed, unique solutions are detected and presented in the exact formulas for the associated boundary layer equations. Temperature field influenced by the microrotation is also mathematically resolved in the cases of constant wall temperature, constant heat flux and Newtonian heating. To discover the salient physical features of many mechanisms acting on the considered problem, it is adequate to have the analytical velocity and temperature fields and also closed-form skin friction/couple stress/heat transfer coefficients, all as given in the current paper. For instance, the practically significant rate of heat transfer is represented by a single formula valid for all three temperature cases.     

The micropolar fluid model for blood flow through a tapered artery with a stenosis

A micropolar model for axisymmetric blood flow through an axially nonsymmetreic but radially symmetric mild stenosis tapered artery is presented. To estimate the effect of the stenosis shape, a suitable geometry has been considered such that the axial shape of the stenosis can be changed easily just by varying a parameter （referred to as the shape parameter）. The model is also used to study the effect of the taper angle Ф. Flow parameters such as the velocity, the resistance to flow （the resistance impedance）, the wall shear stress distribution in the stenotic region and its magnitude at the maximum height of the stenosis （stenosis throat） have been computed for different values of the shape parameter n, the taper angle Ф, the coupling number N and the micropolar parameter m. It is shown that the resistance to flow decreases with increasing the shape parameter n and the micropolar parameter m while it increases with increasing the coupling number N. So, the magnitude of the resistance impedance is higher for a micropolar fluid than that for a Newtonian fluid model. Finally, the velocity profile, the wall shear stress distribution in the stenotic region and its magnitude at the maximum height of the stenosis are discussed for different values of the parameters involved on the problem.     

Effect of viscous dissipation on mixed convection heat transfer in a vertical tube with uniform wall heat flux

 The laminar and parallel flow of a Newtonian fluid in a vertical cylindrical duct with circular cross section has been analysed. Both the viscous dissipation effect and the buoyancy effect have been taken into account. The momentum balance equation and the energy balance equation have been solved by means of a perturbation method, in the case of a uniform heat flux prescribed at the wall of the duct. The velocity distribution, the temperature distribution, the Nusselt number and the Fanning friction factor have been evaluated analytically. Moreover, the velocity and temperature of the fluid have been compared with those obtained in two special cases: forced convection with viscous dissipation (i.e. negligible buoyancy effect); mixed convection with negligible effects of viscous dissipation. Received on 26 June 2000     

Free convection from a vertical permeable circular cone with non-uniform surface heat flux

 In this paper, the problem of laminar free convection from a vertical permeable circular cone maintained with non-uniform surface heat flux is considered. The governing boundary layer equations are reduced non-similar boundary layer equations with surface heat flux proportional to x ⁿ (where x is the distance measured from the leading edge). The solutions of the reduced equations are obtained by using three distinct solution methodologies; namely, (i) perturbation solution for small transpiration parameter, ξ, (ii) asymptotic solution for large ξ, and (iii) the finite difference solutions for all ξ. The solutions are presented in terms of local skin-friction and local Nusselt number for smaller values of Prandtl number and heat flux gradient and are displayed in tabular form as well as graphically. Effects of pertinent parameters on velocity and temperature profiles are also shown graphically. Solutions obtained by finite difference method are also compared with the perturbation solutions for small and large ξ and found to be in excellent agreement. Received on 1 October 1999     

Effect of Hall current on MHD natural convection flow from vertical permeable flat plate with uniform surface heat flux

The effect of the Hall current on the magnetohydrodynamic (MHD) natural convection flow from a vertical permeable flat plate with a uniform heat flux is analyzed in the presence of a transverse magnetic field. It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field. The boundary layer equations are reduced to a suitable form by employing the free variable formulation (FVF) and the stream function formulation (SFF). The parabolic equations obtained from FVF are numerically integrated with the help of a straightforward finite difference method. Moreover, the nonsimilar system of equations obtained from SFF is solved by using a local nonsimilarity method, for the whole range of the local transpiration parameter ζ. Consideration is also given to the regions where the local transpiration parameter ζ is small or large enough. However, in these particular regions, solutions are acquired with the aid of a regular perturbation method. The effects of the magnetic field M and the Hall parameter m on the local skin friction coefficient and the local Nusselt number coefficient are graphically shown for smaller values of the Prandtl number Pr (= 0.005, 0.01, 0.05). Furthermore, the velocity and temperature profiles are also drawn from various values of the local transpiration parameter ζ.     

Effect of Hall current on MHD natural convection flow from vertical permeable fiat plate with uniform surface heat flux

The effect of the Hall current on the magnetohydrodynamic (MHD) natural convection flow from a vertical permeable flat plate with a uniform heat flux is analyzed in the presence of a transverse magnetic field.It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field.The boundary layer equations are reduced to a suitable form by employing the free variable formulation (FVF) and the stream function formulation (SFF).The parabolic equations obtained from FVF are numerically integrated with the help of a straightforward finite difference method.Moreover,the nonsimilar system of equations obtained from SFF is solved by using a local nonsimilarity method,for the whole range of the local transpiration parameter ζ.Consideration is also given to the regions where the local transpiration parameter ζis small or large enough.However,in these particular regions,solutions are acquired with the aid of a regular perturbation method.The effects of the magnetic field M and the Hall parameter m on the local skin friction coefficient and the local Nusselt number coefficient are graphically shown for smaller values of the Prandtl number Pr (=0.005,0.01,0.05).Furthermore,the velocity and temperature profiles are also drawn from various values of the local transpiration parameterζ.     

Combined convection on a vertical slender cylinder in a micropolar fluid

An analysis is presented for the steady state mixed convective boundary layer flow of a micropolar fluid along vertical slender cylinders. The governing equations have been solved numerically. Results for the friction factor, Nusselt number as well as the details of flow and temperature fields are displayed for a range of values of the transverse curvature and material parameters for the micropolar fluid. It is observed that micropolar fluids display drag reduction as well as heat transfer rate reduction when compared to Newtonian fluids.

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Mischkonvektion an einem senkrechten schlanken Zylinder in einem mikropolaren Fluid

Zusammenfassung Die Untersuchung bezieht sich auf stationäre Mischkonvektion in der Grenzschicht einer mikropolaren Flüssigkeit entlang eines senkrechten schlanken Zylinders. Die bestimmenden Gleichungen wurden numerisch gelöst. In einem gewissen Bereich des Krümmungsverhältnisses und der Stoffparameter des mikropolaren Fluids werden Ergebnisse für den Widerstandsbeiwert, die Nusselt-Zahl sowie Besonderheiten des Strömungs- und Temperaturfeldes mitgeteilt. Wie sich zeigte, weisen mikropolare Fluide gegenüber Newtonschen Fluiden sowohl geringeren Widerstand als auch niedrigere Wärmeübertragungsintensität auf.

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Effect of Hall current on MHD natural convection flow from vertical permeable flat plate with uniform surface heat flux

The effect of the Hall current on the magnetohydrodynamic (MHD) natural convection flow from a vertical permeable flat plate with a uniform heat flux is analyzed in the presence of a transverse magnetic field.It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field.The boundary layer equations are reduced to a suitable form by employing the free variable formulation (FVF) and the stream function formulation (SFF).The parabolic equations obtained from FVF are numer...