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.     

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.     

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.     

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 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...     

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.     

Laminar natural convection over a slender vertical frustrum of a cone with constant wall heat flux

The problem of laminar natural convection flow over a slender frustrum of a cone with constant wall heat flux is treated in this paper. The governing differential equations are solved by a combination of quasilinearization and finite-difference methods. Numerical solutions are obtained for Prandtl numbers from 0.1 to 100 for a range of values of transverse curvature parameter. It is found that the effect of transverse curvature is of great significance in such flows.

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Laminare natürliche Konvektion über einem dünnen, senkrechten Kegelstumpf mit konstantem Wand wär mestrom

Zusammenfassung In dieser Arbeit wird das Problem der laminaren, natürlichen Konvektionsströmung öber einem dünnen Kegelstumpf mit konstantem Wandwärmestrom behandelt. Die maßgeblichen Differentialgleichungen werden mit Hilfe einer Kombination von Quasilinearisierung und Differenzenverfahren gelöst. Numerische Lösungen werden für die Prandtl ' sehen Zahlen zwischen 0. l und 100 innerhalb eines Bereiches von Querkrüm mungswerten erhalten. Es wird gezeigt, daß der Einfluß der Querkrümmung in solchen Strömungen von großer Bedeutung ist.

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Nomenclature A,B,C constants in the transformation, defined in Eq.(14) - f dependent variable, defined in Eq. (7) - g dependent variable, defined in Eq. (7) - g_e gravitational acceleration - k heat conductivity - k_n -grid - L characteristic length - Nu Nusselt number - Pr Prandtl number - q_w wall heat flux - r radial distance from the axis of the cone - R_TVC transverse curvature ratio, defined in Eq.(28) - Re Reynolds number - T temperature - u,v velocity components in the x- and y-directions, respectively - x,y rectangular coordinates Greek Letters dimensionless temperature, defined in Eq.(4) - bulk modulus - cone angle - dynamic viscosity - stream function - , transformed independent variables, defined in Eq. (7) - transverse curvature parameter     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

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     

Natural convection flow over a vertical frustum of a cone for constant wall heat flux

Laminar natural convection flow and heat transfer over a vertical frustum of a cone has been studied. The governing boundary layer equations are solved using local non-similarity method for constant wall heat flux. The local similarity and the local non-similarity two and three-equation models are constructed and the resulting equations are solved numerically. Results obtained from two and three-equation models are in good agreement. The numerical values of the flow and temperature functions required to calculate the surface skin friction and heat transfer rate are reported for various values of Prandtl numbers.     

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     

Friction and heat transfer with simultaneous convection on a permeable surface

The velocity and heat transfer fields near a vertical permeable surface with simultaneous convection are investigated. A solution is found for the boundary layer equations with known laws of surface temperature and flow velocity change. The transformed boundary layer equations contain the parameter G/R², which determines the effect of free convection on friction and heat transfer for constrained motion. Calculations of friction and heat transfer as functions of draft (suction) with simultaneous convection are presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 96–100, March–April, 1973.     

Influence of magnetic field and thermal radiation by natural convection past vertical cone subjected to variable surface heat flux

A numerical study is performed to examine the heat transfer characteristics of natural convection past a vertical cone under the combined effects of magnetic field and thermal radiation.The surface of the cone is subjected to a variable surface heat flux.The fluid considered is a gray,absorbing-emitting radiation but a non-scattering medium.With approximate transformations,the boundary layer equations governing the flow are reduced to non-dimensional equations valid in the free convection regime.The dimensionless governing equations are solved by an implicit finite difference method of Crank-Nicolson type which is fast convergent,accurate,and unconditionally stable.Numerical results are obtained and presented for velocity,temperature,local and average wall shear stress,and local and average Nusselt number in air and water.The present results are compared with the previous published work and are found to be in excellent agreement.     

Free convection on a vertical plate with uniform and constant heat flux in a thermally stratified micropolar fluid

This paper analyzes flow and heat transfer characteristics of the free convection on a vertical plate with uniform and constant heat flux in a thermally stratified micropolar fluid. The dimensionless forms of boundary layer equations and their associated boundary conditions have been derived and the numerical results have been obtained using the method of cubic spline collocation with a finite difference scheme. The effects of the micropolar and stratification parameters on the dimensionless wall temperature, skin friction parameter and wall couple stress are discussed.     

Free convection effects on a vertical cone with variable viscosity and thermal conductivity

The present paper deals with a flow of a viscous incompressible fluid along a heated vertical cone, with due allowance for variations of viscosity and thermal diffusivity with temperature. The fluid viscosity is assumed to be an exponential function of temperature, and the thermal diffusivity is assumed to be a linear function of temperature. The governing equations for laminar free convection of the fluid are transformed into dimensionless partial differential equations, which are solved by a finite difference method with the Crank–Nicolson implicit scheme. Dependences of the flow parameters on the fluid viscosity and thermal conductivity are obtained.     

Transient free convection from a vertical plate subjected to a change in surface heat flux in porous media

In this paper we examine the general transient natural convection response arising due to a sudden change of the level of uniform flux dissipation rate from a vertical surface which is embedded in a porous medium. From an analytical investigation of the governing boundary-layer equations both a series solution which is valid at small values of the non-dimensional time and a solution which is valid at large times, when the transport of energy is steady, are derived. A numerical, transient formulation of the full unsteady boundary-layer equations is developed using an explicit finite-difference scheme. The numerical temperature profiles are observed to closely follow the small time solution initially and evolve along a curve which approaches the steady-state solution asymptotically. Results are presented to illustrate the occurrence of transients from both an increase and a decrease in the levels of existing energy inputs.     

Similarity solutions in free convection boundary-layer flows adjacent to vertical permeable surfaces in porous media: II prescribed surface heat flux

The equation which governs the similarity solution for free convection boundary-layer flow along a vertical permeable surface with prescribed surface heating and mass transfer rate is discussed. The solution is seen to depend on two non-dimensional parameters;m, the power-law exponent, and γ, the mass transfer parameter. It is shown that solutions exist for allm>?1 for γ>0 (fluid injection) whereas for γ<0 (fluid withdrawal), solution exist form>m ₀(γ), wherem ₀ is determined as a function of γ. Solutions for large mass transfer rates are obtained, for both γ>0 and γ<0. For γ>0 the form of the asymptotic solution for γ large is seen to depend on the value ofm. Solutions form large are derived, these are seen to be different depending on whether γ is positive or negative.