MHD non-Darcian flow through a non-isothermal vertical surface embedded in a porous medium with radiation

The effect of MHD on steady two-dimensional laminar mixed flow about a vertical porous surface is numerically analyzed. Also the effects of radiation and heat generation and absorption are considered. A power law variation of temperature along the vertical wall is assumed. The nonlinear boundary-layer equations were transformed and the resulting differential equations were solved by an implicit finite difference scheme (Keller box method). Numerical results for the velocity distribution and the temperature distribution are presented for various values of Prandtl number Pr, magnetic parameter, porous medium parameter and internal heat generation or absorption coefficient. Further validation with previous works is carried out.     

Radiation-conduction interaction on mixed convection from a horizontal circular cylinder

A mixed convection flow of an optically dense viscous incompressible fluid along a horizontal circular cylinder has been studied with the effect of radiation when the surface temperature is uniform. Using appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity form. Solutions of the governing equations are obtained employing the implicit finite difference method. Effects of varying the pertinent parameters, such as, the Planck number, R _w the surface temperature parameter, θ_w and the buoyancy parameter, α on the local skin-friction and local heat transfer coefficients are shown graphically as well as in tabular form against the curvature parameter ξ, while taking Prandtl number Pr = 1.0. It is found that an increase of R _d,θ_w or α leads to increases in the values of the local skin-friction and the local rate of heat transfer coefficients. At the stagnation point asymptotic solutions for large value of α are also obtained and the effect of the other pertinent parameters on the formation of the flow separation are studied. Received on 28 July 1998     

On thermal boundary layer of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection

 Heat transfer characteristics of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection were investigated. Similarity solutions of the laminar boundary layer equations describing heat transfer and fluid flow in a quiescent fluid were obtained and solved numerically. Velocity and temperature profiles as well as the Nusselt number, Nu, were studied for two thermal boundary conditions; uniform surface temperature and variable surface temperature, for different parameters; Prandtl number Pr, temperature exponent b, velocity exponent m, injection parameter d and power-law index n. It was found that decreasing injection parameter d, and power-law index n and increasing Prandtl number Pr and surface temperature exponent b enhance the heat transfer coefficient. Received on 27 April 2000     

Heat and mass transfer for micropolar flow with radiation effect past a nonlinearly stretching sheet

In this study, an analysis has been performed for heat and mass transfer with radiation effect of a steady laminar boundary-layer flow of a micropolar flow past a nonlinearly stretching sheet. Parameters n, K, k ₀, Pr, Ec, and Sc represent the dominance of the nonlinearly effect, material effect, radiation effect, heat and mass transfer effects which have presented in governing equations, respectively. The similar transformation, the finite-difference method and Runge–Kutta method have been used to analyze the present problem. The numerical solutions of the flow velocity distributions, temperature profiles, the wall unknown values of θ′(0) and ϕ′(0) for calculating the heat and mass transfer of the similar boundary-layer flow are carried out as functions of n, Ec, k ₀, Pr, Sc. The value of n, k ₀, Pr and Sc parameters are important factors in this study. It will produce greater heat transfer efficiency with a larger value of those parameters, but the viscous dissipation parameter Ec and material parameter K may reduce the heat transfer efficiency. On the other hand, for mass transfer, the value of Sc parameter is important factor in this study. It will produce greater heat transfer efficiency with a larger value of Sc.     

Laminar mixed convection from a continuously moving vertical surface with suction or injection

The boundary layer flow over a uniformly moving vertical surface with suction or injection is studied when the buoyancy forces assist or oppose the flow. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity boundary conditions. The effect is of various governing parameters, such as Prandtl number Pr, temperature exponent n, injection parameter d, and the mixed convection parameter λ=Gr/Re², which determine the velocity and temperature distributions and the heat transfer coefficient, are studied. The heat transfer coefficient increases as λ assisting the flow for all d at Pr=0.72 however, for n=−1 it decreases sharply with λ. On the other hand, increasing λ has no effect on heat transfer coefficient for Pr=10 at n=0, and 1 for almost all values of d studied. However, for n=−1 it has similar effect as for Pr=0.72. It is also found that Nusselt number increases as n increases for fixed λ and d. Received on 26 March 1997     

Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid

In this paper, the problem of free convection boundary layer flow on a solid sphere in a micropolar fluid with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations in the form of partial differential equations are solved numerically using an implicit finite-difference scheme. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity, angular velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the material or micropolar parameter K, the Prandtl number Pr and the conjugate parameter γ are analyzed and discussed.     

Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux

 The effect of uniform suction on the steady two-dimensional laminar forced flow of a viscous incompressible fluid of temperature dependent viscosity past a wedge with uniform surface heat flux is considered. The governing equations for the flow are obtained by using suitable transformations and are solved by using an implicit finite difference method. Perturbation solutions are also obtained near the leading edge and in the downstream regime. The results are obtained in terms of the local skin friction coefficient and the rate of heat transfer for various values of the pertinent parameters, such as the Prandtl number, Pr, the velocity gradient parameter, m, the local suction parameter, ξ, and the viscosity variation parameter, ɛ. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The effect of ξ, m and ɛ on the dimensionless velocity profiles and viscosity distribution are also presented graphically for Pr = 0.7 and 7.0, which are the appropriate values for gases and water respectively. Received on 22 July 1999     

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     

MHD flow and heat transfer for the upper-convected Maxwell fluid over a stretching sheet

In the present work, the effect of MHD flow and heat transfer within a boundary layer flow on an upper-convected Maxwell (UCM) fluid over a stretching sheet is examined. The governing boundary layer equations of motion and heat transfer are non-dimensionalized using suitable similarity variables and the resulting transformed, ordinary differential equations are then solved numerically by shooting technique with fourth order Runge–Kutta method. For a UCM fluid, a thinning of the boundary layer and a drop in wall skin friction coefficient is predicted to occur for higher the elastic number. The objective of the present work is to investigate the effect of Maxwell parameter β, magnetic parameter Mn and Prandtl number Pr on the temperature field above the sheet.     

Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet

An analysis is made for the steady two-dimensional magneto-hydrodynamic flow of an incompressible viscous and electrically conducting fluid over a stretching vertical sheet in its own plane. The stretching velocity, the surface temperature and the transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The transformed boundary layer equations are solved numerically for some values of the involved parameters, namely the magnetic parameter M, the velocity exponent parameter m, the temperature exponent parameter n and the buoyancy parameter λ, while the Prandtl number Pr is fixed, namely Pr = 1, using a finite difference scheme known as the Keller-box method. Similarity solutions are obtained in the presence of the buoyancy force if n = 2m−1. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. It is found that both the skin friction coefficient and the local Nusselt number decrease as the magnetic parameter M increases for fixed λ and m. For m = 0.2 (i.e. n = −0.6), although the sheet and the fluid are at different temperatures, there is no local heat transfer at the surface of the sheet except at the singular point of the origin (fixed point).     

Squeezed flow and heat transfer over a porous surface for viscous fluid

Flow and heat transfer over a permeable sensor surface placed in a squeezing channel is analyzed. A constant transpiration through the sensor surface is assumed. Locally non-similar momentum and energy equations are solved by three different methods, against the transpiration parameter τ, for different values of the squeezing parameter b, and Prandtl number Pr. From the investigation, it is found that when the channel being squeezed, the skin-friction reduces but the heat transfer coefficient increases. Increase in the value of the squeezing parameter onsets reverse flow at the sensor surface when fluid is being injected and the affect is enhanced with the increase of injection through the surface. It is further observed that increase of suction of fluid through the sensor thins the thermal and the momentum boundary layer regions, whereas injection of fluid leads to thickening of both the thermal and the momentum boundary layer regions. Heat transfer from the surface of the sensor increases with the increase of the value of Pr for the entire range of surface mass-flux parameter τ. M. A. Hossain is on leave of absence from University of Dhaka.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

g-Jitter induced free convection near a stagnation point

 The effect of a small but fluctuating gravitational field, characteristic of g-jitter, on the flow near the forward stagnation point of a two-dimensional symmetric body resulting from a step change in its surface temperature has been considered in this paper. The transformed equations are solved numerically by a very efficient finite-difference method known as the Keller-box technique to investigate the effects on the shear stress and rate of heat transfer of variations in the Prandtl number, Pr, the forcing amplitude, a, and the forcing frequency, ω. It has been found that these parameters affect considerably the shear stress and the rate of heat transfer. Received on 28 February 2000     

Forced convection of a power-law fluid in a porous channel – numerical solutions

A detailed numerical study of laminar forced convection in a porous channel which contains a fibrous medium saturated with a power-law fluid was performed. Hydrodynamic and heat transfer results are presented for a configuration that has uniform heat flux or uniform temperature heating at the walls. The flow in the porous medium was modeled using the modified Brinkman-Forchheimer-extended Darcy model for power law fluids in which the non-Darcy effects of inertia and boundary were considered. Parametric studies were conducted to examine the effects of Darcy number, power law index, inertia parameter and Prandtl number. The results indicate that when the power law index is decreased, the velocity gradient near the walls increases but these effects are reduced gradually as the Darcy number decreases until the Darcy regime (Da≤10⁻⁶) is reached in which case the effects of power law index become negligible. As the power law index is decreased, the thermal boundary layer thickness decreases significantly only in the non-Darcy regime. Consequently, as the power law index decreases, the fully developed Nusselt number increases considerably in the non-Darcy regime whereas in the Darcy regime the change in Nusselt number is very small. As the Prandtl number increases, the local Nusselt number increases and this effect is more significant for shear thinning fluids (n<1.0). Received on 2 March 1998     

Thermal radiation effects on the natural convection flow over an isothermal horizontal plate

The natural convection boundary layer flow with conduction-radiation interaction of a viscous incompressible fluid along an isothermal horizontal surface has been studied. The equations valid in the upstream, downstream as well as in the entire regime are obtained. Solutions of the non-similar equations governing the flow for the entire regime and the downstream regime are obtained by employing an efficient implicit finite difference approximation together with the Keller box method, for a Prandtl number of 0.73. Also, the effects of the pertinent parameters, R _d, the radiation-conduction parameter and θ_w, the surface heating parameter are shown graphically in terms of the local skin-friction and the local rate of heat transfer. Comparison of the results obtained for the upstream and the downstream regimes shows good agreement over the entire regime. Effects of R _d and θ_w are also shown on the streamlines and the isotherms. Received on 15 December 1998     

Heat transfer in a power-law fluid film over a unsteady stretching sheet

In this paper the problem of momentum and heat transfer in a thin liquid film of power-law fluid on an unsteady stretching surface has been studied. Numerical solutions are obtained for some representative values of the unsteadiness parameter S and the power-law index n for a wide range of the generalized Prandtl number, 0.001 ≤ Pr ≤ 1000. Typical temperature and velocity profiles, the dimensionless film thickness, free-surface temperature, and the surface heat fluxes are presented at selected controlling parameters. The results show that increasing the value of n tends to increase the boundary-layer thickness and broadens the temperature distributions. The free-surface temperature of a shear thinning fluid is larger than that of a Newtonian fluid, but the opposite trend is true for a shear thickening fluid. For small generalized Prandtl numbers, the surface heat flux increases with a decrease in n, but the impacts of n on the heat transfer diminish for Pr greater than a moderate value (approximately 1 ≤ Pr ≤ 10, depending on the magnitude of S).     

Scaling Transformations for Boundary Layer Flow near the Stagnation-Point on a Heated Permeable Stretching Surface in a Porous Medium Saturated with a Nanofluid and Heat Generation/Absorption Effects

In this article, a similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption is theoretically studied. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions via Lie-group analysis. Copper (Cu) with water as its base fluid has been considered and representative results have been obtained for the nanoparticle volume fraction parameter f{\phi} in the range 0 ￡ f ￡ 0.2{0\leq \phi \leq 0.2} with the Prandtl number of Pr = 6.8 for the water working fluid. Velocity and temperature profiles as well as the skin friction coefficient and the local Nusselt number are determined numerically. The influence of pertinent parameters such as nanofluid volume fraction parameter, the ratio of free stream velocity and stretching velocity parameter, the permeability parameter, suction/blowing parameter, and heat source/sink parameter on the flow and heat transfer characteristics is discussed. Comparisons with published results are also presented. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable to change the flow pattern.     

Combined effect of viscous dissipation and thermal radiation on fluid flows over a non-linearly stretched permeable wall

An analysis is presented for the steady non-linear viscous flow of an incompressible viscous fluid over a horizontal surface of variable temperature with a power-law velocity under the influences of suction/blowing, viscous dissipation and thermal radiation. Numerical results are illustrated by means of tables and graphs. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation. The effects of the stretching parameter n, suction/blowing parameter b, Prandtl number σ, Eckert number E_c(E_c ^* )E_{c}(E_{c}^{ *} ) and radiation parameter N _R are discussed. Two cases are studied, namely, (i) Prescribed surface temperature (PST case) and, (ii) Prescribed heat flux at the sheet (PHF case).     

Numerical study of Prandtl effect on MHD flow at a lid‐driven porous cavity

In this paper, the lattice Boltzmann method is used to study the Prandtl number effect on flow structure and heat transfer rates in a magnetohydrodynamic flow mixed convection in a lid‐driven cavity filled with a porous medium. The right and left walls are at constant but different temperatures (θ_h and θ_c), while the other walls are adiabatic. Gallium and salt water (0.02 < Pr < 13.4) are used as samples of the electroconducting fluids in the cavity. Typical sets of streamlines and isotherms are presented to analyze the flow patterns set up by the competition among the forced flow created by the lid‐driven wall, the buoyancy force of the fluid and the magnetic force of the applied magnetic field. Mathematical formulations in the porous media were constructed based on the Brinkman–Forchheimer model, while the multidistribution‐function model was used for the magnetic field effect. Numerical results were obtained and the effects of the Prandtl number and the other effective parameters such as Richardson, Hartman, and Darcy numbers were investigated. It was found that the fluid fluctuations within the cavity were reduced by increasing the Hartman number. A similar pattern was observed for the Darcy number reduction. Heat transfer was essentially dominated by the conduction for the low Prandtl number and forced convection dominated as the Prandtl number increased. Also, the average Nusselt number was raised by increasing the Prandtl number. It was discovered that a remarkable heat transfer enhancement of up to 28% could be reached by increasing the Prandtl number (from 0.02 to 13.4) at constant Richardson and Darcy numbers. Copyright © 2011 John Wiley & Sons, Ltd.     

MHD mixed convective heat transfer flow about an inclined plate

Mixed convection heat transfer about a semi-infinite inclined plate in the presence of magneto and thermal radiation effects is studied. The fluid is assumed to be incompressible and dense. The nonlinear coupled parabolic partial differential equations governing the flow are transformed into the non-similar boundary layer equations, which are then solved numerically using the Keller box method. The effects of the mixed convection parameter R _i, the angle of inclination α, the magnetic parameter M and the radiation–conduction parameter R _d on the velocity and temperature profiles as well as on the local skin friction and local heat transfer parameters. For some specific values of the governing parameters, the results are compared with those available in the literature and a fairly good agreement is obtained.