Effects of Soret Dufour,chemical reaction and thermal radiation on MHD non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet

A study has been carried out to analyze the combined effects of Soret (thermal-diffusion) and Dufour (diffusion-thermo) on unsteady MHD non-Darcy mixed convection over a stretching sheet embedded in a saturated porous medium in the presence of thermal radiation, viscous dissipation and first-order chemical reaction. Energy equation takes into account of viscous dissipation, thermal radiation and Soret effects. The governing differential equations are transformed into a set of non-linear coupled ordinary differential equations and solved using similarity analysis with numerical technique using appropriate boundary conditions for various physical parameters. The numerical solution for the governing nonlinear boundary value problem is based on shooting algorithm with Runge–Kutta–Fehlberg integration scheme over the entire range of physical parameters. The effects of various physical parameters on the dimensionless velocity, temperature and concentration profiles are depicted graphically and analyzed in detail. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results for local skin-friction, local Nusselt number, and local Sherwood number are tabulated for different physical parameters.     

Soret and Dufour effects on MHD slip flow with thermal radiation over a porous rotating infinite disk

In this investigation, thermal radiation effect over an electrically conducting, Newtonian fluid in a steady laminar magnetohydrodynamic convective flow over a porous rotating infinite disk with the consideration of heat and mass transfer in the presence of Soret and Dufour diffusion effects is investigated. The partial differential equations governing the problem under consideration are transformed by a similarity transformation into a system of ordinary differential equations which are solved numerically using fourth order Runge–Kutta based shooting method. The effects of the magnetic interaction parameter, slip flow parameter, Soret number, Dufour number, Schmidt number, radiation parameter, Prandtl number and suction parameter on the fluid velocity, temperature and concentration distributions in the regime are depicted graphically and are analyzed in detail. The corresponding skin-friction coefficients, the Nusselt number and the Sherwood number are also calculated and displayed in tables showing the effects of various parameters on them.     

Similarity solution in MHD: Effects of thermal diffusion and diffusion thermo on free convective heat and mass transfer over a stretching surface considering suction or injection

An analysis is carried out to study free convective heat and mass transfer of an incompressible, electrically conducting fluid over a stretching sheet in the presence of suction and injection with thermal-diffusion (Soret) and diffusion-thermo (Dufour) effects. The similarity solutions are obtained using scaling transformations. Furthermore, the similarity equations are solved numerically by using shooting technique with fourth-order Runge–Kutta integration scheme. A comparison with previously published work is performed and the results are found to be in good agreement. Numerical results of the local skin friction coefficient, the local Nusselt number and the local Sherwood number as well as the velocity, the temperature and the concentration profiles are presented for different physical parameters. The result indicates: (i) for fluids with medium molecular weight (H₂, air), Dufour and Soret effects should not be neglected; and (ii) the suction and injection parameter has significant impact in controlling the rate of heat transfer in the boundary layer.     

Hydromagnetic non-Darcy flow and heat transfer over a stretching sheet in the presence of thermal radiation and Ohmic dissipation

Non-Darcy flow and heat characteristics over a stretching sheet is presented here by taking into account of Ohmic dissipation and thermal radiation effects. The governing fundamental equations are first transformed into system of ordinary differential equations using self-similarity transformation and they are then solved numerically by using the fifth-order Runge–Kutta–Fehlberg method with shooting technique for some values of the physical parameters. Important features of the flow and heat transfer characteristic for different values of thermal radiation, magnetic and electric fields are analyzed and discussed. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results for the velocity and temperature profiles for a prescribed magnetic field and electric field parameter as well as the development of the local skin-friction coefficient and local Nusselt number with radiation parameters are reported graphically for various parametric conditions to show interesting aspects of the numerical solution.     

Combined effects of non-uniform heat source/sink and thermal radiation on heat transfer over an unsteady stretching permeable surface

The present paper is concerned with the study of flow and heat transfer characteristics in the unsteady laminar boundary layer flow of an incompressible viscous fluid over continuously stretching permeable surface in the presence of a non-uniform heat source/sink and thermal radiation. The unsteadiness in the flow and temperature fields is because of the time-dependent stretching velocity and surface temperature. Similarity transformations are used to convert the governing time-dependent nonlinear boundary layer equations for momentum and thermal energy are reduced to a system of nonlinear ordinary differential equations containing Prandtl number, non-uniform heat source/sink parameter, thermal radiation and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge–Kutta–Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the unsteadiness parameter, thermal radiation, suction/injection parameter, non-uniform heat source/sink parameter on flow and heat transfer characteristics as well as on the local Nusselt number are shown graphically.     

Mixed convection heat transfer in the boundary layers on an exponentially stretching surface with magnetic field

An analysis has been carried out to describe mixed convection heat transfer in the boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution in the presence of magnetic field, viscous dissipation and internal heat generation/absorption. Approximate analytical similarity solutions of the highly non-linear momentum and energy equations are obtained. The present results are found to be in excellent agreement with previously published work on various special cases of the problem. Numerical results for temperature distribution and the local Nusselt number have been obtained for different values of the governing parameters. The numerical solutions are obtained by considering an exponential dependent stretching velocity and prescribed boundary temperature on the flow directional coordinate. The effects of various physical parameters like Prandtl number, Hartman number, Grashof number on dimensionless heat transfer characteristics are discussed in detail. In particular, it has been found that increase in Prandtl number decreases the skin-friction coefficient at the stretching surface, while increase in the strength of the magnetic field leads to increase in the local Nusselt number.     

竖直平板嵌入非Darcy多孔介质时,数值研究流过平板的不稳定MHD自然对流中的Dufour和Soret效应

就竖直平板嵌入非Darcy多孔介质中,导电流体流过平板时作不稳定的二维磁流体(MHD)双扩散对流,数值研究了Dufour和Soret效应对流动的影响.用Crank-Nicolson型的隐式有限差分法,按三对角矩阵处理,求解无量纲的非线性控制方程.详细地研究了问题中出现的各种参数对不稳定无量纲的速度、温度和浓度曲线的影响.进一步地,给出并分析了表面摩擦因数、Nus-selt数和Sherwood数随时间的变化.研究结果表明,不稳定速度、温度和浓度分布曲线,受Dufour和Soret的影响十分显著.随着Dufour数的增加或者Soret数的减小,表面摩擦因数和Sherwood数都在减小,而Nusselt数在增加.研究发现,当磁场参数增加时,边界层中的速度和温度在减小.     

Effects of radiation and variable viscosity on unsteady MHD flow of a rotating fluid from stretching surface in porous medium

This work is focused on the study of unsteady magnetohydrodynamics boundary-layer flow and heat transfer for a viscous laminar incompressible electrically conducting and rotating fluid due to a stretching surface embedded in a saturated porous medium with a temperature-dependent viscosity in the presence of a magnetic field and thermal radiation effects. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. With appropriate transformations, the unsteady MHD boundary layer equations are reduced to local nonsimilarity equations. Numerical solutions of these equations are obtained by using the Runge–Kutta integration scheme as well as the local nonsimilarity method with second order truncation. Comparisons with previously published work have been conducted and the results are found to be in excellent agreement. A parametric study of the physical parameters is conducted and a representative set of numerical results for the velocity in primary and secondary flows as well as the local skin-friction coefficients and the local Nusselt number are illustrated graphically to show interesting features of Darcy number, viscosity-variation, magnetic field, rotation of the fluid, and conduction radiation parameters.     

Effects of slip on steady Bödewadt flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady flow and heat transfer arising due to the rotation of a non-Newtonian fluid at a larger distance from a stationary disk is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The momentum equation gives rise to a highly nonlinear boundary value problem. Numerical solution of the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip, non-Newtonian fluid characteristics and the magnetic interaction parameter on the momentum boundary layer and thermal boundary layer are discussed in detail and shown graphically. It is observed that slip has prominent effects on the velocity and temperature fields.     

Group method analysis of melting effect on MHD mixed convection flow from radiate vertical plate embedded in a saturated porous media

The linear transformation group approach is developed to simulate problem of hydromagnetic heat transfer by mixed convection along vertical plate in a liquid saturated porous medium in the presence of melting and thermal radiation effects for opposing external flow. The application of a one-parameter transformation group reduces the number of independent variables by one so that the governing partial differential equations with the boundary conditions reduce to an ordinary differential equations with appropriate corresponding conditions. The Runge-Kutta shooting method is used to solve the determining equations of the set of nonlinear ordinary differential equations. are presented in the form of the temperature and flow fields in the melting region within the boundary layer for different parameters entering into the analysis. Also the effects of the pertinent parameters on the rate of the heat transfer in terms of the local Nusselt number at the solid–liquid interface are also discussed.     

Influence of double stratification on MHD free convection with Soret and Dufour effects in a Darcian porous media

The heat and mass transfer characteristics of natural convection about a vertical surface embedded in a doubly stratified saturated porous medium subjected to a magnetic field is studied, taking into account the Dufour and Soret effects. It is found that the similarity solution exists for the case of uniform surface heat and mass flux conditions when the thermal and solutal stratification of the medium are assumed to have power function form x^1/3. The resulting set of ordinary coupled nonlinear differential equations is solved numerically using shooting technique. Nusselt and Sherwood numbers are tabulated for various values of the involved parameters. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effects of partial slip,viscous dissipation and Joule heating on Von Kármán flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady Von Kármán flow and heat transfer of an electrically conducting non-Newtonian fluid is extended to the case where the disk surface admits partial slip. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The momentum equations give rise to highly non-linear boundary value problem. Numerical solutions for the governing non-linear equations are obtained over the entire range of the physical parameters. The effects of slip, magnetic parameter and non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. Emphasis has been laid to study the effects of viscous dissipation and Joule heating on the thermal boundary layer. It is interesting to find that the non-Newtonian cross-viscous parameter has an opposite effect to that of the slip and the magnetic parameter on the velocity and the temperature fields.     

UNSTEADY NATURAL CONVECTIVE BOUNDARY LAYER FLOW AND HEAT TRANSFER OF FRACTIONAL SECOND-GRADE NANOFLUIDS WITH DIFFERENT PARTICLE SHAPES

The present study is concerned with unsteady natural convective boundary layer flow and heat transfer of fractional second-grade nanofuids for different particle shapes. Nonlinear boundary layer governing equations are formulated with time fractional derivatives in the momentum equation. The governing boundary layer equations of continuity, momentum and energy are reduced by dimensionless variable. Numerical solutions of the momentum and energy equations are obtained by the finite difference method combined with L1-algorithm. The quantites of physical interest are graphically presented and discussed in details. It is found that particle shape, fractional derivative parameter and the Grashof number have profound influences on the the flow and heat transfer.     

MHD non-Darcy mixed convective diffusion of species over a stretching sheet embedded in a porous medium with non-uniform heat source/sink, variable viscosity and Soret effect

The effects of temperature dependent viscosity and non-uniform heat source/sink on non-Darcy MHD mixed convection boundary layer flow over a vertical stretching sheet embedded in a fluid-saturated porous media is studied in this paper. Boundary layer equations are transformed into ordinary differential equations using self-similarity transformation which are then solved numerically using fifth-order Runge-Kutta-Fehlberg method with shooting technique for various values of the governing parameters. The effects of variable viscosity, porosity, electric field parameter, non-uniform heat source/sink parameters, Soret number and Schmidt number on concentration profiles are analyzed and discussed. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results for variation of the local Sherwood number with buoyancy parameter, Schmidt number, and Soret number are reported graphically to show some interesting aspects of the physical parameters.     

Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity,non-uniform heat source and radiation

An analysis has been carried out to study the magnetohydrodynamic boundary layer flow and heat transfer characteristics of a non-Newtonian viscoelastic fluid over a flat sheet with a linear velocity in the presence of thermal radiation and non-uniform heat source. The thermal conductivity is assumed to vary as a linear function of temperature. The basic equations governing the flow and heat transfer are in the form of partial differential equations, the same have been reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformation. The transformed equations are solved analytically by regular perturbation method. Numerical solution of the problem is also obtained by the efficient shooting method, which agrees well with the analytical solution. The effects of various physical parameters such as viscoelastic parameter, Chandrasekhar number, Prandtl number, variable thermal conductivity parameter, Eckert number, thermal radiation parameter and non-uniform heat source/sink parameters which determine the temperature profiles are shown in several plots and the heat transfer coefficient is tabulated for a range of values of said parameters. Some important findings reported in this work reveals that combined effect of variable thermal conductivity, radiation and non-uniform heat source have significant impact in controlling the rate of heat transfer in the boundary layer region.     

Mixed convective heat and mass transfer in an asymmetric channel with peristalsis

This paper describes the fluid mechanics effects of mixed convective heat and mass transfer in an asymmetric channel with peristalsis. The flow is examined in a wave frame of reference moving with the velocity of the wave. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The momentum, energy and concentration equations have been linearized under long wavelength approximation. The analytical solutions for temperature, concentration, velocity and stream function are obtained. The effects of various parameters such as local temperature Grashof number, local mass Grashof number and geometrical parameters on flow variables have been discussed numerically and explained graphically.     

Effect of variable viscosity on MHD non-Darcy mixed convective heat transfer over a stretching sheet embedded in a porous medium with non-uniform heat source/sink

An analysis has been presented to investigate the effect of temperature-dependent viscosity on non-Darcy MHD mixed convective heat transfer past a porous medium by taking into account of Ohmic dissipation and non-uniform heat source/sink. Thermal boundary layer equation takes into account of viscous dissipation and Ohmic dissipation due to transverse magnetic field and electric field. The governing fundamental equations are first transformed into system of ordinary differential equations using self-similarity transformation and are solved numerically by using the fifth-order Runge–Kutta–Fehlberg method with shooting technique for various values of the physical parameters. The effects of variable viscosity, porosity, Eckert number, Prandtl number, magnetic field, electric field and non-uniform heat source/sink parameters on velocity and temperature profiles are analyzed and discussed. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results on the development of the local skin-friction co-efficient and local Nusselt number with non-uniform heat source/sink are tabulated for various physical parameters to show the interesting aspects of the solution.     

Influence of temperature-dependent viscosity and thermal radiation on MHD forced convection over a non-isothermal wedge

An analysis has been carried out to study heat transfer characteristics of an incompressible Newtonian electrically conducting and heat generating/absorbing fluid having temperature-dependent viscosity over a non-isothermal wedge in the presence of thermal radiation. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The wedge surface is assumed to be permeable so as to allow for possible wall suction or injection. The effects of viscous dissipation, Joule heating, stress work and thermal radiation are included in the model. The governing differential equations are derived and transformed using a non-similarity transformation. The transformed equations are solved numerically by applying a fifth-order Runge-Kutta-Fehlberg scheme with shooting technique. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results for the velocity and temperature profiles for a prescribed magnetic field parameter as well as the development of the local skin-friction coefficient and local Nusselt number with the magnetic field and radiation parameters are presented graphically and in tabulated form to elucidate the influence of the various physical parameters.     

MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, f_w. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer.     

Group method analysis of combined heat and mass transfer by MHD non-Darcy non-Newtonian natural convection adjacent to horizontal cylinder in a saturated porous medium

The group theoretic method is applied for solving problem of combined magneto-hydrodynamic heat and mass transfer of non-Darcy natural convection about an impermeable horizontal cylinder in a non-Newtonian power law fluid embedded in porous medium under coupled thermal and mass diffusion, inertia resistance, magnetic field, thermal radiation effects. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The ordinary differential equations are solved numerically for the velocity using shooting method. The effects of magnetic parameter M, Ergun number Er, power law (viscosity) index n, buoyancy ratio N, radiation parameter R_d, Prandtl number Pr and Lewis number Le on the velocity, temperature fields within the boundary layer, heat and mass transfer are presented graphically and discussed.