Magnetohydrodynamic laminar boundary-layer flow over a porous wedge with thermophoresis particle deposition in the presence of variable stream conditions

An analysis is presented to investigate the effects of thermophoresis variable viscosity on MHD mixed convective heat and mass transfer of a viscous, incompressible and electrically conducting fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform porous medium in order to allow for possible fluid wall suction or injection. The governing boundary layer equations are written into a dimensionless form by local non-similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically by using the R.K. Gill and shooting methods. Favorable comparison with previously published work is performed. Numerical results for the dimensionless velocity, temperature and concentration profiles are obtained and displayed graphically for pertinent parameters to show interesting aspects of the solution.     

Thermophoresis and variable viscosity effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of chemical reaction

An analysis is presented to investigate the effects of thermophoresis and variable viscosity on MHD mixed convective heat and mass transfer of a viscous, incompressible and electrically conducting fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform porous medium in order to allow for possible fluid wall suction or injection. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically by using the R.K. Gill and shooting methods. Favorable comparison with previously published work is performed. Numerical results for the dimensionless velocity, temperature and concentration profiles as well as for the skin friction, heat and mass transfer and deposition rate are obtained and displayed graphically for pertinent parameters to show interesting aspects of the solution.     

Impact of thermophoresis particle deposition and chemical reaction on unsteady non-Darcy mixed convective flow over a porous wedge in the presence of temperature-dependent viscosity

An analysis is presented to investigate the effect of thermophoresis particle deposition and temperature dependent viscosity on unsteady non-Darcy mixed convective heat and mass transfer of a viscous and incompressible fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform non-Darcian porous medium in order to allow for possible fluid wall suction or injection. The governing partial differential equations of the problem, subjected to their boundary conditions, are solved numerically by applying an efficient solution scheme for local nonsimilarity boundary layer analysis. Numerical calculations are carried out for different values of dimensionless parameters arising in the problem. The results are compared with available ones in the literature and excellent agreement is obtained. An analysis of the obtained results shows that the flow field is influenced appreciably by the chemical reaction and thermophoresis particle deposition.     

Local non-similarity solution to impact of chemical reaction on MHD mixed convection heat and mass transfer flow over porous wedge in the presence of suction / injection

Combined heat and mass transfer on free, forced, and mixed convection flow along a porous wedge with magnetic effect in the presence of chemical reaction is investigated. The flow field characteristics are analyzed by the Runge-Kutta-Gill scheme with the shooting method as well as the local non-similarity method up to the third level of truncation, which are used to reduce the governing partial differential equations into nine ordinary differential equations. The governing boundary layer equations are converted to a dimensionless form by Falkner-Skan transformations. Because of the effect of suction/injection on the wall of the wedge with buoyancy force and variable wall temperature, the flow field is locally non-similar. Numerical calculations up to the third order level of truncation are carried out as a special case for different values of dimensionless parameters. Effects of the magnetic field strength in the presence of chemical reaction with variable wall temperature and concentration on the dimensionless velocity, temperature and concentration profiles are shown graphically.     

Local Nonsimilarity Solution for the Impact of a Chemical Reaction in an MHD Mixed Convection Heat and Mass Transfer Flow over a Porous Wedge in the Presence Of Suction/Injection

Combined heat and mass transfer in free, forced, and mixed convection flows along a porous wedge with a magnetic effect in the presence of a chemical reaction is investigated. The flow field characteristics are analyzed with the Runge—Kutta—Gill method in conjunction with the shooting method, and local nonsimilarity method. The governing boundary-layer equations are written in a dimensionless form with the use of the Falkner—Skan transformations. Owing to the effect of the buoyancy force, the power law of temperature and concentration, and suction/injection on the wall of the wedge, the flow field is locally nonsimilar. Numerical calculations up to the third-order level of truncation are carried out for different values of dimensionless parameters as a special case. Effects of the magnetic field strength in the presence of a chemical reaction with a variable wall temperature and concentration on the dimensionless velocity, temperature, and concentration profiles are shown graphically. Comparisons with previously published works are performed, and excellent agreement between the results is obtained.     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

Lie group analysis for thermophoretic and radiative augmentation of heat and mass transfer in a Brinkman-Darcy flow over a flat surface with heat generation

A boundary layer analysis is presented to investigate numerically the effects of radiation,thermophoresis and the dimensionless heat generation or absorption on hydromagnetic flow with heat and mass transfer over a flat surface in a porous medium.The boundary layer equations are transformed to non-linear ordinary differential equations using scaling group of transformations and they are solved numerically by using the fourth order Runge-Kutta method with shooting technique for some values of physical parameters.Comparisons with previously published work are performed and the results are found to be in very good agreement.Many results are obtained and a representative set is displayed graphically to illustrate the influence of the various parameters on the dimensionless velocity,temperature and concentration profiles as well as the local skin-friction coefficient,wall heat transfer,particle deposition rate and wall thermophoretic deposition velocity.The results show that the magnetic field induces acceleration of the flow,rather than deceleration(as in classical magnetohydrodynamics(MHD) boundary layer flow) but to reduce temperature and increase concentration of particles in boundary layer.Also,there is a strong dependency of the concentration in the boundary layer on both the Schmidt number and mass transfer parameter.     

State space approach to unsteady magnetohydrodynamics natural convection heat and mass transfer through a porous medium saturated with a viscoelastic fluid

A technique of the state space approach and the inversion of the Laplace transform method are applied to dimensionless equations of an unsteady one-dimensional boundary-layer flow due to heat and mass transfer through a porous medium saturated with a viscoelastic fluid bounded by an infinite vertical plate in the presence of a uniform magnetic field is described. Complete analytical solutions for the temperature, concentration, velocity, and induced magnetic and electric fields are presented. The inversion of the Laplace transforms is carried out by using a numerical approach. The proposed method is used to solve two problems: boundary-layer flow in a viscoelastic fluid near a vertical wall subjected to the initial conditions of a stepwise temperature and concentration and viscoelastic fluid flow between two vertical walls. The solutions are found to be dependent on the governing parameters including the Prandtl number, the Schmidt number, the Grashof number, reaction rate coefficient, viscoelastic parameter, and permeability of the porous medium. Effects of these major parameters on the transport behavior are investigated methodically, and typical results are illustrated to reveal the tendency of the solutions. Representative results are presented for the velocity, temperature, concentration, and induced magnetic and electric field distributions, as well as the local skin-friction coefficient and the local Nusselt and Sherwood numbers.     

Non-Newtonian effects of second-order fluids on double-layered porous Rayleigh-step bearings

Theoretical study of double-layered porous Rayleigh-step bearings with second-order fluid as lubricant is presented. An approximate method for the solutions of the governing fluid film equations for a porous region is proposed. The expressions for the pressure distribution, load capacity and frictional coefficient are obtained in compact form. Calculations of the dimensionless load capacity, frictional force and frictional coefficients are presented for specific values of the material parameters. It is found that the double porous layer yields an increase in the load capacity and ensures decrease in frictional force at the porous lining as compared with the conventional porous Rayleigh-step bearings. The maximum dimensionless load-carrying capacity is found to occur at a slightly larger step ratio as compared with the conventional porous Rayleigh-step bearings.     

Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.     

Local nonsimilarity solution for the impact of the buoyancy force on heat and mass transfer in a flow over a porous wedge with a heat source in the presence of suction/injection

Combined heat and mass transfer in free, forced and mixed convection flows along a porous wedge with internal heat generation in the presence of uniform suction or injection is investigated. The boundary-layer analysis is formulated in terms of the combined thermal and solute buoyancy effect. The flow field characteristics are analyzed using the Runge-Kutta-Gill method, the shooting method, and the local nonsimilarity method. Due to the effect of the buoyancy force, power law of temperature and concentration, and suction/injection on the wall of the wedge, the flow field is locally nonsimilar. Numerical calculations up to third-order level of truncation are carried out for different values of dimensionless parameters as a special case. The effects of the buoyancy force, suction, heat generation, and variable wall temperature and concentration on the dimensionless velocity, temperature, and concentration profiles are studied. The results obtained are found to be in good agreement with previously published works.     

No slip boundary effects in non-Darcian mixed convection from a vertical wall in saturated porous media. Analytical solution

An analytical study is made for wall effects in non-Darcy mixed convection from vertical impermeable surfaces embedded in a saturated porous medium. The governing equations are transformed into a dimensionless form by non-similar transformation to cover both forced and natural convection dominated regimes. Two different dimensionless parameters that measure the strength of mixed convection were found in both regimes. The parameters of forced convection dominated regime can be related to those of natural convection dominated regime. An approximate analytical solution for the governing equations was obtained. Temperature and velocity profiles for both regimes are presented. Received on 9 September 1997     

The Falkner–Skan Wedge Flows of Power-Law Fluids Embedded in a Porous Medium

The non-Darcy flow characteristics of power-law non-Newtonian fluids past a wedge embedded in a porous medium have been studied. The governing equations are converted to a system of first-order ordinary differential equations by means of a local similarity transformation and have been solved numerically, for a number of parameter combinations of wedge angle parameter m, power-law index of the non-Newtonian fluids n, first-order resistance A and second-order resistance B, using a fourth-order Runge–Kutta integration scheme with the Newton–Raphson shooting method. Velocity and shear stress at the body surface are presented for a range of the above parameters. These results are also compared with the corresponding flow problems for a Newtonian fluid. Numerical results show that for the case of the constant wedge angle and material parameter A, the local skin friction coefficient is lower for a dilatant fluid as compared with the pseudo-plastic or Newtonian fluids.     

Thermal Stratification Effects on Hiemenz Flow of Nanofluid Over a Porous Wedge Sheet in the Presence of Suction/Injection Due to Solar Energy: Lie Group Transformation

The objective of the present work is to investigate theoretically the Hiemenz flow and heat transfer of an incompressible viscous nanofluid past a porous wedge sheet in the presence of thermal stratification due to solar energy (incident radiation). The wall of the wedge is embedded in a uniform Darcian porous medium to allow for possible fluid wall suction or injection and has a power–law variation of the wall temperature. The partial differential equations governing the problem under consideration are transformed by a special form of Lie symmetry group transformations viz., one-parameter group of transformation into a system of ordinary differential equations which are solved numerically by Runge–Kutta–Gill-based shooting method. The conclusion is drawn that the flow field and temperature are significantly influenced by convective radiation, thermal stratification, buoyancy force, and porosity of the sheet.     

Effects of thermophoresis and radiation on laminar flow along a semi-infinite vertical plate

The present contribution deals with the thermophoresis particle deposition and thermal radiation effects on the flow, heat and mass transfer characteristics in a viscous fluid over a semi-infinite vertical porous plate. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically by means of the fourth-order Runge–Kutta method with a shooting technique. The effects of different parameters on the dimensionless velocity, temperature, and concentration profiles are shown graphically. In addition, results for the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number are tabulated and discussed.     

Natural convection of non-Newtonian power-law fluid over axisymmetric and two-dimensional bodies of arbitrary shape in fluid-saturated porous media

Numerical analysis of the free convection coupled heat and mass transfer is presented for non-Newtonian power-law fluids with the yield stress flowing over a two-dimensional or axisymmetric body of an arbitrary shape in a fluid-saturated porous medium. The governing boundary layer equations and boundary conditions are cast into a dimensionless form by the similarity transformation. The resulting system of equations is solved by a finite difference method. The parameters studied are the rheological constants, the buoyancy ratio, and the Lewis number. Representative velocity, temperature, and concentration profiles are presented and discussed. It is found that the results depend strongly on the values of the yield stress parameter and the power-law index of the non-Newtonian fluid.     

Natural convection of non-Newtonian power-law fluid over axisymmetric and two-dimensional bodies of arbitrary shape in fluid-saturated porous media

Numerical analysis of the free convection coupled heat and mass transfer is presented for non-Newtonian power-law fluids with the yield stress flowing over a two-dimensional or axisymmetric body of an arbitrary shape in a fluid-saturated porous medium. The governing boundary layer equations and boundary conditions are cast into a dimensionless form by the similarity transformation. The resulting system of equations is solved by a finite difference method. The parameters studied are the rheological constants, the buoyancy ratio, and the Lewis number. Representative velocity, temperature, and concentration profiles are presented and discussed. It is found that the results depend strongly on the values of the yield stress parameter and the power-law index of the non-Newtonian fluid.     

Effect of an Inserted Porous Layer Located at a Wall of a Parallel Plate Channel on Forced Convection Heat Transfer

A theoretical study is performed on heat and fluid flow in partially porous medium filled parallel plate channel. A uniform symmetrical heat flux is imposed onto the boundaries of the channel partially filled with porous medium. The dimensional forms of the governing equations are solved numerically for different permeability and effective thermal conductivity ratios. Then, the governing equations are made dimensionless and solved analytically. The results of two approaches are compared and an excellent agreement is observed, indicating correctness of the both solutions. An overall Nusselt number is defined based on overall thermal conductivity and difference between the average temperature of walls and mean temperature to compare heat transfer in different channels with different porous layer thickness, Darcy number, and thermal conductivity ratio. Moreover, individual Nusselt numbers for upper and lower walls are also defined and obtained. The obtained results show that the maximum overall Nusselt number is achieved for thermal conductivity ratio of 1. At specific values of Darcy number and thermal conductivity ratio, individual Nusselt numbers approach to infinity since the value of wall temperatures approaches to mean temperature.     

Natural convection on one side of a vertical wall embedded in a Brinkman-porous medium coupled with film condensation on the other side

A theoretical study of conjugate natural convection and film condensation in porous media is reported. The natural convection phenomenon takes place along one side of a vertical impermeable wall and the condensation phenomenon along the other side. This wall constitutes the interface between two spaces filled with fluid-saturated porous media. The flow in both porous spaces is modelled on the basis of the Brinkman-modified Darcy momentum equation which satisfies the condition of zero velocity on a solid boundary. The temperature and flow fields in the porous medium are completely determined in the natural convection side as well as in the condensation side of the wall. In addition, the dependence of the wall heat flux and temperature distributions on height and on a number of dimensionless groups relevant to the problem is thoroughly documented. Important results pertinent to the impact of the problem parameters on the overall heat leak from the condensation space to the natural convection space are also reported. These results are presented with the help of the Nusselt number. Finally, the effect of the wall thermal resistance on the heat and fluid flow characteristics of the system is determined.     

纳米流体在伸/缩楔体上的MHD流动数值研究

对纳米流体在伸/缩楔体上的磁流体(MHD)流动进行了数值研究。首先,通过相似变换将控制偏微分方程转化为非线性常微分方程组;然后,利用Matlab软件,借助打靶法,结合四阶五常龙格库塔迭代方案进行数值求解;最后,详细讨论了各控制参数对无量纲速度、温度、浓度、表面摩擦系数、局部Nusselt数和局部Sherwood数的影响。结果表明,楔体在拉伸情况下只有唯一解,理论上不会出现边界层分离;而在一定收缩强度范围内存在双解,边界层流动在壁面处可能会出现边界层分离,壁面抽吸会使边界层分离推迟;楔体在拉伸情况下,磁场参数对表面摩擦系数的影响较大,对局部Nusselt数和局部Sherwood数的影响较小。