Numerical investigation of variable viscosities and thermal stratification effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of a chemical reaction

An analysis is presented to investigate the effects of variable viscosities and thermal stratification on the MHD mixed convective heat and mass transfer of a viscous, incompressible, and electrically conducting fluid past a porous wedge in the presence of a chemical reaction. The wall of the wedge is embedded in a uniform nonDarcian 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 with finite difference methods. Numerical calculations up to the thirdorder level of truncation are carried out for different values of dimensionless parameters. The results are presented graphically, and show that the flow field and other quantities of physical interest are significantly influenced by these parameters. The results are compared with those available in literature, and show excellent agreement.     

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.     

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.     

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.     

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.     

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.     

Numerical simulation of flow of micropolar fluids in a channel with a porous wall

Two‐dimensional steady, laminar, and incompressible flow of a micropolar fluid in a channel with no‐slip at one wall and constant uniform injection through the other wall is considered for different values of the Reynolds number R. The main flow stream is superimposed by constant injection velocity at the porous wall. The micropolar model introduced by Eringen is used to describe the working fluid. An extension of Berman's similarity transformations is used to reduce governing equations to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. An algorithm based on finite difference method is employed to solve these ODEs and Richardson's extrapolation is used to obtain higher order accuracy. It has been found that the magnitude of shear stress increases strictly at the impermeable wall whereas it decreases steadily at the permeable wall, by increasing the injection velocity. The maximum value of streamwise velocity and that of the microrotation both increase with increasing the magnitude of R. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Poiseuille Flow of a Third Grade Fluid in a Porous Medium

Effects of porous medium have been investigated on the steady flow of a third grade fluid between two stationary porous plates. The continuity and momentum equations along with modified Darcy??s law are used for the development of mathematical problem. The governing nonlinear problem is solved by a homotopy analysis method. The dimensionless velocity and shear stresses at the plates are analyzed.     

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     

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.     

The effect of radial diffusion on the performance of a liquid-liquid displacement process

The effect of radial diffusion on the performance of a liquid-liquid displacement process is considered in fluid flow between porous parallel plates and through a porous tube, as examples of a two-zone problem in unsteady-state mass transfer. The double Laplace transformation is applied to the system equations. In obtaining the inversion of the Laplace transformed equations the first inversion (with respect to the transformed dimensionless axial distance) is performed by use of the residue method, and then the second inversion (with respect to the transformed dimensionless time) is performed by use of the numerical Laplace transform technique advanced by Bellman et al. A numerical example is shown and discussed.     

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.     

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.     

Analytic Approximate Solution for a Flow of a Second-Grade Viscoelastic Fluid in a Converging Porous Channel

The problem of a two-dimensional steady flow of a second-grade fluid in a converging porous channel is considered. It is assumed that the fluid is injected into the channel through one wall and sucked from the channel through the other wall at the same velocity, which is inversely proportional to the distance along the wall from the channel origin. The equations governing the flow are reduced to ordinary differential equations. The boundary-value problem described by the latter equations is solved by the homotopy perturbation method. The effects of the Reynolds and crossflow Reynolds number on the flow characteristics are examined.     

Stokes flow past a porous spheroid embedded in another porous medium

A study is made with an analysis of an incompressible viscous fluid flow past a slightly deformed porous sphere embedded in another porous medium. The Brinkman equations for the flow inside and outside the deformed porous sphere in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in small parameter characterizing the deformation. The flow through the porous oblate spheroid embedded in another porous medium is considered as the particular example of the deformed porous sphere embedded in another porous medium. The drag experienced by porous oblate spheroid in another porous medium is also evaluated. The dependence of drag coefficient and dimensionless shearing stress on the permeability parameter, viscosity ratio and deformation parameter for the porous oblate spheroid is presented graphically and discussed. Previous well-known results are then also deduced from the present analysis.     

Unsteady motion of a non-linear viscoelastic fluid

In this paper, we study the unsteady flow of a generalized second grade fluid. Specifically, we solve numerically the linear momentum equations for the flow of this viscoelastic shear-thinning (shear-thickening) fluid surrounding a solid cylindrical rod that is suddenly set into longitudinal and torsional motion. The equations are made dimensionless. The results are presented for the shear stresses at the wall, related to the drag force; these are physical quantities of interest, especially in oil-drilling applications.