This paper deals with the MHD convention non-Darcy flow with heat and mass transfer along the vertical surface in a fluid stratified non porous media in the presence of temperature gradients (Soret effects) and concentration gradients (Dufour effects). The differential equations of the governing flow are expand a in terms of perturbation functions as a result we get set of ordinary differential equations of zero,first and second order so the set of equations are solved by using the implicit finite difference scheme. A parametric study of the physical parameters involve in the problem is conducted and a representative set of numerical results are illustrated graphically
Radiation and Soret Effect on Visco-Elastic MHD Oscillatory Horizontal Channe...IJERA Editor
An analysis of radiation and soret effect on visco-elastic mhd oscillatory flow with heat and mass transfer
through a porous medium bounded by two infinite horizontal parallel porous plates in presence of chemical
reaction and a heat source has been presented when one plate is kept at rest while the other is oscillating in its
own plane. The fluid is considered to be non-Newtonian characterized by Walters liquid (Model B′). The
temperature of the stationary plate is assumed to be constant whereas the temperature of the other plate varies
periodically with time about a steady mean. The equations governing the fluid flow, heat and mass transfer have
been solved analytically. The expressions for velocity, temperature, species concentration, non-dimensional
skin-friction at the plates, the coefficient of rate of heat transfer from the plates to the fluid in terms of Nusselt
number in non-dimensional form, the coefficient of rate of mass transfer from the plates to the fluid in terms of
Sharewood number in non-dimensional form are obtained and illustrated graphically to observe the visco-elastic
effects in combination of other flow parameters involved in the solution. It is noticed that the momentum,
thermal and concentration fields are significantly affected by the visco-elastic parameter.
Effects of Thermal Radiation and Chemical Reaction on MHD Free Convection Flo...IJERA Editor
This paper analyzes the radiation and chemical reaction effects on MHD steady two-dimensional laminar
viscous incompressible radiating boundary layer flow over a flat plate in the presence of internal heat generation
and convective boundary condition. It is assumed that lower surface of the plate is in contact with a hot fluid
while a stream of cold fluid flows steadily over the upper surface with a heat source that decays exponentially.
The Rosseland approximation is used to describe radiative heat transfer as we consider optically thick fluids.
The governing boundary layer equations are transformed into a system of ordinary differential equations using
similarity transformations, which are then solved numerically by employing fourth order Runge-Kutta method
along with shooting technique. The effects of various material parameters on the velocity, temperature and
concentration as well as the skin friction coefficient, the Nusselt number, the Sherwood number and the plate
surface temperature are illustrated and interpreted in physical terms. A comparison of present results with
previously published results shows an excellent agreement.
The steady laminar two dimensional stagnation point flow of an incompressible electrically conducting magneto-micropolar fluid over a permeable stretching surface with heat source/sink and viscous dissipation in the presence of mass transfer and chemical reaction has been studied. Using the similarity transformations, the governing equations have been transformed into a system of ordinary differential equations. These differential equations are highly nonlinear which cannot be solved analytically. Therefore, forth order Runge-Kutta method along with shooting technique has been used for solving it. Numerical results are obtained for the skin-friction coefficient, the local Nusselt number and Sherwood number as well as the velocity, temperature and concentration profiles for different values of the governing parameters, namely, velocity ratio parameter, boundary parameter, material parameter, magnetic parameter, Prandtl number, Eckert number, heat source/sink parameter, Schmidt number and chemical reaction parameter.
Laplace Analysis of Periodic Heat and Mass Transport on a Parabolic Started S...IJERA Editor
An analysis of periodic heat and mass transport of unsteady hydromagnetic flow past a parabolic started motion of the infinite vertical plate immersed in Darcian porous regime in presence of a first order chemical reaction has been presented. Here the plate temperature as well as concentration level near the plate are increased linearly with time. The boundary layer conservation equations have been solved by Laplace transforms technique. It has been observed that both the velocity and concentration are decreased with increasing values of chemical reaction parameter. But the opposite behavior has been found for the flow velocity when the values of free convection as well as porosity parameter are increased. Application of magnetic fields to medical science is growing rapidly, with the development of novel magnetic pumps, hydromagnetic separation devices with chemical engineering and geophysical energy systems.
MHD Mixed Convection Flow from a Vertical Plate Embedded in a Saturated Porou...IJERA Editor
We analysed in this paper the problem of MHD mixed convection flow from a vertical plate embedded in a saturated porous medium in the presence of melting, thermal dispersion, radiation and heat absorption or generation effects for aiding and opposing external flows. Similarity solution for the governing equations is obtained for the flow equations in steady state. The equations are numerically solved by Runge-Kutta fourth order method coupled with shooting technique. The effect of melting and heat absorption or generation under different parametric conditions on velocity, temperature and heat transfer was analyzed for both aiding and opposing flows
Radiation and Soret Effect on Visco-Elastic MHD Oscillatory Horizontal Channe...IJERA Editor
An analysis of radiation and soret effect on visco-elastic mhd oscillatory flow with heat and mass transfer
through a porous medium bounded by two infinite horizontal parallel porous plates in presence of chemical
reaction and a heat source has been presented when one plate is kept at rest while the other is oscillating in its
own plane. The fluid is considered to be non-Newtonian characterized by Walters liquid (Model B′). The
temperature of the stationary plate is assumed to be constant whereas the temperature of the other plate varies
periodically with time about a steady mean. The equations governing the fluid flow, heat and mass transfer have
been solved analytically. The expressions for velocity, temperature, species concentration, non-dimensional
skin-friction at the plates, the coefficient of rate of heat transfer from the plates to the fluid in terms of Nusselt
number in non-dimensional form, the coefficient of rate of mass transfer from the plates to the fluid in terms of
Sharewood number in non-dimensional form are obtained and illustrated graphically to observe the visco-elastic
effects in combination of other flow parameters involved in the solution. It is noticed that the momentum,
thermal and concentration fields are significantly affected by the visco-elastic parameter.
Effects of Thermal Radiation and Chemical Reaction on MHD Free Convection Flo...IJERA Editor
This paper analyzes the radiation and chemical reaction effects on MHD steady two-dimensional laminar
viscous incompressible radiating boundary layer flow over a flat plate in the presence of internal heat generation
and convective boundary condition. It is assumed that lower surface of the plate is in contact with a hot fluid
while a stream of cold fluid flows steadily over the upper surface with a heat source that decays exponentially.
The Rosseland approximation is used to describe radiative heat transfer as we consider optically thick fluids.
The governing boundary layer equations are transformed into a system of ordinary differential equations using
similarity transformations, which are then solved numerically by employing fourth order Runge-Kutta method
along with shooting technique. The effects of various material parameters on the velocity, temperature and
concentration as well as the skin friction coefficient, the Nusselt number, the Sherwood number and the plate
surface temperature are illustrated and interpreted in physical terms. A comparison of present results with
previously published results shows an excellent agreement.
The steady laminar two dimensional stagnation point flow of an incompressible electrically conducting magneto-micropolar fluid over a permeable stretching surface with heat source/sink and viscous dissipation in the presence of mass transfer and chemical reaction has been studied. Using the similarity transformations, the governing equations have been transformed into a system of ordinary differential equations. These differential equations are highly nonlinear which cannot be solved analytically. Therefore, forth order Runge-Kutta method along with shooting technique has been used for solving it. Numerical results are obtained for the skin-friction coefficient, the local Nusselt number and Sherwood number as well as the velocity, temperature and concentration profiles for different values of the governing parameters, namely, velocity ratio parameter, boundary parameter, material parameter, magnetic parameter, Prandtl number, Eckert number, heat source/sink parameter, Schmidt number and chemical reaction parameter.
Laplace Analysis of Periodic Heat and Mass Transport on a Parabolic Started S...IJERA Editor
An analysis of periodic heat and mass transport of unsteady hydromagnetic flow past a parabolic started motion of the infinite vertical plate immersed in Darcian porous regime in presence of a first order chemical reaction has been presented. Here the plate temperature as well as concentration level near the plate are increased linearly with time. The boundary layer conservation equations have been solved by Laplace transforms technique. It has been observed that both the velocity and concentration are decreased with increasing values of chemical reaction parameter. But the opposite behavior has been found for the flow velocity when the values of free convection as well as porosity parameter are increased. Application of magnetic fields to medical science is growing rapidly, with the development of novel magnetic pumps, hydromagnetic separation devices with chemical engineering and geophysical energy systems.
MHD Mixed Convection Flow from a Vertical Plate Embedded in a Saturated Porou...IJERA Editor
We analysed in this paper the problem of MHD mixed convection flow from a vertical plate embedded in a saturated porous medium in the presence of melting, thermal dispersion, radiation and heat absorption or generation effects for aiding and opposing external flows. Similarity solution for the governing equations is obtained for the flow equations in steady state. The equations are numerically solved by Runge-Kutta fourth order method coupled with shooting technique. The effect of melting and heat absorption or generation under different parametric conditions on velocity, temperature and heat transfer was analyzed for both aiding and opposing flows
MHD Stagnation Point Flow of a Jeffrey Fluid Over a
Stretching/Shrinking Sheet Through Porous Medium by Eswara Rao M and Krishna Murthy M* in COJ Electronics & Communications
MHD Convection Flow of Kuvshinski Fluid past an Infinite Vertical Porous Plat...rahulmonikasharma
An unsteady MHD free convective flow of a viscous incompressible visco-elastic fluid through a porous medium in the presence of radiation and chemical reaction effects are considered. The governing non-linear partial differential equations of the flow, heat and mass transfer are transformed into ordinary differential equations by using similarity transformations and then solved by two term perturbation technique. The effects of various flow parameters on velocity, temperature and concentration profiles as well as the skin-friction, Nusselt number and Shear wood number are discussed qualitatively and discussed graphically.
Effects of Variable Fluid Properties and MHD on Mixed Convection Heat Transfe...IOSR Journals
The effects of variable Fluid Properties like variation of permeability, porosity, thermal conductivity and magnetic field on Mixed Convection Heat transfer from Vertical Heated Plate Embedded in a Sparsely Packed Porous Medium have been approached numerically. The boundary layer flow in the porous medium is governed by Lapwood – Forchheimer – Brinkman extended Darcy model and the Lorentz force. The natures of these equations are highly non-linear and coupled each other. The non-linear differential equations are non-dimensionalised using the non-dimensional parameter involving Grashoff number Gr, Prandtl number Pr, Hartmann number M, Eckert number E and so on. Similarity transformations are employed and the resulting ordinary differential equations are solved numerically by using shooting algorithm with Runge – Kutta and Newton – Raphson method to obtain velocity and temperature distributions. Besides, skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence in decreasing the flow field, whereas its influence is reversed on the rate of heat transfer for all values of permeability considered. The effect of Magnetic field is diminution with velocity of the fluid flow. Further, the obtained results under the two limiting conditions were found to be in good agreement with the existing results
Effects Of Higher Order Chemical Reactions and Slip Boundary Conditions on Na...IJERA Editor
We investigate the effects of higher order chemical reactions on coupled heat and mass transfer from a permeable stretching sheet in nanofluid flow with slip boundary conditions. Nanofluids play an important role in emerging nanotechnology and biotechnology industries where they find application in heat transfer processes. We study the effects of the order of the chemical reaction and chemical reaction parameter. The highly nonlinear and coupled governing differential equations were solved numerically. The solution depends on Brownian motion number, thermophoresis number, thermal convective parameter, mass convective parameter and double diffusive buoyancy parameter. The effects of various engineering parameters are investigated.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Heat transefer of a dusty fluid over a stretching sheet with internal heat ge...eSAT Journals
Abstract This paper pertains to investigate the heat transfer characteristics of two dimensional two phase flow in a porous medium over a stretching sheet with internal heat generation. The novelty of the present study is to consider the permeability parameter, space and temperature dependent internal heat generation along with various parameters like Froud number ,heat source/sink parameter, Grashof number, Prandtl number, Eckert number, Volume fraction, fluid interaction parameter etc. The method of solution involves similarity transformation which reduces the partial differential equations into non linear ordinary differential equations. These non linear ordinary differential equations have been solved by applying Runge-Kutta 4-th order method with help of shooting technique. The temperature profiles for different values of flow parameters are presented in figures. It is observed from all the figures that the boundary conditions are satisfied asymptotically in all the cases which supporting the accuracy of the numerical results. All the figures shows that increasing values of any parameter increase the thermal boundary layer except the prandtl number and permeability parameter. AMS classification 76T10, 76T15 Keywords: Heat source/sink, Volume fraction, Fluid – particle interaction parameter, Dusty fluid, Boundary layer flow, Stretching sheet, Eckert number, Grashof number, Prandtle number, Shooting techniques, Froud number, permeability parameter.
MHD Flow past a Vertical Oscillating Plate with Radiation and Chemical Reacti...iosrjce
An analysis is performed to study the effect of Thermal radiation and first order Chemical reaction on unsteady natural Convective flow of a viscous incompressible Conducting fluid past over an infinite isothermal vertical oscillating plate in Porous medium. The dimensionless governing equations are solved using the Laplace transform technique. The velocity, temperature and concentration are studied for different
parameters like the magnetic field parameter, radiation parameter, chemical reaction parameter, thermal
Grashof number, Schmidt number, phase angle and time. It is observed that the velocity increases with decreasing magnetic field parameter or radiation parameter. It is also observed that the velocity increases with decreasing magnetic field parameter, radiation parameter and phase angle
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Unsteady MHD Flow Past A Semi-Infinite Vertical Plate With Heat Source/ Sink:...IJERA Editor
In the present paper a numerical attempt is made to study the combined effects of heat source and sink on unsteady laminar boundary layer flow of a viscous, incompressible, electrically conducting fluid along a semiinfinite vertical plate. A magnetic field of uniform strength is applied normal to the flow. The governing boundary layer equations are solved numerically, using Crank-Nicolson method. Graphical results of velocity and temperature fields, tabular values of Skin-friction and Nusselt are presented and discussed at various parametric conditions. From this study, it is found that the velocity and temperature of the fluid increase in the presence of heat source but they decrease in the presence of heat absorption parameter.
Boundary layer flow and heat transfer of a dusty fluid over a vertical permea...eSAT Journals
Abstract
The steady boundary layer free convective flow of a dusty fluid past a vertical permeable stretching surface is studied .The governing equations are converted into first order ordinary differential equations using similarity transformations. These equations are solved numerically by using Runge kutta forth order method. The effects of physical parameters like fluid-particle interaction parameter, local Grashof number, suction parameters, Prandtl number, radiation parameter and Eckert number on the flow and heat transfer characteristics are computed and presented graphically. Also the rate of heat transfer at the surface is discussed. The present results are compared with the previous study and there is a good agreement.
AMS classification 76T10, 76T15
Keywords: Volume fraction, Interaction parameter, Dusty fluid, Thermal radiation, suction parameter, steady flow and heat transfer, Boundary layer flow, Numerical solution.
first order chemical reaction and hall effect on mhd flow past an infinite ve...INFOGAIN PUBLICATION
Combined study of first order chemical reaction and Hall current on MHD flow past an accelerated infinite vertical plate in the presence of rotating fluid with variable mass diffusion has been analyzed. The effects of Hall parameter (m), Hartmann number (M), an imposed rotation parameter (Ω non-dimensional angular velocity),Thermal Grashof’s number(Gr) and mass Grashof’s number (Gc) on axial and transverse velocity profiles are presented graphically. It is found that when , the transverse velocity component vanishes and axial velocity attains a maximum value.
Soret Effect And Effect Of Radiation On Transient Mhd Free Convective Flow Ov...inventionjournals
The present paper is concerned to analyze the radiation, Magneto hydrodynamic and soret effects on unsteady flow heat and mass transfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinite vertical porous plate through porous media the porous plate is subjected to a transverse variable suction velocity. The transient, non linear and coupled dimensionless governing equations for this investigation are solved analytically using perturbation technique about a small parameter . the effects of governing parameters on the flow variables are discussed graphically.
STEADY FLOW OF A VISCOUS FLUID THROUGH A SATURATED POROUS MEDIUM AT A CONSTAN...Journal For Research
In this paper the Steady flow of a viscous fluid through a porous medium over a fixed horizontal, impermeable and thermally insulated bottom. The flow through the porous medium satisfies the general momentum and energy equations are obtained when the temperature on the fixed bottom and on free surface prescibed. By using Galerkin Method, the expression for Velocity and Drag force are obtained. The Galerkin Method endowed with distinct features that account for its superiority over competing methods. The effect of different parameters on Velocity and Drag force are discussed with the help of graphs.
The expansion of power generation in Saudi Arabia is essential in order to meet the expected growth of its electricity demand. Due to the availability of high solar irradiation, vast rainless area and longtime sun light, Saudi Arabia is one of the most suitable countries to utilize solar energy resources in greater extend. Kingdom has planned to increase the production of solar power in order to meet a considerable share of country’s future energy demand. Numerous installation and research works are going on nowadays in the kingdom in order to attain its targets of solar power capacity in the coming years. Hence latest updates of country’s solar industry are essential for further research and R&D works in this field. Saudi Arabia’s current status and future possibility of solar industry are discussed in this paper.
Performance of RGB and L Base Supervised Classification Technique Using Multi...IJERA Editor
In the present growth of sensor technology is to improve the new chance and applications in GIS. This enhances the technology law a new method that should not focus on real time available products, but it must automatically lead to new ones. The aim of the paper is to make a maximum use of remote sensing data and GIS techniques to access land use and land cover classification in the Kiliyar sub basin sector in palar river of northen part of Tamil Nadu.IRS P6 LISS III is merged data to perform the classification using ERDAS Imaging. The RGB and L base supervised classification was based up on a Multispectral analysis, land use and land cover information‟s (maps and existing reports), which involves advanced technology and complex data processing to find detailed imagery in the study region. Ground surface reflects more radar energy emitted by the sensor from the study region, which makes it easy to distinguish between the water body, hilly, agriculture, settlement and wetland.
MHD Stagnation Point Flow of a Jeffrey Fluid Over a
Stretching/Shrinking Sheet Through Porous Medium by Eswara Rao M and Krishna Murthy M* in COJ Electronics & Communications
MHD Convection Flow of Kuvshinski Fluid past an Infinite Vertical Porous Plat...rahulmonikasharma
An unsteady MHD free convective flow of a viscous incompressible visco-elastic fluid through a porous medium in the presence of radiation and chemical reaction effects are considered. The governing non-linear partial differential equations of the flow, heat and mass transfer are transformed into ordinary differential equations by using similarity transformations and then solved by two term perturbation technique. The effects of various flow parameters on velocity, temperature and concentration profiles as well as the skin-friction, Nusselt number and Shear wood number are discussed qualitatively and discussed graphically.
Effects of Variable Fluid Properties and MHD on Mixed Convection Heat Transfe...IOSR Journals
The effects of variable Fluid Properties like variation of permeability, porosity, thermal conductivity and magnetic field on Mixed Convection Heat transfer from Vertical Heated Plate Embedded in a Sparsely Packed Porous Medium have been approached numerically. The boundary layer flow in the porous medium is governed by Lapwood – Forchheimer – Brinkman extended Darcy model and the Lorentz force. The natures of these equations are highly non-linear and coupled each other. The non-linear differential equations are non-dimensionalised using the non-dimensional parameter involving Grashoff number Gr, Prandtl number Pr, Hartmann number M, Eckert number E and so on. Similarity transformations are employed and the resulting ordinary differential equations are solved numerically by using shooting algorithm with Runge – Kutta and Newton – Raphson method to obtain velocity and temperature distributions. Besides, skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence in decreasing the flow field, whereas its influence is reversed on the rate of heat transfer for all values of permeability considered. The effect of Magnetic field is diminution with velocity of the fluid flow. Further, the obtained results under the two limiting conditions were found to be in good agreement with the existing results
Effects Of Higher Order Chemical Reactions and Slip Boundary Conditions on Na...IJERA Editor
We investigate the effects of higher order chemical reactions on coupled heat and mass transfer from a permeable stretching sheet in nanofluid flow with slip boundary conditions. Nanofluids play an important role in emerging nanotechnology and biotechnology industries where they find application in heat transfer processes. We study the effects of the order of the chemical reaction and chemical reaction parameter. The highly nonlinear and coupled governing differential equations were solved numerically. The solution depends on Brownian motion number, thermophoresis number, thermal convective parameter, mass convective parameter and double diffusive buoyancy parameter. The effects of various engineering parameters are investigated.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Heat transefer of a dusty fluid over a stretching sheet with internal heat ge...eSAT Journals
Abstract This paper pertains to investigate the heat transfer characteristics of two dimensional two phase flow in a porous medium over a stretching sheet with internal heat generation. The novelty of the present study is to consider the permeability parameter, space and temperature dependent internal heat generation along with various parameters like Froud number ,heat source/sink parameter, Grashof number, Prandtl number, Eckert number, Volume fraction, fluid interaction parameter etc. The method of solution involves similarity transformation which reduces the partial differential equations into non linear ordinary differential equations. These non linear ordinary differential equations have been solved by applying Runge-Kutta 4-th order method with help of shooting technique. The temperature profiles for different values of flow parameters are presented in figures. It is observed from all the figures that the boundary conditions are satisfied asymptotically in all the cases which supporting the accuracy of the numerical results. All the figures shows that increasing values of any parameter increase the thermal boundary layer except the prandtl number and permeability parameter. AMS classification 76T10, 76T15 Keywords: Heat source/sink, Volume fraction, Fluid – particle interaction parameter, Dusty fluid, Boundary layer flow, Stretching sheet, Eckert number, Grashof number, Prandtle number, Shooting techniques, Froud number, permeability parameter.
MHD Flow past a Vertical Oscillating Plate with Radiation and Chemical Reacti...iosrjce
An analysis is performed to study the effect of Thermal radiation and first order Chemical reaction on unsteady natural Convective flow of a viscous incompressible Conducting fluid past over an infinite isothermal vertical oscillating plate in Porous medium. The dimensionless governing equations are solved using the Laplace transform technique. The velocity, temperature and concentration are studied for different
parameters like the magnetic field parameter, radiation parameter, chemical reaction parameter, thermal
Grashof number, Schmidt number, phase angle and time. It is observed that the velocity increases with decreasing magnetic field parameter or radiation parameter. It is also observed that the velocity increases with decreasing magnetic field parameter, radiation parameter and phase angle
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Unsteady MHD Flow Past A Semi-Infinite Vertical Plate With Heat Source/ Sink:...IJERA Editor
In the present paper a numerical attempt is made to study the combined effects of heat source and sink on unsteady laminar boundary layer flow of a viscous, incompressible, electrically conducting fluid along a semiinfinite vertical plate. A magnetic field of uniform strength is applied normal to the flow. The governing boundary layer equations are solved numerically, using Crank-Nicolson method. Graphical results of velocity and temperature fields, tabular values of Skin-friction and Nusselt are presented and discussed at various parametric conditions. From this study, it is found that the velocity and temperature of the fluid increase in the presence of heat source but they decrease in the presence of heat absorption parameter.
Boundary layer flow and heat transfer of a dusty fluid over a vertical permea...eSAT Journals
Abstract
The steady boundary layer free convective flow of a dusty fluid past a vertical permeable stretching surface is studied .The governing equations are converted into first order ordinary differential equations using similarity transformations. These equations are solved numerically by using Runge kutta forth order method. The effects of physical parameters like fluid-particle interaction parameter, local Grashof number, suction parameters, Prandtl number, radiation parameter and Eckert number on the flow and heat transfer characteristics are computed and presented graphically. Also the rate of heat transfer at the surface is discussed. The present results are compared with the previous study and there is a good agreement.
AMS classification 76T10, 76T15
Keywords: Volume fraction, Interaction parameter, Dusty fluid, Thermal radiation, suction parameter, steady flow and heat transfer, Boundary layer flow, Numerical solution.
first order chemical reaction and hall effect on mhd flow past an infinite ve...INFOGAIN PUBLICATION
Combined study of first order chemical reaction and Hall current on MHD flow past an accelerated infinite vertical plate in the presence of rotating fluid with variable mass diffusion has been analyzed. The effects of Hall parameter (m), Hartmann number (M), an imposed rotation parameter (Ω non-dimensional angular velocity),Thermal Grashof’s number(Gr) and mass Grashof’s number (Gc) on axial and transverse velocity profiles are presented graphically. It is found that when , the transverse velocity component vanishes and axial velocity attains a maximum value.
Soret Effect And Effect Of Radiation On Transient Mhd Free Convective Flow Ov...inventionjournals
The present paper is concerned to analyze the radiation, Magneto hydrodynamic and soret effects on unsteady flow heat and mass transfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinite vertical porous plate through porous media the porous plate is subjected to a transverse variable suction velocity. The transient, non linear and coupled dimensionless governing equations for this investigation are solved analytically using perturbation technique about a small parameter . the effects of governing parameters on the flow variables are discussed graphically.
STEADY FLOW OF A VISCOUS FLUID THROUGH A SATURATED POROUS MEDIUM AT A CONSTAN...Journal For Research
In this paper the Steady flow of a viscous fluid through a porous medium over a fixed horizontal, impermeable and thermally insulated bottom. The flow through the porous medium satisfies the general momentum and energy equations are obtained when the temperature on the fixed bottom and on free surface prescibed. By using Galerkin Method, the expression for Velocity and Drag force are obtained. The Galerkin Method endowed with distinct features that account for its superiority over competing methods. The effect of different parameters on Velocity and Drag force are discussed with the help of graphs.
The expansion of power generation in Saudi Arabia is essential in order to meet the expected growth of its electricity demand. Due to the availability of high solar irradiation, vast rainless area and longtime sun light, Saudi Arabia is one of the most suitable countries to utilize solar energy resources in greater extend. Kingdom has planned to increase the production of solar power in order to meet a considerable share of country’s future energy demand. Numerous installation and research works are going on nowadays in the kingdom in order to attain its targets of solar power capacity in the coming years. Hence latest updates of country’s solar industry are essential for further research and R&D works in this field. Saudi Arabia’s current status and future possibility of solar industry are discussed in this paper.
Performance of RGB and L Base Supervised Classification Technique Using Multi...IJERA Editor
In the present growth of sensor technology is to improve the new chance and applications in GIS. This enhances the technology law a new method that should not focus on real time available products, but it must automatically lead to new ones. The aim of the paper is to make a maximum use of remote sensing data and GIS techniques to access land use and land cover classification in the Kiliyar sub basin sector in palar river of northen part of Tamil Nadu.IRS P6 LISS III is merged data to perform the classification using ERDAS Imaging. The RGB and L base supervised classification was based up on a Multispectral analysis, land use and land cover information‟s (maps and existing reports), which involves advanced technology and complex data processing to find detailed imagery in the study region. Ground surface reflects more radar energy emitted by the sensor from the study region, which makes it easy to distinguish between the water body, hilly, agriculture, settlement and wetland.
Cloud computing is an on demand service in which shared resources, information, software and other devices are provided to the end user as per their requirement at a specific time. A cloud consists of several elements such as clients, datacenters and distributed servers. There are n number of clients and end users involved in cloud environment. These clients may make requests to the cloud system simultaneously, making it difficult for the cloud to manage the entire load at a time. The load can be CPU load, memory load, delay or network load. This might cause inconvenience to the clients as there may be delay in the response time or it might affect the performance and efficiency of the cloud environment. So, the concept of load balancing is very important in cloud computing to improve the efficiency of the cloud. Good load balancing makes cloud computing more efficient and improves user satisfaction. This paper gives an approach to balance the incoming load in cloud environment by making partitions of the public cloud.
HTML5 e CSS3 já não é mais futuro. Nesta talk é abordado um overview de novas tecnologias que já estão em desenvolvimento e tornarão realidade nos próximos anos. Dentre elas: DOM Level 4, CSS Selectors Level 4 & 5, Media Query 4, HTML 5.1, Ecmascript 6 & 7+.
Apresentada em: 14 de Dezembro de 2013 no Front in Recife
Local: Faculdade Marista
Palestrantes: Luiz Tiago & Guilherme Farias
Soret Effect And Effect Of Radiation On Transient Mhd Free Convective Flow Ov...inventionjournals
The present paper is concerned to analyze the radiation, Magneto hydrodynamic and soret effects on unsteady flow heat and mass transfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinite vertical porous plate through porous media the porous plate is subjected to a transverse variable suction velocity. The transient, non linear and coupled dimensionless governing equations for this investigation are solved analytically using perturbation technique about a small parameter . the effects of governing parameters on the flow variables are discussed graphically.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Radiation Effects on MHD Free Convective Rotating Flow with Hall EffectsIJERA Editor
In this paper, we have studied the unsteady an incompressible MHD rotating free convection flow of Viscoelastic fluid through a porous medium with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter. The expressions for the velocity, temperature, concentration have been derived analytically and also its behaviour is computationally discussed with reference to different flow parameters with the help of graphs. The skin friction, the Nusselt number and the Sherwood number are also obtained and their behaviour discussed.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Effect of Thermo-Diffusion and Chemical Reaction on Mixed Convective Heat And...IJERA Editor
A finite element study of combined heat and mass transfer flow through a porous medium in a circular cylindrical annulus with Soret and Dufour effects in the presence of heat sources has been analyzed. The coupled velocity, energy, and diffusion equations are solved numerically by using Galerkin- finite element technique. Shear stress, Nusslet number and Sherwood number are evaluated numerically for different values of the governing parameters under consideration and are shown in tabular form.
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MIXED CONVECTIVE HEAT AND MASS TRANSFER MHD FLOW PAST AN UNSTEADY STRETCHING ...IAEME Publication
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unsteady porous stretching sheet is studied in the present analysis. The flow is subjected to
magnetic field normal to the vertical plate in a saturated porous medium. The governing non-linear
partial differential equations have been reduced to ordinary differential equations using suitable
similarity transformation variables. The resultant equations which are coupled and highly nonlinear
are solved by standard Runge-Kutta fourth order numerical technique via shooting method.
The momentum, temperature and concentration field distributions are analyzed and discussed
numerically and presented pictorially through graphs. Numerical values for skin friction
coefficient, local Nusselt number and Sherwood number at the plate in the presence of magnetic
field and porous medium are derived and discussed for various values of physical parameters and
are presented in table. Finally the present results are compared with previously published results
and found to be well in agreement.
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Homotopy Analysis to Soret and Dufour Effects on Heat and Mass Transfer of a ...iosrjce
The objective of this paper is to study the Soret and Dufour effects on the free convection boundary
layer flow of an incompressible, viscous and chemically reacting fluid over a vertical plate in the presence of
viscous dissipation. The governing partial differential equations are converted to a set of ordinary differential
equations using suitable similarity transformations. The resulting equations are solved analytically using
homotopy analysis method (HAM). The convergence of obtained analytical solutions is explicitly discussed. The
effects of various parameters on dimensionless velocity, temperature and concentration profiles are discussed
with the help of graphs. The numerical values of skin friction, Nusselt number and Sherwood number for
different parameters are presented in tabular form. Our results are compared with the previously published
results and are found to be in good agreement.
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The new method of enhancing heat transfer through tri- hybrid nanofluid is discussed in the current study and represented in differential equation system.
Effect of Chemical Reaction and Radiation Absorption on Unsteady Convective H...IJMER
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Investigation of the Effect of Nanoparticles Mean Diameter on Turbulent Mixed...A Behzadmehr
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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Dufour and Soret Effects on Convective Heat and Mass Transfer in Non-Darcy Doubly Stratified Porous Media
1. Srinivas Maripala Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 9( Version 4), September 2014, pp.17-26
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Dufour and Soret Effects on Convective Heat and Mass Transfer in Non-Darcy Doubly Stratified Porous Media Srinivas Maripala1, N.Kishan2 1Dept of Mathematics, Sreenidhi Institute of Science and Technology, Hyd. 2 Department of mathematics, Osmania University, Hyd. Abstract This paper deals with the MHD convention non-Darcy flow with heat and mass transfer along the vertical surface in a fluid stratified non porous media in the presence of temperature gradients (Soret effects) and concentration gradients (Dufour effects). The differential equations of the governing flow are expand a in terms of perturbation functions as a result we get set of ordinary differential equations of zero,first and second order so the set of equations are solved by using the implicit finite difference scheme. A parametric study of the physical parameters involve in the problem is conducted and a representative set of numerical results are illustrated graphically Key words: MHD, heat transfer , mass transfer double stratification , boundary layer, porous mediua, convection ,dufour number, soret number, finite difference method.
I. Introduction
Simultaneous heat and mass transfer from different geometries embedded in porous media has many engineering and geophysical applications such as geothermal reservoirs, drying of porous solids, thermal insulation, enhanced oil recovery, packed- bed catalytic reactors, cooling of nuclear reactors, and underground energy transport. Cheng and Minkowycz [1] have presented similarity solutions for free thermal convection from a vertical plate in a fluid-saturated porous medium. The problem of combined thermal convection from a semi-infinite vertical plate in the presence or absence of a porous medium has been studied by many authors (see, for example [2–5]). Nakayama and Koyama [4] have studied pure, combined and forced convection in Darcian and non-Darcian porous media. Lai and Kulacki [6] has investigated coupled heat and mass transfer by mixed convection from an isothermal vertical plate in a porous medium. Hsieh et al. [5] has presented non-similar solutions for combined convection in porous media. Chamkha [7] has investigated hydro magnetic natural convection from a isothermal inclined surface adjacent to a thermally stratified porous medium.There has been a renewed interest in studying magneto hydrodynamic (MHD) flow and heat transfer in porous and non-porous media due to the effect of magnetic fields on the boundary layer flow control and on the performance of many systems using electrically conducting fluids. In addition, this type of flow has attracted the interest of many investigators in view of its applications in many engineering problems such as MHD generators, plasma studies, nuclear reactors, geothermal energy extractions. For example, Rapits et al. [8] have analyzed hydro magnetic free convection flow through a porous medium between two parallel plates. Aldoss et al. [9] have studied mixed convection from a vertical plate embedded in a porous medium in the presence of a magnetic field. Gribben [10] has studied boundary-layer flow over a semi-infinite plate with an aligned magnetic field in the presence of a pressure gradient. He has obtained solutions for large and small magnetic Prandtl numbers using the method of matched asymptotic expansion. Soundalegkar [11] obtained approximate solutions for two-dimensional flow of an incompressible, viscous fluid past an infinite porous vertical plate with constant suction velocity, the difference between the temperature of the plate and the free steam is moderately large causing the free convection currents. Raptis and Kafousias [12] have studied the influence of a magnetic field on steady free convection flow through a porous medium bounded by an infinite vertical plate with constant suction velocity. Raptis [13] has studied mathematically the case of time-varying two- dimensional natural convective flow of an incompressible, electrically conducting fluid along an infinite vertical porous plate embedded in a porous medium. Bian et al. [14] have reported on the effect of an electromagnetic field on natural convection in an inclined porous medium. Buoyancy-driven convection in a rectangular enclosure with a transverse magnetic field has been considered by Garandet et al. [15] and Khanafer and Chamkha [16].
It is well-known that in double-diffusive (e.g. thermohaline) convection the coupling between the transport of heat and mass takes place because the density q of the fluid mixture depends on both
RESEARCH ARTICLE OPEN ACCESS
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temperature T and concentration C. For suffi- ciently small isobaric changes in temperature and concentration the mixture density q depends linearly on both T and C (Nield and Bejan [17]). In some circumstances there is direct coupling between T and C. This is when cross-diffusion (Soret and Dufour effects) is not negligible. The energy flux caused by a composition gradient was discovered in 1873 by Dufour and was correspondingly reffered to the Dufour effect. It was also called the diffusion-thermo effect. On the other hand, mass flux can also be created by a temperature gradient, as was established by Soret. This is the thermal-diffusion effect. The Soret effect has been also used for isotope separation and in mixture between gases with very light molecular weight ðH2;HeÞ and of medium molecular weight (H2, air) (Postelnicu [18]). In many studies Soret and Dufour effects are neglected, on the basis that they are of a smaller order of magnitude than the effects described by Fourier’s and Fick’s laws. There are, however, exceptions. Eckert and Drake [19] have presented several cases when the Dufour effect cannot be neglected. Platten and Legros [20] state that in most liquid mixtures the Dufour effect is inoperate, but that this may not be the case in gases. Mojtabi and Charrier-Mojtabi [21] confirm this by noting that in liquids the Dufour coefficient is an order of magnitude smaller than the Soret effect. Benano-Molly et al. [22] have studied the problem of thermal diffusion in binary fluid mixture, lying within a porous medium and subjected to a horizontal thermal gradient and have shown that multiple convection-roll flow patterns can develop depending on the values of the Soret number. They conclude that for saturated porous media, the phenomenon of cross diffusion is further complicated because of the interaction between the fluid and the porous matrix and because accurate values of the cross-diffusion coefficients are not available. The onset of Soret- driven convection in an infinite cell filled with a porous medium saturated by a binary fluid was studied by Sovran et al. [23]. However, Soret and Dufour effects have been found to appreciably influence the flow field in free convection boundary layer over a vertical surface embedded in a fluid- saturated porous medium (Postelnicu [18] and Anghel et al. [24]).
In certain porous media applications such as those involving heat removal from nuclear fuel debris, underground disposal of radioactive waste material, storage of food stuffs, and exothermic and/or endothermic chemical reactions and issociating fluids in packed-bed reactors, the working fluid heat generation (source) or absorption (sink) effects are important. Representative studies dealing with these effects have been reported by such authors as Acharya and Goldstein [25], Vajravelu and Nayfeh [26] and Chamkha [27,28]. In the combined heat and mass transfer studies in porous media, Murthy et al .[29] reported that the temperature and concentration became negative in the boundary layer depending on the relative intensity thermal and solutal stratification. These are a few articles to quote from the literature where the temperature/concentration or non dimensional heat/mass transfer coefficients becoming negative. Angirasa et al . [30] presented and analysis of combined heat and mass transfer in thermally stratified porous enclosure. Srinivasachrya et.al [31] studied the MHD and Radiation on Non- Darcy Mixed Convection, Ramakrishna and Hemalatha analyzed the Non-Darcy Mixed convection with Thermal Disipersion in a satrated porous medium. Rathish kumar and Shalini[32] presented the free convection heat and mass from a vertical wavy surface in a doubly stratified external porous layer.
Fig.I. Free convective heat and mass transfer from a semi infinite vertical wall in a fluid saturated doubly stratified doubly stratified porous medium.
The results presented in[29,33] have given the impetus for investigating free convection heat and mass transfer characteristics in a non-Darcian fluid saturated doubly stratified purpose medium .Unlike in [29] and [33], the medium is assumed to be stratified linearly with the height, which is more physically realistic case. As a result of this, mathematical model ceases to have a similar solution. In the light of the results obtained [34] Lakshmi Narayana and P.A.,Murthy has extended the analysis be more significant given be clear from the section, “Results and Discussion.” This type of investigation is useful in understanding heat and mass transfer characteristics around a hot radioactive subsurface storage site or around a cooling magnetic intrusion, where the theory of convection heat and mass transport is involved. In this paper we have investigated the MHD effects on steady free convective heat and mass transfer from a vertical surface embedded in a doubly stratified non-Darcy porous medium by using the implicit finite difference
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scheme. The results were compared with [35] when the Dufour and souret number is absent. Governing Equations Consider the flow configuration shown in figure.I. The flow is moderate (i.e., order of Reynolds number is greater than one), so pressure drop is proportional to the linear combination of fluid velocity and the square of the velocity (Forchheimer flow model). Viscous resistance due to the solid boundary is neglected under the assumption that the medium is with low permeability. A magnetic field is applied in the y-direction. The vertical wall temperature and concentration are assumed to be constant and the porous medium is assumed to be vertically linearly stratified with respect to both heat and concentration as indicated in Fig.I. The boundary layer equations along with the equation of continuity may be written as 휕푢 휕푥 + 휕푣 휕푦 =0 (1) [1+ 휎퐵02퐾 휌푣 ] 휕푢 휕푦 + 푐√퐾 푣 휕푢2 휕푦 = 훽푇퐾푔 푣 휕푇 휕푦 + 훽퐶퐾푔 푣 휕퐶 휕푦 (2) u 휕푇 휕푥 +푣 휕푇 휕푦 =훼 휕2푇 휕푦2 + 퐷푚퐾푇 푐푠푐푝 휕2퐶 휕푦2 (3) 푢 휕퐶 휕푥 +푣 휕퐶 휕푦 =D∂2C∂y2 + 퐷푚퐾푇 푇푚 휕2푇 휕푦2 (4) and the boundary conditions are 푦=0: 푣=0, T=푇푤, C = 퐶푤,} (5) 푦→∞: 푢→0, T=푇∞,0+퐴푥, C = 퐶∞,0+퐵푥 Here x and y are the Cartesian coordinate , 푢 and 푣 are the averaged velocity components in x and y directions respectively ,T and C are the dimensional temperature and concentration, respectively, 훽푇 and 훽퐶 are the coefficients of thermal and solutal expansion, respectively, c is the Forchheimer constant, ν is the kinematic viscosity of the fluid, K is the permeability, g is the acceleration due to gravity and α, D are the thermal and solutal diffusivities of the medium. 퐵0 is the strength of the maagetic field, 휎 is the electrical conductivity of the fluid, 흆 is the density, v is the kinematic viscosity. Here, A and B are constants ,varied to alter the intensity of stratification in the medium . Thicknesses of thermal and solutal boundary layers (훿푇,훿푐) which are shown in Fig. 1, are depending on the strength of the respective field and for all calculations we have taken the boundary layer thickness as the maximum of these values .The subscripts w, (∞, 0 )and ∞ indicate the conditions at the wall, at some reference point in the medium, and at the outer edge of the boundary layer respectively.
Making use of the following transformation which is derived using order magnitudes
Η = 푦 푥 R푎푥 12 , f (x,η)= 휓 훼푅푎푥 12 , (6) θ(x,η) = 푇−푇푥,0 푇푊−푇∞,0 - 퐴푥 푇푊−푇∞,0, (7) 휑(x,η) = 퐶−퐶∞,표 퐶푊−퐶∞,0 - 퐵푥 퐶푤−퐶∞,0, The governing eqns.(1)-(4) become (1+퐻푎2)푓"+2퐹푐푓ʹ푓"=휃ʹ+푁휑ʹ (8) 휃ʹʹ+ 12 푓휃ʹ+ 퐷푓휑″ = ε푓ʹ+ ε(푓ʹ휕휃 휕휀 −휃ʹ휕푓 휕휀 ) (9) 휑ʹʹ+ 12 퐿푒푓휑ʹ+ Le Sr 휃″ = LeSrε푓ʹ+Le ε(푓ʹ휕휑 휕휀 −휑ʹ휕푓 휕휀 ) (10) And the boundary conditions (5) transform into η=0: f(ε,0)=0, θ(ε,0)=1, 휑(ε,0)=1 η→∞ : 푓ʹ(휀,∞) =0, θ(ε,∞)=1, 휑(ε,∞)=0 11)
The various parameters involved in the present study which influence the system are the Darcy- Rayleigh number 푅푎푥= 퐾푔훽푇 푇푤−푇∞,0 푥 /훼푣 which is defined with reference to the thermal conditions, the inertia parameter 퐹푐 which is defined as 푐√퐾퐾푔훽푇 푇푤−푇∞,0 /푣2 the diffusivity ratio Le=α/D and the buoyancy ratio N = [훽푐(퐶푤− 퐶∞,0)]/[훽푇 푇푤−푇∞,0 ].When the Forchheimer coefficient c is zero, 퐹푐 will become zero, the inertia effects are absent in the medium , and the study becomes double stratification in the Darcy porous medium. Thermal buoyancy acts always vertically upwards, the species buoyancy may act in either direction depending on the relative molecular weight. So N>0 indicates aiding buoyancy where both the thermal and solutal buoyancies are in the same directions and N<0 indicates opposing buoyancy where the solutal buoyancy is in the opposite direction to the thermal buoyancy. .when N=0 ,the flow is driven by thermal buoyancy alone . The thermal stratification parameter is given by 휀= 퐴푥/ 푇푤−푇∞,0 . Here we are defining a quantity called the stratification ratio 푆푟=퐵 푇푤−푇∞,0 / 퐴(퐶푤−퐶∞,0) which is a constant , it indicates the relative strengths of solutal and thermal stratification in the medium. As 푇푤−푇∞,0 and 퐶푤− 퐶∞,0 are constants, when Sr>1, solutal stratification is more than the thermal stratification in the medium and vice versa. The product 푆푟.휀=퐵푥/(퐶푤− 퐶∞,0) Will be the solutal stratification parameter. 퐻푎2= 1+ 휎퐵02퐾 휌푣 is the magnetic parameter.
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Expanding the stream function, temperature distribution and concentration distribution in terms of perturbation functions f(ε,η)= −1 푚∞ 푚=0휀푚푓푚(휂) θ(ε,η)= −1 푚∞ 푚=0휀푚휃푚(휂) (12) φ(ε,η)= −1 푚∞ 푚=0휀푚휑푚(휂) and substituting Eq.(12)into Eqs.(8)-(11) and equating the coefficients of various powers of ε to zero, we have the following sets of ordinary differential equations for zeroth , first ,and second order in as following : (1+퐻푎2)푓0ʹʹ+2퐹푐푓0ʹ푓0ʹʹ=휃0ʹ+푁휑0ʹ 휃0ʹʹ+ 12 푓0휃0ʹ + 퐷푓 휑0″ = 0 (13) 휑0ʹʹ+ 12 퐿푒푓0휑0ʹ + Le Sr 휃0ʹʹ = 0 Subject to the boundary conditions 푓0 0 =0 , 휃0 0 =1 휑0 0 =1, 푓0ʹ ∞ =0,휃0 ∞ =0 ; 휑0 ∞ =0, (14) (1+퐻푎2)푓1ʹʹ+2퐹푐 푓0ʹ푓1ʹʹ+푓1ʹ푓0ʹʹ =휃1ʹ+푁휑1ʹ 휃1ʹʹ+ 1 2 푓1휃0ʹ+푓0휃1ʹ +푓0ʹ+(−푓0ʹ휃1+푓1휃0ʹ)=0 (15) 휑1ʹʹ+ 12 퐿푒 푓1휑0ʹ+푓0휑1ʹ +퐿푒푆푟푓0ʹ+퐿푒 −푓0ʹ휑1+푓1휑0ʹ +퐿푒푆푟푓0ʹ=0 Subject to the boundary conditions 푓1 0 =0, 휃1 0 =0, 휑1 0 =0 (16) 푓1ʹ ∞ =0, 휃1 ∞ =0 ; 휑1 ∞ =0, And (1+퐻푎2) 푓2ʹʹ+2퐹푐(푓0ʹ푓2ʹʹ+푓1ʹ푓1ʹʹ+푓2ʹ푓0ʹʹ)=휃2ʹ+푁ø2ʹ 휃2ʹʹ+ 12 푓0휃2ʹ+ 32 푓1휃1ʹ+ 52 푓2휃0ʹ−푓1ʹ휃1−2푓0ʹ휃2+푓1ʹ + 퐷푓 휑2ʹʹ = 0 (17) 휑2ʹʹ+ 12 퐿푒(푓2휑0ʹ+푓1휑1ʹ+푓0휑2ʹ)+퐿푒푆푟푓1ʹ−퐿푒(푓1ʹ휑1+2푓0ʹ휑2−푓1휑1ʹ−2푓2휑0ʹ ) + Le 퐿푒푆푟푓0ʹ= 0 Subject to the boundary conditions 푓2 0 =0,휃2 0 =0, 휑2 0 =0 푓2ʹ ∞ =0,휃2 ∞ =0 ; 휑2 ∞ =0, (18) Results and discussion In order to see the effects of various parameters on velocity(ƒ), Temperature(θ) and Concentration(φ). profiles we display Fig1-6. The values of Nusselt number and the Sherwood number for various values of embedded parameters are also display in figures. Fig.1 shows the effect of Thermal stratification parameter ε on velocity, temperature and concentration profiles. In the figure1.(i). variations of non-dimensional velocity profiles are present . From the figure it is seen that the increase of Thermal stratification parameter ε, the velocity profiles decreases up to some value of η inside boundary layer (0<η<5) after that the reverse phenomenon is observed for both the cases of Fc=0, Fc=1. Fig.1(ii). Illustrates the effect of Thermal stratification parameter ε on the Temperature profiles, it is noticed that as Thermal stratification parameter ε, increases, the temperature field decreases, as seen in the earlier cases for away from the plate the effect is not that much significant when Fc=0 and Fc=1. The effect of the Thermal stratification parameter ε on the concentration profiles is illustrate in fig.1.(iii). It is observed that as the Thermal stratification parameter ε increases the concentration appear to be decreasing not much of effect in the concentration is noticed in the boundary layer region and in the area far away from the plate . however significant variation is observed at a moderate distance from the plate when Fc=o and Fc=1
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Fig.2. shows the the effects of Diffusivity ratio Le on velocity(ƒ), Temperature(θ) and Concentration(φ). Fig.2(i). illustrates that with the increase of diffusivity ratio Le, the velocity profiles decrees up to some value of η after that the reverse phenomenon is observe for the both the cases of Fc=0 and Fc=1 Fig.2(ii). Shows the effects Diffusivity ratio Le the various values of Fc=0 and Fc=1 on temperature profiles. Note that that the temperature filed and the boundary layer thickness increases with Diffusivity ratio Le increases when Fc=0.it is seen in the Fig.2.ii.(a). and the reverse phenomenon we observe in the fig2.ii(b). when Fc=1. Fig.2.iii. displays the concentration profiles from this figure we seen that the concentration and the boundary layer thickness increases with Diffusivity ratio Le increases when FC=0 and Fc=1
0
0.5
1
1.5
2
2.5
3
0
5
10
15
ε=0
ε=0.01
ε=0.05
ƒ
Fc=0,N=0.5,Le=1,Df=1,Sr=0,Ha=0
1.i.a) Variationsof Non-Dimensional Velocity Profiles
η
0
0.5
1
1.5
2
2.5
0
2
4
6
8
10
12
ε=0
ε=0.01
ε=0.05
Fc=1,N=0.5,Le=1,Df=1,Sr=0,Ha=0
η
1.i.b) Variationsof Non-Dimensional Velocity Profiles
ƒ
0
0.5
1
1.5
2
0
1
2
3
4
ε = 0
ε = 0.01
ε = 0.03
η
η
Fc=0,N=0.5,Le=1,Df=1,Sr=0,Ha=0
1.ii.a) Variationsof Non-Dimensional Tempreature Profiles
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
1
2
3
4
ε = 0
ε = 0.01
ε = 0.03
Fc=1,N=0.5,Le=1,Df=1,Sr=0,Ha=0
η
η
1.ii.b) Variationsof Non-Dimensional Tempreature Profiles
0
0.2
0.4
0.6
0.8
1
0
1
2
3
4
5
6
ε = 0
ε = 0.01
ε = 0.03
Fc=0,N=0.5,Le=1,Df=1,Sr=0,Ha=0
1'iii.a)Variationsof Non-Dimensional Concentration Profiles
η
η
0
0.2
0.4
0.6
0.8
1
0
2
4
6
ε = 0
ε = 0.01
ε = 0.03
Fc=1,N=0.5,Le=1,Df=1,Sr=0,Ha=0
1.iii.b)Variationsof Non-Dimensional Concentration Profiles
η
η
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
5
10
15
Le = 0
Le = 1
Le = 2
η
ƒ
Fc=0,ε=0.01,Df=0.1,Sr=0.1,Ha=0
2.i.a) Variationsof Non-Dimensional Velocity Profiles
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0
5
10
15
Le = 0
Le = 1
Le = 2
η
ƒ
Fc=1,N=0.5,ε=0.01,Df=0.1,Sr=0.1,Ha=0
2.i.b) Variationsof Non-Dimensional Velocity Profiles
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Fig.3.illustrates for the different values of Buoyancy ratio N effects on velocity(ƒ), Temperature(θ) and Concentration(φ).In fig3.(i). the effect of Buoyancy ratio N for aiding Buoyancy ( N>0) is to increases the velocity profiles, where as the effect of opposing buoyancy (N<0) is to decreases the velocity profiles . the effect of Buoyancy ratio N on Temperature profiles from fig.3.(ii). Observed that when N increases the temperature profiles will decreases for Fc=0 and Fc=1 same phenomenon we observe in the concentration profiles is evident from figrure3.(iii).
0
0.2
0.4
0.6
0.8
1
0
5
10
15
Le = 0
Le = 1
Le = 2
Fc=0,N=0.5,ε=0.01,Df=0.1,Sr=0.1,Ha=0
2.ii.a) Variationsof Non-Dimensional Tempreature Profiles
η
η
0
0.2
0.4
0.6
0.8
1
-1
4
9
14
Le = 0
Le = 1
Le = 2
Fc=1,N=0.5,ε=0.01,Df=0.1,Sr=0.1,Ha=0
2.ii.b) Variationsof Non-Dimensional Tempreature Profiles
η
η
0
0.2
0.4
0.6
0.8
1
0
5
10
15
Le = 0
Le = 1
Le = 2
2.iii.a) Variationsof Non-Dimensional Concentration Profiles
Fc=0,N=0.5,ε=0.01,Df=0.1,Sr=0.1,Ha=0
Ф
η
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
5
10
15
Le = 0
Le = 1
Le = 2
Fc=1,N=0.5,ε=0.01,Df=0.1,Sr=0.1,Ha=0
2.iii.b) Variationsof Non-Dimensional Concentration Profiles
η
η
0
0.2
0.4
0.6
0.8
1
1.2
0
5
10
15
N = -0.2
N = -0.1
N= 0
N= 0.1
N= 0.2
ƒ
Fc=0,,Le=0.1,Df=0.1,Sr=0.1,ε=0.01,Ha=0
η
3.i.a) Variationsof Non-Dimensional Velocity Profiles
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
5
10
15
N = -0.2
N = -0.1
N= 0
N= 0.1
N= 0.2
3.i.b) Variationsof Non-Dimensional Velocity Profiles
Fc=1,ε=0.01,Le=0.1,Df=0.1,Sr=0.1,Ha=0
ƒ
η
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
5
10
15
N = -0.2
N = -0.1
N = 0
N = 0.1
N = 0.2
Fc=0,ε=0.01,Le=0.1,Df=0.1,Sr=0.1,Ha=0
3.ii.a) Variationsof Non-Dimensional Tempreature Profiles
η
η
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1
4
9
14
N = -0.2
N = -0.1
N = 0
N = 0.1
N = 0.2
Fc=1,ε=0.01,Le=0.1,Df=0.1,Sr=0.1,Ha=0
3.ii.b) Variationsof Non-Dimensional Tempreature Profiles
η
η
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In fig 4 it is observed that the Duffer number Df effect is slidely reduces the velocity profiles is evident from fig.4.i. we have seen in the fig.4.(ii). The effect of Duffer number Df on the Temperature profiles it is noticed that as the Df increases a increasing trend in the temperature filed is noticed . not much of significant contribution of Df is noticed as we move far away from the plate when Fc=0 and Fc=1. The reverse phenomenon we observe in fig.4.iii. of concentration profiles.
0
0.2
0.4
0.6
0.8
1
0
5
10
15
N = -0.5
N = 0
N = 0.5
3.iii.a) Variationsof Non-Dimensional Concentration Profiles
Fc=0,Le=0.1,ε=0.01,Df=0.1,Sr=0.1,Ha=0
η
η
0
0.2
0.4
0.6
0.8
1
0
5
10
15
N = -0.5
N = 0
N = 0.5
Fc=1,Le=0.1,ε=0.01,,Df=0.1,Sr=0.1,Ha=0
η
η
3.iii.b) Variationsof Non-Dimensional Concentration Profiles
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0
5
10
15
Df = 0
Df = 0.1
Df = 0.3
Fc=0,N=0.5,Le=0.1,ε=0.01,Sr=0.1,Ha=0
4.i.a) Variationsof Non-Dimensional Velocity Profiles
ƒ
η
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
5
10
15
Df = 0
Df = 0.1
Df = 0.3
η
Fc=1,N=0.5,Le=0.1,ε=0.01,Sr=0.1,Ha=0
ƒ
4.i.b) Variationsof Non-Dimensional Velocity Profiles
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
5
10
15
Df = 0
Df = 0.1
Df = 0.3
Fc=0,N=0.5,Le=0.1,ε=0.01,Sr=0.1,Ha=0
4.ii.a) Variationsof Non-Dimensional Tempreature Profiles
η
η
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
5
10
15
Df = 0
Df = 0.1
Df = 0.3
4.ii.b) Variationsof Non-Dimensional Tempreature Profiles
Fc=1,N=0.5,Le=0.1,ε=0.01,Sr=0.1,Ha=0
η
η
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
5
10
15
Df = 0
Df = 0.1
Df =0. 3
4.iii.a) Variationsof Non-Dimensional ConcentrationProfiles
Fc=0,N=0.5,Le=0.1,ε=0.01,Sr=0.1,Ha=0
η
η
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
5
10
15
Df = 0
Df =0. 1
Df =0. 3
Fc=1,N=0.5,Le=0.1,ε=0.01,Sr=0.1,Ha=0
4.iii.b) Variationsof Non-Dimensional Concentration Profiles
Ф
η
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In the figure 5 , the velocity(ƒ), Temperature(θ) and Concentration(φ) graphs are shown graphically. It is observed that from the figure the effect of stratification number Sr , is in to increases the velocity profiles. The reverse phenomenon observes after the some number η. Fig.5.ii. shows that the effect of Sr on temperature profiles. We observed that when Fc=0, the temperature profile decreases when Sr=1 increase when Sr=3. Compare with Sr=0 but Fc=1, in this case temperature profiles decreasing with increasing of Sr. It is clear from the figure 5.iii. the effect of Sr is to increases the concentration profiles.
The effect of magnetic parameter on on ƒ,θ and φ explains graphically from the fig.6. we observe in these fig. the magnetic parameter increases the velocity profiles but decrease the temperature profiles and concentration profiles. It explains fig.6.i.,fig.6.ii.,fig.6.iii. respectively. In the case of aiding buoyancy the heat transfer coefficient increases with the increases of Sr is observed from figure . the mass transfer coefficient is shown in fig.7 for different values of Le and Sr the mass transform coefficient become negative and increases with the increasing value of Sr and it decreases of with the increases of Loweies number Le. From fig.8 it is evedient that with the effect of magnetic parameter the concentration profiles increases.
0
0.5
1
1.5
2
0
5
10
15
Sr =0
Sr = 1
Sr = 3
5.i.a) Variationsof Non-Dimensional Velocity Profiles
Fc=0,N=0.5,Le=0.1,Df=0.1,ε=0.01,Ha=0
ƒ
η
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
5
10
15
Sr =0
Sr = 1
Sr = 3
η
Fc=1,N=0.5,Le=0.1,Df=0.1,ε=0.01,Ha=0
ƒ
5.i.b) Variationsof Non-Dimensional Velocity Profiles
0
0.2
0.4
0.6
0.8
1
0
5
10
15
Sr = 0
Sr = 1
Sr = 3
5.ii.a) Variationsof Non-Dimensional Tempreature Profiles
Fc=0,N=0.5,Le=0.1,Df=0.1,ε=0.01,Ha=0
η
η
0
0.2
0.4
0.6
0.8
1
0
5
10
15
Sr = 0
Sr = 1
Sr = 3
Fc=1,N=0.5,Le=0.1,Df=0.1,ε=0.01,Ha=0
5.ii.b) Variationsof Non-Dimensional Tempreature Profiles
η
η
0
0.2
0.4
0.6
0.8
1
1.2
0
5
10
15
Sr = 0
Sr = 1
Sr = 3
Fc=0,N=0.5,Le=0.1,Df=0.1,ε=0.01,Ha=0
5.iii.a) Variationsof Non-Dimensional Concentration Profiles
η
η
0
0.2
0.4
0.6
0.8
1
1.2
0
5
10
15
Sr = 0
Sr = 1
Sr = 3
5.iii.b) Variationsof Non-Dimensional Concentration Profiles
Fc=1,N=0.5,Le=0.1,Df=0.1,ε=0.01,Ha=0
η
η
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0
1
2
3
4
5
6
7
0
0.2
0.4
0.6
0.8
1
Le=0.1,Sr=0.1
Le=0.1,sr=1
Le=0.1,Sr=5
(7).Variationof nondimensional heat transfer coefficient with εfor
fixed Fc,N,Df,Ha
ε
Nux/Rax
Fc=0,N=2
-5
-4
-3
-2
-1
0
0
0.2
0.4
0.6
0.8
1
Le=1,Sr=0.1
Le=0.1,Sr=5
Le=0.1,Sr=1
Le=0.1,sr=0.1
ε
Shx/Rax
(8).Varation of nondimentional mass transfer coefficientwith
ε for fixed Fc,N,Df,Ha
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[33.] Rathish Kumar, B. V., and Shalini, G., 2005, “Combined Influence of Mass and Thermal Stratification on Double Diffusion non-Darcian Natural Convection from a Wavy Vertical Wall to Porous Media,” ASME J. Heat Transfer,127, pp. 637–642
[34.] Chamkha, A. J., and Khaled, A.-R. A., 2000, “Hydro-Magnetic Simultaneous Heat and Mass Transfer by Mixed Convection from a Vertical Plate Embedded in a Stratified Porous Medium With Thermal Dispersion Effects,” Int. J. Heat Mass Transfer, 36, pp. 63–70
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[36.] N.Kishan , Srinivas Maripala and C.Srinivas Reddy .2011, “MHD EFFECTS ON FREE CONVECTIVE HEAT AND MASS TRANSRFER IN A DOUBLY STRATIFIED NON-DARCY POROUS MEDIUM”, Vol. 3 No.12 December 2011