This document summarizes a research paper that analyzes the influence of chemical reaction and heat source on MHD free convection boundary layer flow of a radiation absorbing fluid through a porous medium past a semi-infinite vertical plate. The governing equations for momentum, energy and species transport are presented. Dimensionless variables and parameters are introduced. Boundary conditions at the plate and in the free stream are specified. The Rosseland approximation is used to model radiation heat transfer. The effects of various physical parameters on velocity, temperature and concentration fields are studied through graphs.
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.
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
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.
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.
This document summarizes a research paper that analyzes unsteady magnetohydrodynamic (MHD) free convection heat and mass transfer flow through a non-homogeneous porous medium bounded by an infinite porous vertical plate. The flow considers effects of radiation, chemical reaction, temperature gradient dependent heat source, and exponentially decreasing permeability and suction velocity. Governing equations are derived and solved approximately using perturbation techniques. Results are discussed for cooling of the plate, showing effects of parameters on velocity, skin friction, and heat/mass transfer rates through graphs and tables.
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.
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.
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.
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.
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
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.
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.
This document summarizes a research paper that analyzes unsteady magnetohydrodynamic (MHD) free convection heat and mass transfer flow through a non-homogeneous porous medium bounded by an infinite porous vertical plate. The flow considers effects of radiation, chemical reaction, temperature gradient dependent heat source, and exponentially decreasing permeability and suction velocity. Governing equations are derived and solved approximately using perturbation techniques. Results are discussed for cooling of the plate, showing effects of parameters on velocity, skin friction, and heat/mass transfer rates through graphs and tables.
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.
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.
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.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
The International Journal of Engineering and Science (The 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.
This document summarizes a research paper that analyzes the effects of Soret and Dufour on an unsteady magnetohydrodynamic free convective fluid flow past a vertical porous plate with suction or injection. The paper presents the governing equations and solves them numerically using finite difference methods. It discusses the effects of various parameters like Soret number, Dufour number, suction/injection on velocity, temperature and concentration profiles. Tables show values of skin friction coefficient, Nusselt number and Sherwood number for different physical parameters. Previous studies examining combined heat and mass transfer with effects of radiation, magnetic fields, viscosity variations are also reviewed.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Thermal Radiation and Viscous Dissipation Effects on an Oscillatory Heat and ...rahulmonikasharma
An anticipated outcome that is intended chapter is to investigate effects of magnetic field on an oscillatory flow of a viscoelastic fluid with thermal radiation, viscous dissipation with Ohmic heating which bounded by a vertical plane surface, have been studied. Analytical solutions for the quasi – linear hyperbolic partial differential equations are obtained by perturbation technique. Solutions for velocity and temperature distributions are discussed for various values of physical parameters involving in the problem. The effects of cooling and heating of a viscoelastic fluid compared to the Newtonian fluid have been discussed.
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: Crimson Publishers_ Journal of electronics and communication engineering impact factor
In this analysis the MHD stagnation point flow of Jeffrey fluid over a stretching/shrinking sheet through porous medium is studied. The governing partial differential equations are transformed into nonlinear ordinary differential equation using the similarity transformations and are solved shooting technique. The effects of governing parameters on the velocity, the temperature and the concentration while the skin friction coefficients, the rate of heat transfer are studied graphically
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
MIXED CONVECTIVE HEAT AND MASS TRANSFER MHD FLOW PAST AN UNSTEADY STRETCHING ...IAEME Publication
Mixed two dimensional convection heat and mass transfer flow with suction, viscous
dissipation, heat source/sink effect, mass diffusion including Soret and Dufour effects due to an
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.
This document summarizes a study on the effects of magnetohydrodynamic mixed convection of a micropolar fluid near a stagnation point on a vertical stretching sheet, accounting for radiation and mass transfer. The governing equations are transformed into ordinary differential equations using similarity transformations and solved numerically. Parameters such as material property, radiation, magnetic field, and velocity ratio are varied to analyze their effects on velocity, temperature, concentration, skin friction, heat and mass transfer rates. It is observed that the micropolar fluid can reduce drag forces and act as a cooling agent, and that radiation effects are important for flows at high temperatures.
The Effect of Radiation on the Convective Heat and Mass Transfer Flow of a Vi...inventionjournals
The Effect of Radiation on the Convective Heat and Mass Transfer Flow of a Viscous Electrically Conducted Fluid in a Horizontal Rotating Channel in the Presence of Constant Heat Sources.
Minimal M-gs Open and Maximal M-gs Closed Sets In Interior Minimal Spaceinventionjournals
The main objective of this paper is to study the notions of Minimal M-GS Closed set, Maximal M-GS Open set, Minimal M-GS Open set and Maximal M-GS Closed set and their basic properties in Interior Minimal Space.
Key Words: Maximal GS-closed set, Maximal GS-open set, Minimal GS-closed set, Minimal GS-open set
The Study of Heat Generation and Viscous Dissipation on Mhd Heat And Mass Dif...IOSR Journals
The present work is devoted to the numerical study of magneto hydrodynamic (MHD) natural convection flow of heat and mass transfer past a plate taking into account viscous dissipation and internal heat generation. The governing equations and the associated boundary conditions for this analysis are made non dimensional forms using a set of dimensionless variables. Thus, the non dimensional governing equations are solved numerically using finite difference method Crank-Nicolson’s scheme. Numerical outcomes are found for different values of the magnetic parameter, Modified Grashof number, Prandtl number, Eckert number, heat generation parameter and Schmidt number for the velocity and the temperature within the boundary layer as well as the skin friction coefficients and the rate of heat and mass transfer along the surface. Results are presented graphically with detailed discussion.
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
MHD Free Convection and Mass Transfer Flow past a Vertical Flat PlateIJMER
1) This document discusses a study of magnetohydrodynamic (MHD) free convection and mass transfer flow past a vertical flat plate. The study considers the effects of heat source, thermal diffusion, large suction, and a uniform magnetic field applied normal to the flow.
2) Governing equations for momentum, energy, and concentration are introduced and solved using perturbation techniques. Expressions are obtained for velocity, temperature, concentration, skin friction, and rates of heat and mass transfer.
3) Results are discussed through graphs and tables to observe the effects of various parameters like magnetic field, heat source, suction, and thermal diffusion on flow, temperature, concentration, and transport properties.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
This document presents a study of unsteady magnetopolar free convection flow through a porous medium with radiation and variable suction. The governing equations for this problem are derived and solved using a series expansion method. Results are presented graphically showing the effects of various parameters like permeability, magnetic field, thermal Grashof number, and slip on velocity, angular velocity, temperature, concentration, skin friction, and heat transfer rate for the fluids air and water. Key findings include that decreasing thermal Grashof number decreases skin friction in air but increases it in water, and decreasing the temperature jump parameter increases the heat transfer rate.
Effects of Variable Viscosity and Thermal Conductivity on MHD Free Convection...IRJET Journal
This document summarizes a numerical study that examines the effects of temperature dependent viscosity and thermal conductivity on the steady, laminar boundary layer flow of a dusty fluid near a vertically stretching sheet. The study considers magnetohydrodynamic (MHD) free convection flow with heat generation. The governing nonlinear partial differential equations are reduced to ordinary differential equations using similarity transformations and then solved numerically using a shooting method. Results for velocity, temperature, species concentration, and skin friction, Nusselt, and Sherwood numbers are obtained and analyzed for various physical parameters, providing insights into heat and mass transfer with practical applications in industries like material processing and manufacturing.
Chemical Reaction Effects on Free Convective Flow of a Polar Fluid from a Ver...IOSR Journals
This article deals with a study of two dimensional free convective flow of a polar fluid through a porous medium due to combined effects of thermal and mass diffusion in presence of a chemical reaction of first order. The objective of the present investigation is to analyze the free convective flow in the presence of prescribed wall heat flux and mass flux condition. The governing partial differential equations are non-dimensionalized and transformed into a system of non-similar equations. The resulting coupled nonlinear partial differential equations are solved numerically under appropriate transformed boundary conditions using an implicit finite difference scheme in combination with quasilinearisation technique. Computations are performed for a wide range of values of the various governing flow parameters of the velocity, angular velocity, temperature and species concentration profiles and results are presented graphically. The numerical results for local skin friction coefficient, couple stress coefficient, local Nusselt number and local Sherwood number are also presented. The obtained results are compared with previously published work and were to be in excellent agreement. The study reveals that the flow characteristics are profoundly influenced by the polar effects
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Effect of Radiation on Mixed Convection Flow of a Non-Newtonian Nan fluid ove...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
This document summarizes a study on mixed convective heat and mass transfer flow of a visco-elastic fluid over a stretching sheet. The study considers the effects of chemical reaction, temperature gradient dependent heat sink, and magnetic field. The governing equations for momentum, energy and diffusion are presented and non-linear similarity transformations are applied to reduce the equations to an ordinary differential equation system. The system is solved numerically using a Runge-Kutta method with shooting techniques to determine skin friction coefficient, Nusselt number, and Sherwood number.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
The International Journal of Engineering and Science (The 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.
This document summarizes a research paper that analyzes the effects of Soret and Dufour on an unsteady magnetohydrodynamic free convective fluid flow past a vertical porous plate with suction or injection. The paper presents the governing equations and solves them numerically using finite difference methods. It discusses the effects of various parameters like Soret number, Dufour number, suction/injection on velocity, temperature and concentration profiles. Tables show values of skin friction coefficient, Nusselt number and Sherwood number for different physical parameters. Previous studies examining combined heat and mass transfer with effects of radiation, magnetic fields, viscosity variations are also reviewed.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Thermal Radiation and Viscous Dissipation Effects on an Oscillatory Heat and ...rahulmonikasharma
An anticipated outcome that is intended chapter is to investigate effects of magnetic field on an oscillatory flow of a viscoelastic fluid with thermal radiation, viscous dissipation with Ohmic heating which bounded by a vertical plane surface, have been studied. Analytical solutions for the quasi – linear hyperbolic partial differential equations are obtained by perturbation technique. Solutions for velocity and temperature distributions are discussed for various values of physical parameters involving in the problem. The effects of cooling and heating of a viscoelastic fluid compared to the Newtonian fluid have been discussed.
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: Crimson Publishers_ Journal of electronics and communication engineering impact factor
In this analysis the MHD stagnation point flow of Jeffrey fluid over a stretching/shrinking sheet through porous medium is studied. The governing partial differential equations are transformed into nonlinear ordinary differential equation using the similarity transformations and are solved shooting technique. The effects of governing parameters on the velocity, the temperature and the concentration while the skin friction coefficients, the rate of heat transfer are studied graphically
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
MIXED CONVECTIVE HEAT AND MASS TRANSFER MHD FLOW PAST AN UNSTEADY STRETCHING ...IAEME Publication
Mixed two dimensional convection heat and mass transfer flow with suction, viscous
dissipation, heat source/sink effect, mass diffusion including Soret and Dufour effects due to an
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.
This document summarizes a study on the effects of magnetohydrodynamic mixed convection of a micropolar fluid near a stagnation point on a vertical stretching sheet, accounting for radiation and mass transfer. The governing equations are transformed into ordinary differential equations using similarity transformations and solved numerically. Parameters such as material property, radiation, magnetic field, and velocity ratio are varied to analyze their effects on velocity, temperature, concentration, skin friction, heat and mass transfer rates. It is observed that the micropolar fluid can reduce drag forces and act as a cooling agent, and that radiation effects are important for flows at high temperatures.
The Effect of Radiation on the Convective Heat and Mass Transfer Flow of a Vi...inventionjournals
The Effect of Radiation on the Convective Heat and Mass Transfer Flow of a Viscous Electrically Conducted Fluid in a Horizontal Rotating Channel in the Presence of Constant Heat Sources.
Minimal M-gs Open and Maximal M-gs Closed Sets In Interior Minimal Spaceinventionjournals
The main objective of this paper is to study the notions of Minimal M-GS Closed set, Maximal M-GS Open set, Minimal M-GS Open set and Maximal M-GS Closed set and their basic properties in Interior Minimal Space.
Key Words: Maximal GS-closed set, Maximal GS-open set, Minimal GS-closed set, Minimal GS-open set
The Study of Heat Generation and Viscous Dissipation on Mhd Heat And Mass Dif...IOSR Journals
The present work is devoted to the numerical study of magneto hydrodynamic (MHD) natural convection flow of heat and mass transfer past a plate taking into account viscous dissipation and internal heat generation. The governing equations and the associated boundary conditions for this analysis are made non dimensional forms using a set of dimensionless variables. Thus, the non dimensional governing equations are solved numerically using finite difference method Crank-Nicolson’s scheme. Numerical outcomes are found for different values of the magnetic parameter, Modified Grashof number, Prandtl number, Eckert number, heat generation parameter and Schmidt number for the velocity and the temperature within the boundary layer as well as the skin friction coefficients and the rate of heat and mass transfer along the surface. Results are presented graphically with detailed discussion.
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
MHD Free Convection and Mass Transfer Flow past a Vertical Flat PlateIJMER
1) This document discusses a study of magnetohydrodynamic (MHD) free convection and mass transfer flow past a vertical flat plate. The study considers the effects of heat source, thermal diffusion, large suction, and a uniform magnetic field applied normal to the flow.
2) Governing equations for momentum, energy, and concentration are introduced and solved using perturbation techniques. Expressions are obtained for velocity, temperature, concentration, skin friction, and rates of heat and mass transfer.
3) Results are discussed through graphs and tables to observe the effects of various parameters like magnetic field, heat source, suction, and thermal diffusion on flow, temperature, concentration, and transport properties.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
This document presents a study of unsteady magnetopolar free convection flow through a porous medium with radiation and variable suction. The governing equations for this problem are derived and solved using a series expansion method. Results are presented graphically showing the effects of various parameters like permeability, magnetic field, thermal Grashof number, and slip on velocity, angular velocity, temperature, concentration, skin friction, and heat transfer rate for the fluids air and water. Key findings include that decreasing thermal Grashof number decreases skin friction in air but increases it in water, and decreasing the temperature jump parameter increases the heat transfer rate.
Effects of Variable Viscosity and Thermal Conductivity on MHD Free Convection...IRJET Journal
This document summarizes a numerical study that examines the effects of temperature dependent viscosity and thermal conductivity on the steady, laminar boundary layer flow of a dusty fluid near a vertically stretching sheet. The study considers magnetohydrodynamic (MHD) free convection flow with heat generation. The governing nonlinear partial differential equations are reduced to ordinary differential equations using similarity transformations and then solved numerically using a shooting method. Results for velocity, temperature, species concentration, and skin friction, Nusselt, and Sherwood numbers are obtained and analyzed for various physical parameters, providing insights into heat and mass transfer with practical applications in industries like material processing and manufacturing.
Chemical Reaction Effects on Free Convective Flow of a Polar Fluid from a Ver...IOSR Journals
This article deals with a study of two dimensional free convective flow of a polar fluid through a porous medium due to combined effects of thermal and mass diffusion in presence of a chemical reaction of first order. The objective of the present investigation is to analyze the free convective flow in the presence of prescribed wall heat flux and mass flux condition. The governing partial differential equations are non-dimensionalized and transformed into a system of non-similar equations. The resulting coupled nonlinear partial differential equations are solved numerically under appropriate transformed boundary conditions using an implicit finite difference scheme in combination with quasilinearisation technique. Computations are performed for a wide range of values of the various governing flow parameters of the velocity, angular velocity, temperature and species concentration profiles and results are presented graphically. The numerical results for local skin friction coefficient, couple stress coefficient, local Nusselt number and local Sherwood number are also presented. The obtained results are compared with previously published work and were to be in excellent agreement. The study reveals that the flow characteristics are profoundly influenced by the polar effects
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Effect of Radiation on Mixed Convection Flow of a Non-Newtonian Nan fluid ove...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
This document summarizes a study on mixed convective heat and mass transfer flow of a visco-elastic fluid over a stretching sheet. The study considers the effects of chemical reaction, temperature gradient dependent heat sink, and magnetic field. The governing equations for momentum, energy and diffusion are presented and non-linear similarity transformations are applied to reduce the equations to an ordinary differential equation system. The system is solved numerically using a Runge-Kutta method with shooting techniques to determine skin friction coefficient, Nusselt number, and Sherwood number.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Low power architecture of logic gates using adiabatic techniquesnooriasukmaningtyas
The growing significance of portable systems to limit power consumption in ultra-large-scale-integration chips of very high density, has recently led to rapid and inventive progresses in low-power design. The most effective technique is adiabatic logic circuit design in energy-efficient hardware. This paper presents two adiabatic approaches for the design of low power circuits, modified positive feedback adiabatic logic (modified PFAL) and the other is direct current diode based positive feedback adiabatic logic (DC-DB PFAL). Logic gates are the preliminary components in any digital circuit design. By improving the performance of basic gates, one can improvise the whole system performance. In this paper proposed circuit design of the low power architecture of OR/NOR, AND/NAND, and XOR/XNOR gates are presented using the said approaches and their results are analyzed for powerdissipation, delay, power-delay-product and rise time and compared with the other adiabatic techniques along with the conventional complementary metal oxide semiconductor (CMOS) designs reported in the literature. It has been found that the designs with DC-DB PFAL technique outperform with the percentage improvement of 65% for NOR gate and 7% for NAND gate and 34% for XNOR gate over the modified PFAL techniques at 10 MHz respectively.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Embedded machine learning-based road conditions and driving behavior monitoring
asianjournal.pdf
1. See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/274898987
INFLUENCE OF CHEMICAL REACTION AND HEAT SOURCE ON MHD FREE
CONVECTION BOUNDARY LAYER FLOW OF RADIATION ABSORBING
KUVSHINSHIKI FLUID IN POROUS MEDIUM
Article · January 2015
CITATION
1
READS
135
1 author:
Mohammed Ibrahim
GITAM University
53 PUBLICATIONS 457 CITATIONS
SEE PROFILE
All content following this page was uploaded by Mohammed Ibrahim on 14 April 2015.
The user has requested enhancement of the downloaded file.
2. Asian Journal of Mathematics and Computer
Research
3(2): 87-103, 2015
International Knowledge Press
www.ikpress.org
_____________________________________
*Corresponding author: ibrahimsvu@gmail.com;
INFLUENCE OF CHEMICAL REACTION AND HEAT
SOURCE ON MHD FREE CONVECTION BOUNDARY
LAYER FLOW OF RADIATION ABSORBING
KUVSHINSHIKI FLUID IN POROUS MEDIUM
S. MOHAMMED IBRAHIM1*
AND K. SUNEETHA1
1
Department of Mathematics, Priyadarshini College of Engineering & Technology, Nellore,
Andhra Pradesh, India.
AUTHORS’ CONTRIBUTIONS
This work was carried out in collaboration between both authors. Author SMI conceived of the study,
directed analyses and led the writing of the paper. Author KS assisted with the conceptualization, analysis,
and writing. Both authors contributed to the interpretation of results, and review of successive drafts of the
manuscript.
Received: 17 February 2015
Accepted: 14 March 2015
Published: 11 April 2015
_______________________________________________________________________________
ABSTRACT
The purpose of this research paper is an unsteady MHD two-dimensional free convection flow of a viscous,
incompressible, radiating, chemically reacting, radiation absorbing Kuvshinshiki fluid through a porous
medium past a semi-infinite vertical plate are investigated in presence of heat source. The non dimensional
governing equations are solved by a regular perturbation law. The expressions for velocity, temperature and
concentration fields are obtained. The effects of various physical parameters on the above flow quantities are
studied through graphs. Finally, the values of the local skin-friction coefficient, rate of heat transfer and rate
of mass transfer are also shown in tabular form.
Keywords: MHD; unsteady; boundary layer; porous medium; radiation; heat source; chemical reaction.
1 Introduction
A porous medium is characterized by a partitioning of the total volume into solid matrix and pore space,
with the latter being filled by one or more fluids. The convection problem in a porous medium has important
applications in geothermal reservoirs and geothermal extractions. The process of heat and mass transfer is
encountered in aeronautics, fluid fuel nuclear reactor, chemical process industries and many engineering
applications in which the fluid is the working medium. The wide range of technological and industrial
applications has stimulated considerable amount of interest in the study of heat and mass transfer in
convection flows. Free convective flow past a vertical plate has been studied extensively by Ostrach [1].
Comprehensive discussions and or reviews are found in literature [2-5].
Original Research Article
3. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
88
The study of hydromagnetic free convection flow finds applications in science and engineering, in areas
such as geophysical exploration, solar physics and astrophysical studies. Chamkha [6] proposed the unsteady
magnetohydrodynamic convective heat and mass transfer past a semi-infinite vertical permeable moving
plate. A problem of convective heat transfer over a continuously moving plate embedded in a non-Darcian
porous medium with magnetic field was analyzed by Abo-Eldahab and Gendy [7]. Sivaiah et al. [8] studied
the heat and mass transfer effects on MHD free convective flow past a vertical porous plate.
The effects of radiation on free convection on the accelerated flow of a viscous incompressible fluid past an
infinite vertical porous plate has many important technological applications in the astrophysical, geophysical
and engineering problem. Radiation and free convection flow past a moving plate was considered by Raptis
and Perdikis [9]. Ghosh et al. [10] studied the thermal radiation effects on unsteady MHD free convective
flow along an inclined plane. Abo-Eldahab and Gendy [11] reported that thermal radiation effects on
convective heat transfer in an electrically conducting fluid past a stretching surface in presence of variable
viscosity and uniform free-stream.
In the above mentioned studies the heat source/sink effect is ignored. Due to its great applicability to
ceramic tiles production problems, the study of heat transfer in the presence of a heat source/sink has
acquired newer dimensions. A number of analytical studies have been carried made of for various forms of
heat generation in [12–15]. Singh [16] analyzed MHD free convection and mass transfer flow with heat
source and thermal diffusion. Heat source effect on MHD convective flow from a sphere to a non-Darcian
porous medium was carried by Beg et al. [17].
Chemical reactions usually accompany a large amount of exothermic and endothermic reactions. These
characteristics can be easily seen in a lot of industrial processes. Recently, it has been realized that it is not
always permissible to neglect the convection effects in porous constructed chemical reactors [18]. The
reaction produced in a porous medium was extraordinarily in common, such as the topic of PEM fuel cells
modules and the polluted underground water because of discharging the toxic substance, etc. Kandaswamy
et al. [19,20] studied the effects of chemical reaction and radiation on boundary layer flow over a porous
wedge in presence of suction or injection. The effect of chemical reaction on heat and mass transfer in a
boundary layer flow has been studied under different conditions by several researchers [21–26].
In all the above studies the fluid considered is Newtonian. Most of the practical problems involve non-
Newtonian fluids types. The study of non Newtonian fluid flow has gained the attention of engineers and
scientist in recent times due to its important application in various branches of science, engineering and
technology: particularly in chemical and nuclear industries, material processing, geophysics, and bio-
engineering. In view of these applications an extensive range of mathematical models have been developed
to simulate the diverse hydrodynamic behavior of these non-Newtonian fluids. Radiation effects on MHD
free convection flow of Kuvshinshiki fluid past a vertical porous plate in porous medium in presence of mass
transfer was studied by Vidyasagar and Ramana [27]. Gupta et al. [28] proposed the heat and mass transfer
effects on MHD free convective flow of Kuvshinshiki fluid past a vertical plate. Mohan Krishna et al. [29]
investigated the effects of chemical reaction and radiation on MHD free convective flow of Kuvshinshiki
fluid past through a vertical porous plate in presence of heat generation. The influence of the soret number
on unsteady MHD of Kuvshinshiki fluid flow past a vertical porous plate with variable suction, heat and
mass transfer was analyzed by Jimoh et al. [30]. Very recently Vidya Sagar et al. [31] studied the unsteady
MHD free convective boundary layer flow of radiation absorbing Kuvshinshiki fluid through porous
medium. Chemical reaction and Soret effects on unsteady MHD flow of a viscoelastic fluid past an
impulsively started infinite vertical plate in presence of heat generation was studied by Mohammed Ibrahim
and Suneetha [32]. MHD free convection flow of a visco-elastic (Kuvshiniski type) dusty gas through a
semi-infinite plate moving with velocity decreasing exponentially with time and radiative heat transfer was
investigated by Prakash, et al. [33]. Umamaheswar, et al. [34] studied an unsteady MHD free convective
visco-elastic fluid flow bounded by an infinite inclined porous plate in the presence of heat source.
Motivated by the above studies in this article we have analyzed the effect of heat source on an unsteady
MHD free convective boundary layer flow of radiation absorbing Kuvshinshiki fluid through porous
medium past a semi-infinite vertical plate.
4. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
89
2 Formulation of the Problem
We have considered an unsteady MHD two dimensional free convective flow of a viscous, incompressible,
radiating, chemically reacting and radiation absorbing Kuvshinski fluid through a porous medium past a
semi-infinite vertical plate in presence of heat source. The x-axis be taken along the vertical plate in the
upward direction in the direction of the flow and y- axis is taken perpendicular to it. It is assumed that,
initially, the plate and the fluid are at the same temperature
*
T and concentration
*
C in the entire of the
fluid. The effects of soret and Dofour are neglected, as the level of foreign mass is assumed to be very low.
The radiative heat flux in x-direction is considered to be negligible in comparison to that of y axis. The fluid
considered here is gray, emitting and absorbing radiation but non scattering medium. The presence of
viscous dissipation cannot be neglected and also the presence of chemical reaction of first order and the
influence of radiation absorption are considered. All the fluid properties are considered to be constant except
the influence of the density variation caused by the temperature changes, in the body force term. It is also
assumed that the induced magnetic field is neglected in comparison with applied magnetic field, as the
magnetic Reynolds number is very small. Now, under the above assumptions, the governing equations of
such type of flow, momentum, energy and species equations [31].
Continuity Equation
*
*
0
v
y
(1)
Momentum Equation
2
2
* * 2 *
* * * * * * * * *
0
* * * * *
*
1 1
B
u u u
v g T T g C C u
t t y k t
y
(2)
Energy Equation
2
2 *
* * 2 * *
* * * * * *
0
1
* * * * *
*
1
1 r
p p p p p
Q
q R
T T k T u
v C C T T
t t y C C y C y C C
y
(3)
Species equation
2
* * 2 *
* * * *
1
* * * *
1
C C C
v D k C C
t t y y
(4)
The boundary conditions at the wall and in the free stream are
* *
*
0 1 t n
u v e
,
* *
w
T T
,
* *
w
C C
at
*
y =0 (5)
*
0
u ,
*
0
T ,
*
0
C as
*
y
The equation (1) gives
*
0
v v
(6)
5. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
90
where 0
v is the constant suction velocity. The radiative heat flux
*
r
q using the Rosseland approximation
diffusion model for radiation heat transfer is expressed as
4
* *
*
* *
4
3
r
T
q
k y
(7)
Where
*
and
*
k are respectively the Stream-Boltzmann constant and the main absorption coefficient. We
assume that the temperature difference with in the flow is sufficiently small and in Taylor series about
*
T
and neglecting higher order terms, thus
4 3 4
* * * *
4 3
T T T T
(8)
In view of equations (8) and (9) the equation (3) reduced to the following form
3
2 2
2 * *
* * 2 * * 2 *
* * * * * *
0
1
* * * * *
* *
16
1
3
p p p p p
Q
T R
T T k T u T
v C C T T
t t y C C y C k C C
y y
(9)
Introducing the following dimensionless variable and parameters,
3 * *
* * 2 * * * * * *
* *
0 0
2 * * * * * 3
0 0 0
* * * * 2 2 * 2 2
0 0 0 0
3 2 * *
0 0
2 * *
1
2 * *
0
4
, , , , , , , ,
,Pr , , , , , ,
,
w
w w
w p
p w
w
w
g T T
y v t v T T C C T
u n
u y t n R Gr
v v T T C C k k v
g C C C k v v v
Gm Sc k M Ec
v k D v C T T
R C C
Ra K
kv T T
2
1 0
0
2 2
0
,
k Qkv
r Q
v
(10)
Into set of equations (2) - (5), we obtain
2 2
1 1
2 2
u u u u
M u Gr Gm
t t y y
(11)
2
2 2
1
2 2
Pr Pr Pr Pr a
u
N Ec R Q
t t y y y
(12)
2 2
2 2
Sc Sc Sc KrSc
t t y y
(13)
Where 1 1 1 1
1 4
, 1 , 1
3
R
M M N M
k
The corresponding boundary conditions in non-dimensional form are
6. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
91
1 , 1, 1,
nt
u e
at y=0,
0, 0, 0
u
as y (14)
2.1 Solution of the Problem
Equations (11) - (13) are coupled, non-linear differential equations and these cannot be solved in closed-
form. However, these equations can be reduced to a set of ordinary differential equations, which can be
solved analytically. This can be done by representing the velocity, temperature and concentration of the fluid
in the neighborhood of the plate as
2
0 1
( ) ( ) 0( )..............
nt
u u y u y e
2
0 1
( ) ( ) 0( )..............
nt
y y e
(15)
2
0 1
( ) ( ) 0( )..............
nt
y y e
Substituting (15) into set of equations (11)-(13) and equating the harmonic terms and non-harmonic terms,
and neglecting the higher order terms of, we obtain
0 0 1 0 0 0
u u M u Gr Gm
(16)
1 1 2 1 1 1
u u M u Gr Gm
(17)
2
1 0 0 0 0 0
Pr Pr ( )
N Q Ec u Ra
(18)
1 1 1 2 0 0 1 1
Pr 2Pr
N N Ecu u Ra
(19)
0 0 0 0
Sc ScKr
(20)
1 1 1 1 0
Sc L
(21)
Where prime denotes ordinary differentiation with respect to y and
2 2 2
2 1 1 2 1
, Pr Pr ,
M M n n N n n Q L nSc n Sc KrSc
The corresponding boundary conditions are
0 1 0 1 0 1
1, 1, 1, 0, 1, 0,
u u
at y=0
0 1 0 1 0 1
0, 0, 0, 0, 0, 0,
u u
as y (22)
Solving the equations (20) and (21) subject to the corresponding boundary conditions, we obtain.
1
0
m y
e
1 0
(23)
7. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
92
The set of equations (16)-(19) are still coupled and non-linear ordinary differential equations, whose exact
solutions are not possible. So we expand 0 1 0 1
, , ,
u u in terms of Ec in the following form, as the Eckert
number is very small for incompressible flows.
To solve these equations, assuming the Eckert number E to be small, we write.
0 01 02
1 11 12
0 01 02
1 11 12
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
u y u y Ecu y
u y u y Ecu y
y y Ec y
y y Ec y
(24)
Substituting the equations (24) into equations (16)-(19), equating the coefficients of Ec to zero and
neglecting the higher order terms, we obtain
01 01 1 01 01 0
u u M u Gr Gm
(25)
11 11 2 11 11 1
u u M u Gr Gm
(26)
1 01 01 01 0
Pr
N Q Ra
(27)
1 11 11 2 11 1
Pr
N N Ra
(28)
02 02 1 02 02
u u M u Gr
(29)
12 12 2 12 12
u u M u Gr
(30)
2
1 02 02 02 01
Pr Pr
N Q u
(31)
1 12 12 2 12 01 11
Pr 2Pr
N N u u
(32)
The corresponding boundary conditions are
01 02 11 12
01 02 11 12
1, 0, 1, 0
1, 0, 0, 0
u u u u
at y=0
01 02 11 12
01 02 11 12
0, 0, 0, 0
0, 0, 0, 0
u u u u
as y (33)
The analytical solutions of equations (25)-(32) under the boundary conditions (33) are given by
3 1
01 2 1
m y m y
A e Ae
(34)
5 3 1
01 5 3 4
m y m y m y
u A e A e A e
(35)
8. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
93
6
11
m y
u e
(36)
7 5 3 3
1 1 2
2 2 2
02 12 6 7 8 9 10 11
m y m y m y b y
m y b y b y
A e A e A e A e A e A e A e
(37)
8 5 6
4
12 16 13 14 15
m y b y b y
b y
A e A e A e A e
(38)
9 7 5 3 3
1 1 2
2 2 2
02 24 17 18 19 20 21 22 23
m y m y m y m y b y
m y b y b y
u A e A e A e A e A e A e A e A e
(39)
10 9 7 5 3 3
1 1 2
2 2 2
12 33 25 26 27 28 29 30 31 32
m y m y m y m y m y b y
m y b y b y
u A e A e A e A e A e A e A e A e A e
(40)
The velocity, temperature and concentration distributions in the boundary layer become
( , )
u y t
5 3 1
9 7 5 3 3
1 1 2
01 5 3 4
2 2 2
02 24 17 18 19 20 21 22 23
m y m y m y
m y m y m y m y b y
m y b y b y
u A e A e A e
E u A e A e A e A e A e A e A e A e
6
10 9 7 5 3 3
1 1 2
2 2 2
12 33 25 26 27 28 29 30 31 32
m y
nt
m y m y m y m y m y b y
m y b y b y
e
e
E u A e A e A e A e A e A e A e A e A e
(41)
3 1
7 5 3 3
1 1 2
01 2 1
2 2 2
02 12 6 7 8 9 10 11
( , )
m y m y
m y m y m y b y
m y b y b y
A e Ae
y t
E A e A e A e A e A e A e A e
8 5 6
4
12 16 13 14 15
m y b y b y
b y
nt
e E A e A e A e A e
(42)
1
m y
e
(43)
Where the expressions for the constants are given in the Appendix
The skin-friction, Nusselt number and Sherwood number are important physical parameters for this type of
boundary layer flow.
Knowing the velocity field, the skin-friction at the plate can be obtained, which in non-dimensional form if
given by
0 1
0 0
nt
y y
u u
u
Cf e
y y y
=
=
5 5 3 3 1 4 9 24 17 7 18 5 3 19 20 1 1 21 2 22 3 23
2 2 2
m A m A m A Ec m A A m A m m A A m b A b A b A
6 10 33 9 25 7 26 5 27 3 28 1 29 30 1 31 2 32 3
2 2 2
nt
e m Ec m A m A m A m A m A m A A b A b A b
(44)
Knowing the temperature field, the rate of heat transfer coefficient can be obtained, which in the non-
dimensional form, in terms of the Nusselt number, is given by
9. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
94
0 1
0 0
nt
y y
Nu e
y y y
=
=
3 2 1 1 7 12 6 5 3 7 8 1 1 9 2 10 3 11
8 16 13 4 14 5 15 6
2 2 2
nt
m A m A Ec m A A m m A A m b A b A b A
e Ec m A A b A b A b
(45)
Knowing the concentration field, the rate of mass transfer coefficient can be obtained, which in the non-
dimensional form, in terms of the Sherwood number, is given by
0 1
1
0 0
nt
y y
Sh e m
y y y
(46)
3 Results and Discussion
In order to look into the physical insight of the problem, the expressions obtained in previous section are
studied with help of graphs from Figs. 1 – 9. The effects of different physical parameters viz., the thermal
Grashof number (Gr), the mass Grashof number (Gm), magnetic field parameter (M), radiation parameter
(R), radiation absorption parameter (Ra), heat Source (Q), Schmidt number (Sc) and chemical reaction (Kr)
are studied analytically by choosing arbitrary values.
Fig. 1 shows the velocity profiles for different values of thermal Grashof number Gr. It is observed that, as
Gr increases, velocity also increases. This is due to the buoyancy which is acting on the fluid particles due to
gravitational force that enhances the fluid velocity. A similar effect is identified from Fig. 2, in the presence
of mass Grashof number Gm, which also increases fluid velocity.
Typical variation of velocity profiles for various values of magnetic field parameter M is shown in Fig. 3.
From this figure it is found that velocity gets reduced by the increase of magnetic parameter M. Because the
magnetic force which is applied perpendicular to the plate, retards the flow, which is known as Lorentz
force.
The effect of Prandtl number Pr on the velocity and temperature profiles is presented in Figs. 4(a) and 4(b).
From Fig. 4(a), it is observed that velocity decreases with an increase in Prandtl number Pr. This is
physically true because, the Prandtl number is a dimensionless number which is the ratio of momentum
diffusivity (kinematic viscosity) to thermal diffusivity. In many of the heat transfer problems, the Prandtl
number controls the relative thickness of the momentum and thermal boundary layers. When Pr is small, it
means that the heat diffuses very quickly compared to the velocity (momentum), which means that in the
case of liquid metals the thickness of the thermal boundary layer is much bigger than the velocity boundary
layer. This absolutely coincides with the results that is shown in Fig. 4(b) where the thermal boundary layer
shrinks for higher values of Prandtl number Pr.
The effect of radiation absorption parameter Ra on velocity and temperature profiles was plotted in the Figs.
5(a) and 5(b). The velocity and temperature profiles increase with an increase of radiation absorption
parameter Ra which is clearly represented in Figs. 5(a) and 5(b).
The effects of radiation parameter R on the velocity and temperature profiles are presented in Figs. 6(a) and
6(b). From these figures, we noticed that as the value of radiation parameter increases, the velocity and the
temperature profiles increase. This is because the thermal radiation is associated with high temperature,
thereby increasing the temperature distribution of the fluid flow.
10. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
95
Figs. 7(a) and 7(b) depicts the velocity and temperature profiles for various values of heat source Q. From
these figures we can see that the heat generated buoyancy force increases which induce the flow rate to
increase giving rise to the increase in the velocity and temperature profiles.
For the various values of the Schmidt number Sc, the velocity, temperature and concentration profiles are
presented in Figs. 8(a) – 8(c). The Schmidt number Sc embodies the ratio of the momentum to the mass
diffusivity. This Schmidt number therefore quantifies the relative effectiveness of momentum and mass
transport by diffusion in the hydrodynamic, temperature and concentration boundary layers. It is obvious
that the effect of increasing values of Sc, results in a decreasing velocity, temperature and concentration
distributions across the boundary layer. A similar effect is identified from Figs. 9(a) - 9(c), in the presence of
chemical reaction parameter Kr, which also exhibits decreasing fluid velocity, temperature and
concentration.
Tables 1 - 3 presents the values of skin friction coefficient, Nusselt and Sherwood numbers at numerous
values of parameters. The effects of various physical parameters on skin-friction coefficient, Nuselt number
and Sherwood number are shown in Tables 1 - 3. The behavior of these parameters is self-evident from
Tables. 1, 2, 3 and hence they are not discussed any further due to brevity.
Fig. 1. The graph of u against y for varies
values of Gr
Fig. 2. The graph of u against y for varies values
of Gm
Fig. 3. The graph of u against y for varies
values of M
Fig. 4(a). The graph of u against y for varies
values of Pr
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
y
U
Gr=8,10,12,14
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
y
U
Gm=5,7,9,11
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
M=1.0,1.5,2.0,2.5
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
Pr=0.71,0.91,1.11,1.31
11. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
96
Fig. 4(b). The graph of T against y for varies
values of Pr
Fig.5(a). The graph of u against y for varies
values of Ra
Fig. 5(b). The graph of T against y for varies
values of Ra
Fig. 6(a). The graph of u against y for varies
values of R
Fig. 6(b). The graph of T against y for varies
values of R
Fig. 7(a). The graph of u against y for varies
values of Q
Fig. 7(b). The graph of T against y for varies
values of Q
Fig. 8(a). The graph of u against y for varies
values of Sc
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
Pr=0.71,0.91,1.11,1.31
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
y
U
Ra=0,1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
y
T
Ra=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
R=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
R=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
Q=0.00,0.01,0.05,0.07
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
Q=0.005,0.01,0.015,0.02
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
Sc=0.22,0.44,0.66,0.88
12. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
97
Fig. 8(b). The graph of T against y for varies
values of Sc
Fig. 8(c). The graph of C against y for varies
values of Sc
Fig. 9(a). The graph of u against y for varies
values of Kr
Fig. 9(b). The graph of T against y for varies
values of Kr
Fig. 9(c). The graph of C against y for varies values of Kr
Table 1. Effect of various physical parameters on skin-friction coefficient, Nusselt number and
Sherwood number for Pr = 0.71, R = 0.1, E = 0.01, Ra = 0.1, Q = 0.01, Sc = 0.22, Kr = 0.1
Gr Gm M K Cf Nu Sh
8 5 1.0 0.1 0.3259 0.1462 -0.2947
10 5 1.0 0.1 1.0158 0.2780 -0.2947
12 5 1.0 0.1 1.7243 0.4268 -0.2947
8 7 1.0 0.1 0.9878 0.2793 -0.2947
8 9 1.0 0.1 1.6532 0.4297 -0.22947
8 5 1.2 0.1 0.2537 0.1323 -0.2947
8 5 1.4 0.1 0.1829 0.1190 -0.2947
8 5 1.0 0.4 6.4519 3.2132 -0.2947
8 5 1.0 0.8 11.3294 7.4571 -0.2947
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
Sc=0.22,0.44,0.66,0.88
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
C
Sc=0.22,0.44,0.66,0.88
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
Kr=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
Kr=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
C
Kr=0.1,0.20.3,0.4
13. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
98
Table 2. Effect of various physical parameters on skin-friction coefficient, Nusselt number and
Sherwood number for Gr = 8, Gm = 5, M = 1.0, K = 0.1, Sc = 0.22, Kr = 0.1
Pr R Ra Q Cf Nu Sh
0.71 0.1 0.1 0.01 0.3259 0.1462 -0.2947
0.81 0.1 0.1 0.01 0.1853 -0.3245 -0.2947
0.91 0.1 0.1 0.01 0.1155 -0.4562 -0.2947
0.71 0.3 0.1 0.01 0.2568 -0.3725 -0.2947
0.71 0.4 0.1 0.01 0.2792 -0.3258 -0.22947
0.71 0.1 0.3 0.01 1.1333 2.5174 -0.2947
0.71 0.1 0.5 0.01 2.1960 6.1668 -0.2947
0.71 0.1 0.1 0.005 0.2894 0.0189 -0.2947
0.71 0.1 0.1 0.007 0.3022 0.0619 -0.2947
Table 3. Effect of various physical parameters on skin-friction coefficient, Nusselt number and
Sherwood number for Gr = 8, Gm = 5, M = 1.0, K = 0.1, Pr = 0.71, R = 0.1, E = 0.01, Ra = 0.1, Q = 0.01
Sc Kr Cf Nu Sh
0.22 0.1 0.3259 0.1462 -0.2947
0.24 0.1 0.0996 -0.8856 -0.3160
0.26 0.1 0.14454 -0.5938 -0.3371
0.22 0.4 0.0788 -0.4830 -0.4264
4 Conclusion
In this paper a theoretical study is carried out for an unsteady MHD two dimensional free convention flow of
a viscous incompressible, radiating, chemically reacting, radiation absorbing and heat source Kuvshinshiki
fluid through a porous medium past a semi-infinite vertical plate. The dimensionless equations governing the
flow are solved by perturbation technique. The fundamental parameters which were found to have an
influence on the problem under consideration are heat source, magnetic field parameter, permeability of
porous medium, radiation parameter, radiation absorption parameter, Grashof number, mass Grashof
number, Schmidt number, chemical reaction parameter and Prandtl number. The main conclusions are as
follows.
a. Velocity increases when Grashof number, mass Grashof number, radiation parameter, and heat
source, increase, whereas it decreases when there is an increase in magnetic field parameter,
Schmidt number, chemical reaction parameter, radiation absorption, Prandtl number.
b. Temperature increases with an increase in radiation parameter, radiation absorption parameter, and
heat source whereas it decreases with an increase in Prandtl number.
c. Concentration is observed to decrease when Schmidt number and chemical reaction increase.
d. Skin-friction increases with an increase in Prandtl number, permeability of porous medium, Grashof
number, mass Grashof number whereas it has reverse effect in the case of magnetic field parameter
and Schmidt number.
e. Nusselt number increases with an increase in Pradtl number, radiation absorption parameter and
heat generation, whereas it has reverse effect in the case of radiation parameter.
f. Sherwood number gets decreased when Schmidt number and chemical reaction parameter
increased.
Acknowledgement
Authors are thankful to the referee for his/her valuable suggestions, who helped to improve the quality of
this manuscript.
14. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
99
Competing Interests
Authors have declared that no competing interests exist.
References
[1] Ostrach S. New aspects of natural convection heat transfer. Trans. Am. Soc, Mec. Engrs.
1953;75(75):1287-1290.
[2] Cheng P. Proc, NATO Advanced Study in Natural Convection, (Izmir, Turkey); 1985.
[3] Cheng P, Minkowycz WJ. Freeconvection about a vertical flat plate embedded in a porous medium
with application to heat transfer from a dike. J. Geophys. Res. 1977;82(14):2040-2044.
[4] Ghoshdastidar PS. Heat Transfer, (Oxford University Press, UK); 2004.
[5] Nield DA, Bejan A. Convection in porous media, (Springer, Berlin); 1992.
[6] Chamkha AJ. Unsteady MHD convective heat and mass transfer past a semi-infinite vertical
permeable moving plate with heat absorption. Int. J. Eng. Sci. 2004;42:217–230.
[7] Abo Eldahab EM, Gendy MSE. Convective heat transfer past a continuously moving plate embedded
in a non-darcian porous medium in the presence of a magnetic field. Canadian Journal of Physics.
2001;97:1031-1038, 2001.
[8] Sivaiah M, Nagarajan A, Reddy PS. Heat and mass transfer effects on MHD free convective flow past
a vertical porous plate. The Icfai University Journal of Computational Mathematics. 2009;2(2):14-21.
[9] Raptis A, Perdikis C. Radiation and free convection flow past a moving plate. Int. J. of App. Mech.
And Engg. 1999;4:817-821.
[10] Ghosh SK, Rawat S, Beg OA, Beg TA. Thermal radiation effects on unsteady hydromagnetic gas
flow along an inclined plane with indirect natural convection. Int. J. Appl. Math and Mech.
2010;6(13):41-57.
[11] Abo Eldahab EM, Gendy MSE. Radiation effects on convective heat transfer in an electrically
conducting fluid at a stretching surface with variable viscosity and uniform free-stream. Physica
Scripta. 2000;62:321-325.
[12] Ostrach S. Laminar natural convection flow and heat transfer of fluid with and without heat source in
channel with wall temperature. NACA TN, 2863; 1952.
[13] Ibid. Combined natural and forced convection laminar flow and heat transfer of fluid with and without
heat sources in channels with linearly varying wall temperatures. NACA TN 3141; 1954.
[14] Ibid. Unstable convection in vertical channels with heating from below, including effects of heat
source and frictional heating. NACA TN 3458; 1958.
[15] Raptis AA. Free convection and mass transfer effects on the flow past an infinite moving vertical
porous plate with constant suction and heat source. Astrophy. Space Sci. 1982;86:43.
[16] Singh AK. MHD free convection and mass transfer flow with heat source and thermal diffusion. J. of
Energy, Heat and Mass Transfer. 2001;23:227–249.
15. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
100
[17] Beg OA, Zueco J, Bhargava R, Jakhar HS. Magneto hydrodynamic convection flow from a sphere to
a non-darcian porous medium with heat generation or absorption effects network simulation.
International Journal of Thermal Sciences. 2009;48:1813-1830.
[18] Nield DA, Bejan A. Convection in porous media. 2nd Edition, Springer-Verlag, New York; 1999.
[19] Kandaswamy R, Wahib B, Md Raj, Azme BK. Effects of chemical reaction, heat and mass transfer in
boundary layer flow over a porous wedge with heat radiation in presence of suction or injection.
Theatrical Applied Mechanics Belgrade. 2006;33:123-148.
[20] Kandaswamy R, Periaswamy K, Sivagnana PK. Chemical reaction, heat and mass transfer along a
wedge with heat source and concentration in the presence of suction or injection. International Journal
of Heat and Mass Transfer. 2005;48:1388-1394.
[21] Mahdy A. Effect of chemical reaction and heat generation or absorption on double-diffusive
convection from a vertical truncated cone in a porous media with variable viscosity. International
Communications in Heat and Mass Transfer. 2010;37:548-554.
[22] Muthucumarswamy R, Ganesan P. Diffusion and first order chemical reaction on impulsively started
infinite vertical plate with variable temperature. International Journal of Thermal Sciences.
2002;41:475-479.
[23] Muthucumarswamy R. Chemical reaction effects on vertical oscillating plate with variable
temperature. Chemical Industry and Chemical Engineering Quarterly. 2002;16:167-173.
[24] Patil PM, Kulkarni PS. Effects of chemical reaction on free convective flow of a polar fluid through a
porous medium in the presence of internal heat generation. International Journal of Thermal Sciences.
2008;47:1043-1054.
[25] Raju MC, Varma SVK, Rao RRK. Unsteady MHD free convection and chemically reactive flow past
an infinite vertical porous plate. Journal of Future Engineering and Technology. 2013;8:35-40.
[26] Sudheer Babu M, Satya Narayana PV. Effects of the chemical reaction and radiation absorption on
free convection flow through porous medium with variable suction in the presence of uniform
magnetic field. JP Journal of Heat and Mass Transfer. 2009;3:219-234.
[27] Vidyasagar G, Ramana B. Radiation effect on MHD free convection flow of kuvshinshiki fluid with
mass transfer past a vertical porous plate through porous medium. Asian Journal of Current
Engineering and Maths. 2013;2(3):170-174.
[28] Gupta PC, Varshney NK, Janamejay Singh. Perturbation technique to MHD free convection flow of
Kuvshinshiki fluid with heat and mass transfer past a vertical porous plate. International Journal of
Mathematical Archive. 2011;2(8):1416-1422.
[29] Mohan Krishna P, Sugunamma V, Sandeep N. Effects of Chemical Reaction and Radiation on MHD
Free Convection Flow of Kuvshinshiki Fluid Through a Vertical Porous Plate with Heat Source,
American-Eurasian Journal of Scientific Research. 2013;8(3):135-143.
[30] Jimoh A, Idowu AS, Titiloye EO. Influence of Soret on Unsteady MHD of Kuvshinshiki Fluid Flow
With Heat and Mass Transfer Past a Vertical Porous Plate With Variable Suction. International
Journal on Recent and Innovation Trends in Computing and Communication. 2014;2599–2611.
[31] Vidya Sagar B, Raju MC, Varma SVK, Venkataramana S. Unsteady MHD free convection boundary
layer flow of radiation absorbing kuvshinski fluid through porous medium. Review of Advances in
Physics Theories and Applications. 2014;1(3):48-62.
16. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
101
[32] Mohammed Ibrahim S, Suneetha K. Chemical reaction and Soret effects on unsteady MHD flow of a
viscoleastic fluid past an impulsively started infinite vertical plate with heat source/sink. International
Journal of Mathematics and Computational Science. 2015;1(1):5–14.
[33] Prakash O, Kumar D, Dwivedi YK. MHD free convection flow of a visco-elastic (Kuvshiniski Type)
dusty gas through a semi-infinite plate moving with velocity decreasing exponentially with time and
radiative heat transfer. AIP Advances. 2011;1:022-132.
[34] Umamaheswar M, Varma SVK, Raju MC. Unsteady MHD free convective visco-elastic fluid flow
bounded by an infinite inclined porous plate in the presence of heat source, viscous dissipation and
ohmic heating. International Journal of Advanced Science and Technology. 2013;61:39-52.
17. Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
102
APPENDIX
2
1
4
,
2
Sc Sc ScKr
m
2
1
2
4
2
Sc Sc L
m
,
2
1
3
1
Pr Pr 4
,
2
QN
m
N
2
1 2
4
1
Pr Pr 4
,
2
N N
m
N
1
5
1 1 4
,
2
M
m
2
6
1 1 4
2
M
m
,
2
1
7
1
Pr Pr 4
,
2
QN
m
N
2
1 2
8
1
Pr Pr 4
,
2
N N
m
N
1
1
2
1 1
1 1
,
Pr
Ra
N
A
Q
m m
N N
2 1
1 ,
A A
2
3 2
3 3 1
,
GrA
A
m m M
1
4 2
1 1 1
,
GrA Gm
A
m m M
3 4
5 1 ,
A A A
2 2
5 5 1
6
2
5 5
1 1
Pr /
,
Pr
4 2
A m N
A
Q
m m
N N
2 2
3 3 1
7
2
3 3
1 1
Pr /
,
Pr
4 2
A m N
A
Q
m m
N N
2 2
4 1 1
8
2
1 1
1 1
Pr /
,
Pr
4 2
A m N
A
Q
m m
N N
5 3 3 5 1
9
2
3 5 3 5
1 1
2Pr /
,
Pr
A A m m N
A
Q
m m m m
N N
5 1 3 5 1
10
2
1 5 1 5
1 1
2Pr /
Pr
A Am m N
A
Q
m m m m
N N
4 3 3 4 1
11
2
3 4 3 4
1 1
2Pr /
Pr
A A m m N
A
Q
m m m m
N N
, 12 6 7 8 9 10 11,
A A A A A A A
,
5 5 6 1
13
2 2
5 6 6 5
1 1
2Pr /
,
Pr
A m m N
A
N
m m m m
N N
3 3 6 1
14
2 2
3 6 6 3
1 1
2Pr /
,
Pr
A m m N
A
N
m m m m
N N