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
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
Solution of MHD Effect on Transient Free Convection Flow past a Vertical Plat...iosrjce
IOSR Journal of Applied Physics (IOSR-JAP) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
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 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.
The Effect of Hall Current on an Unsteady MHD Free Convective Flow along a Ve...iosrjce
In this paper we investigate the effect of hall current on an unsteady magnetohydrodynamic free
convective flow along a vertical porous plate with thermal radiation. The dimensionless governing coupled,
non-linear boundary layer partial differential equations are solved by an efficient finite element method for
velocity, temperature, concentration field and also the expression for shearing stress, Nusselt number and
Sherwood number are obtained. The effect of thermal Grashof Number, solutal Grashof Number, Transpiration
cooling parameter, Prandtl Number, Schmidt Number, Hartmann number, Thermal radiation parameter, Eckert
number and Hall parameter on the flow field are shown through graphs and tables.
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 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
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
Solution of MHD Effect on Transient Free Convection Flow past a Vertical Plat...iosrjce
IOSR Journal of Applied Physics (IOSR-JAP) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
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 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.
The Effect of Hall Current on an Unsteady MHD Free Convective Flow along a Ve...iosrjce
In this paper we investigate the effect of hall current on an unsteady magnetohydrodynamic free
convective flow along a vertical porous plate with thermal radiation. The dimensionless governing coupled,
non-linear boundary layer partial differential equations are solved by an efficient finite element method for
velocity, temperature, concentration field and also the expression for shearing stress, Nusselt number and
Sherwood number are obtained. The effect of thermal Grashof Number, solutal Grashof Number, Transpiration
cooling parameter, Prandtl Number, Schmidt Number, Hartmann number, Thermal radiation parameter, Eckert
number and Hall parameter on the flow field are shown through graphs and tables.
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.
Non-Darcy Convective Heat and Mass Transfer Flow in a Vertical Channel with C...IJERA Editor
In this paper, We made an attempt to study thermo-diffusion and dissipation effect on non-Darcy convective
heat and Mass transfer flow of a viscous fluid through a porous medium in a vertical channel with Radiation and
heat sources. The governing equations of flow, heat and mass transfer are solved by using regular perturbation
method with δ, the porosity parameter as a perturbation parameter. The velocity, temperature, concentration,
shear stress and rate of Heat and Mass transfer are evaluated numerically for different variations of parameter.
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.
Effect of Chemical Reaction and Radiation Absorption on Unsteady Convective H...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.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
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.
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.
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.
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.
Effect of Radiation and Thermo-Diffusion on Convective Heat and Mass Transfer...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
International Journal of Computational Engineering Research(IJCER)ijceronline
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.
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.
Dufour and Soret Effects on Convective Heat and Mass Transfer in Non-Darcy Do...IJERA Editor
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.
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.
Radiation and Mass Transfer Effects on MHD Natural Convection Flow over an In...IJMER
A numerical solution for the unsteady, natural convective flow of heat and mass transfer along an inclined plate is presented. The dimensionless unsteady, coupled, and non-linear partial differential conservation equations for the boundary layer regime are solved by an efficient, accurate and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields have been studied for the effect of Magnetic parameter, buoyancy ratio parameter, Prandtl number, radiation parameter and Schmidt number. The local skin-friction, Nusselt number and Sherwood number are also presented and analyzed 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.
Non-Darcy Convective Heat and Mass Transfer Flow in a Vertical Channel with C...IJERA Editor
In this paper, We made an attempt to study thermo-diffusion and dissipation effect on non-Darcy convective
heat and Mass transfer flow of a viscous fluid through a porous medium in a vertical channel with Radiation and
heat sources. The governing equations of flow, heat and mass transfer are solved by using regular perturbation
method with δ, the porosity parameter as a perturbation parameter. The velocity, temperature, concentration,
shear stress and rate of Heat and Mass transfer are evaluated numerically for different variations of parameter.
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.
Effect of Chemical Reaction and Radiation Absorption on Unsteady Convective H...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.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
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.
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.
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.
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.
Effect of Radiation and Thermo-Diffusion on Convective Heat and Mass Transfer...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
International Journal of Computational Engineering Research(IJCER)ijceronline
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.
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.
Dufour and Soret Effects on Convective Heat and Mass Transfer in Non-Darcy Do...IJERA Editor
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.
Radiation and Soret Effect on Visco-Elastic MHD Oscillatory Horizontal Channe...
Similar to Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a Porous Flat Plate with Chemical Reaction and Viscous Dissipation
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.
Radiation and Mass Transfer Effects on MHD Natural Convection Flow over an In...IJMER
A numerical solution for the unsteady, natural convective flow of heat and mass transfer along an inclined plate is presented. The dimensionless unsteady, coupled, and non-linear partial differential conservation equations for the boundary layer regime are solved by an efficient, accurate and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields have been studied for the effect of Magnetic parameter, buoyancy ratio parameter, Prandtl number, radiation parameter and Schmidt number. The local skin-friction, Nusselt number and Sherwood number are also presented and analyzed 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.
Magnetohydrodynamic mixed convection flow and boundary layer control of a nan...IAEME Publication
This research work is focused on the numerical solution of steady MHD mixed convection boundary layer flow of a nanofluid over a semi-infinite flat plate with heat generation/absorption and viscous dissipation effects in the presence of suction and injection. Gyarmati’s variational principle developed on the thermodynamic theory of irreversible processes is employed to solve the problem
numerically. The governing boundary layer equations are approximated as simple polynomial functions, and the functional of the variational principle is constructed.
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 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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Study of Magnetohydrodynamic Fluid Flows and Their Applicationsijtsrd
In this paper a detailed study is presented on magnetohydrodynamic fluid flows with their basic governing equations. Basic concept of magnetohydrodynamic is discussed in detail. The results of various problems done by researchers are presented and addressed properly. The various applications of magnetohydrodynamic fluid flows have been presented. Dr. Vishal Saxena ""Study of Magnetohydrodynamic Fluid Flows and Their Applications"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020, URL: https://www.ijtsrd.com/papers/ijtsrd29931.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/applied-mathematics/29931/study-of-magnetohydrodynamic-fluid-flows-and-their-applications/dr-vishal-saxena
HYDROMAGNETIC MIXED CONVECTION MICRO POLAR FLOW DRIVEN BY A POROUS STRETCHING...IAEME Publication
We analyze a finite element solution for the magneto hydrodynamics mixed convection micro polar flow through a porous medium driven by a porous stretching sheet with uniform suction. The governing partial differential equations are solved numerically by using finite element technique. The
effect of Hartmann number, Darcy parameter and surface condition on velocity, micro rotation and temperature functions has been study.
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.
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.
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.
Similar to Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a Porous Flat Plate with Chemical Reaction and Viscous Dissipation (19)
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
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Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a Porous Flat Plate with Chemical Reaction and Viscous Dissipation
1. International Journal of Engineering Inventions
ISSN: 2278-7461, ISBN: 2319-6491
Volume 2, Issue 1 (January 2013) PP: 22-30
www.ijeijournal.com P a g e | 22
Effects of Hall Current on an Unsteady MHD Flow of
Heat and Mass Transfer along a Porous Flat Plate with
Chemical Reaction and Viscous Dissipation
BalaSidduluMalga1
, Nagaraju.L2
, N.Kishan3
, Ramarao Kotikalapudi4
1
Chaitanya Bharathi Institute Technology, Gandipet, Hyderabad-500075, 2
Research Scholar, Osmania
University, Hyderabad-500007, 3
University College of Science, Osmania University, Hyderabad-500007,
4
Chaitanya Bharathi Institute Technology, Gandipet, Hyderabad-500075,
Abstract:- The influence of Hall current, chemical reaction and viscous dissipation on an unsteady MHD flow
and heat transfer along a porous flat plate with mass transfer is investigated. The transformed governing
equations in the present study were solved numerically by using Galerkin Finite element method. The effects of
Hall current, viscous dissipation, chemical reaction over primary velocity u, secondary velocity w, heat transfer
and mass transfer are analyzed. Effects of magnetic parameter M, Prandtl number Pr, Schmidt number Sc,
Modified Grashof number Gr for heat transfer and Modified Grashof number Gc for mass transfer on the
velocity components and temperature is also examined.
Keywords:- MHD, Hall current, viscous dissipation, chemical reaction, finite element method.
I. INTRODUCTON
The combined heat and mass transfer problems with chemical reaction are of importance in many
processes, and therefore have received considerable amount of attention in recent years. In processes such as
drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and flow in a desert
cooler, the heat and mass transfer occurs simultaneously. Chemical reaction can be codified as either
homogeneous or heterogeneous processes. A homogeneous reaction is one that occurs uniformly through a
given phase; in contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a
phase. A reaction is said to be the first order reaction if the rate of reaction is directly proportional to the
concentration itself. In many chemical engineering processes, a chemical reaction between foreign mass and
fluid occur. Das et.al. [1] Studied the effect of mass transfer on flow past an impulsively started infinite
vertical plate with constant heat flux and chemical reaction. Muthukumarswamy and Ganesan,P [2,3] discussed,
flow past an impulsively started infinite plate with constant heat and mass flux and effect of chemical reaction
on moving isothermal surface with suction.
In the recent years theoretical study of MHD channel flows has been a subject of great interest due to
its widespread applications in designed cooling systems with liquid metals, petroleum industry, purification of
crude oil, polymer technology, centrifugal separation of matter from liquid, MHD generators, pumps,
accelerators and flow meters. In an ionized gas where the density is low and/or the magnetic field is very strong,
the conductivity normal to the magnetic field is reduced to the free spiraling of electrons and ions about the
magnetic lines of force before suffering collision; also, a current is induced in a direction normal to both the
electric and magnetic fields. The phenomenon, well known in the literature, is called the Hall Effect. The study
of magnetohydrodynamic flows with Hall currents has important engineering applications in problems of
magnetohydrodynamic generators and of Hall accelerators as well as in flight magnetohydrodynamics. Sato [4]
has studied the effect of Hall currents on the steady hydromagnetic flow between two parallel plates. Various
workers Sherman and Sutton [5], Hossain [6], Ram [7], Pop [8] studied the Hall effects. This effect can be taken
into account within the range of magnetohydrodynamical approximation. Katagiri [9] studied the steady
incompressible boundary layer flow past a semi-infinite flat plate in a transverse magnetic field at small
magnetic Reynolds number considering with the effect of Hall current. Agarwal et.al [10] gave the solution of
flow and heat transfer of a micro polar fluid over a stretching sheet using finite element technique. Sri Ramulu
et.al [11] studied the Effect of Hall Current on MHD Flow and Heat Transfer along a Porous Flat Plate with
Mass Transfer.
Hossain and Rashid [12] investigated the effect of Hall current on the unsteady free-convection flow of
a viscous incompressible fluid with mass transfer along a vertical porous plate subjected to a time dependent
transpiration velocity when the constant magnetic field is applied normal to the flow. And an analysis of heat
transfer taking dissipation function into account, in boundary layer flow of a hydromagnetic fluid over a
2. Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a…
www.ijeijournal.com P a g e | 23
stretching sheet in the presence of uniform transverse magnetic field has been given by Lodha and Tak [13],
Gupta [14] studied the Hall current effects in the steady hydromagnetic flow of an electrically conducting fluid
past an infinite porous flat plate. Sattar and Maleque [15] studied the effect of viscous dissipation as well as
joule heating on two-dimensional steady MHD free-convection and mass transfer flow with Hall current of an
electrically conducting viscous incompressible fluid past an accelerated infinite vertical porous plate with wall
temperature and concentration. Recently, Bala Siddulu Malga et.al [16] studied the effects of viscous dissipation
on unsteady free convection and mass transfer boundary layer flow past an accelerated infinite vertical porous
flat plate with suction. In this paper, we investigated Hall current, viscous dissipation and chemical reaction of
an unsteady free-convection MHD flow of an electrical conducting viscous incompressible fluid with mass
transfer along an infinite vertical porous plate. We employed Galerkin finite element method to solve the
governing partial differential equations. The effects of various important physical parameters on the flow and
heat, mass transfer are analyzed.
II. MATHEMATICAL FORMULATION
An unsteady free-convection flow of an electrically conducting viscous incompressible fluid with mass
transfer along an infinite vertical porous plate has been considered. The flow is assumed to be in 𝑥′
direction
which is taken along the plate in upward direction. The 𝑦′
axis is taken to be normal to the direction of the plate.
At time𝑡 ≥ 0 , the temperature and the species concentration at the plate are raised to 𝑇𝑤 (≠ 𝑇∞) and 𝐶 𝑤 (≠ 𝐶∞)
and thereafter maintained uniform. The level of species concentration is assumed to be very low and hence
species thermal diffusion and diffusion thermal energy effects can be neglected. A magnetic field of uniform
strength is assumed to be applied transversely to the porous plate. The magnetic Reynolds number of the flow is
taken to be small enough so that the induced magnetic field can be neglected. The equation of conservation of
electric change ∇. 𝐽 = 0 gives 𝑗𝑦 =constant, where 𝐽 = ( 𝑗𝑥 , 𝑗𝑦 , 𝑗𝑧) .It is also assumed that the plate is non-
conducting. This implies 𝑗𝑦 = 0 at the plate and hence zero everywhere. When the strength of magnetic field is
very large the generalized Ohm’s law, in the absence of electric field takes the form
𝐽 +
𝑤 𝑒𝑇 𝑒
𝐵0
𝐽𝐵 = 𝜎(𝜇 𝑒 𝑉𝐵 +
1
𝑒𝑛 𝑒
∇𝑃𝑒) (1)
where V is the velocity vector; 𝜎, the electric conductivity; 𝜇 𝑒, the magnetic permeability; 𝜔𝑒 , the electron
frequency; 𝜏 𝑒, the electron collision time; e, the electron charge; 𝑛 𝑒, the number density of the electron; and
𝑃𝑒, the electron pressure. Under the assumption that the electron pressure (for weakly ionized gas), the thermo-
electric pressure and ion-ship are negligible, equation (1) becomes
jx =
𝜎 𝜇 𝑒 𝐵0
1+𝑚2 (mu − w)
jz =
𝜎 µ𝐵0
1+𝑚2 (u + mw)
(2)
Where u is the x component of V; w, the z component of V and m (=𝜔𝑒 𝜏 𝑒), the Hall parameter. Within
this frame work, the equations which govern the flow under the usual Boussinesq approximation are:
∂V′
y′ = 0 (3)
𝜕𝑢′
𝜕𝑡′
+ 𝑉′
𝜕𝑢′
𝜕𝑦′
= 𝑣
𝜕2 𝑢′
𝜕𝑦′
′ 2 −
σ𝜇 𝑒
2 𝐵0
2
𝜌 1+𝑚2 𝑢′
+ 𝑚𝑤′
+ g𝛽(𝑇 − 𝑇∞) + g𝛽∗
𝐶′
− 𝐶′
∞ (4)
𝜕𝑤′
𝜕𝑡′
+ 𝑉′
𝜕𝑤′
𝜕𝑦′
= 𝑣
𝜕2 𝑤′
𝜕𝑦′2 −
σ𝜇 𝑒
2 𝐵0
2
𝜌 1+𝑚2 𝑤′
− 𝑚𝑢′ (5)
𝜕𝑇
𝜕𝑡′
+ 𝑉′ 𝜕𝑇
𝜕𝑦′
=
𝑘
ρCp
𝜕2 𝑇
𝜕𝑦′ 2 +
μ
ρCp
∂u′
∂y′
2
(6)
𝜕𝐶′
𝜕𝑡′
+ 𝑉
′
𝜕 𝐶′
𝜕 𝑦′
= 𝐷
𝜕2 𝐶′
𝜕𝑦 ′ 2 − 𝐾1
′
c′
(7)
Where g is the acceleration due to gravity; 𝛽, the volumetric coefficient of thermal expansion; 𝛽∗
, the
coefficient of volume expansion with species concentration; T, the temperature of the fluid within the boundary
layer; 𝐶′
, the species concentration; 𝜌, 𝜇, 𝑣, 𝑘, 𝑐 𝑝 are respectively density, viscosity, kinematic viscosity, thermal
conductivity, specific heat at constant pressure; and D, the chemical molecular diffusivity, Chemical reaction
parameter𝐾1
′
. The initial and boundary conditions of the problem are
𝑡′
≤ 0: 𝑢′
= 0, 𝑤′
= 0, 𝑇 = 𝑇∞, 𝐶′
= 𝐶∞ 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑦′
𝑡′
> 0:
𝑢′
= 0, 𝑤 = 0, 𝑇 = 𝑇𝑤 , 𝐶′
= 𝐶 𝑤 𝑎𝑡 𝑦′
= 0
𝑢′
= 0, 𝑤′
= 0, 𝑇 = 𝑇∞, 𝐶′
= 𝐶∞, 𝑎𝑡 𝑦′
→ ∞
(8)
The non-dimensional quantities introduced in the equations (3) to (7) are
3. Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a…
www.ijeijournal.com P a g e | 24
𝑡 =
𝑡′ 𝑈0
2
𝑣
, 𝑦 =
𝑦′ 𝑈0
𝑣
, 𝑢, 𝑉, 𝑤 =
(𝑢′, 𝑉′, 𝑤′)
𝑈0
𝜃 =
(𝑇−𝑇∞)
(𝑇 𝑤 −𝑇∞)
, 𝐶 =
(𝐶′−𝐶∞)
(𝐶 𝑤 −𝐶∞)
, 𝐾1 =
𝐾1
′ 𝑣
𝑉0
2
(9)
Where 𝑈0 is the reference velocity. Therefore, the governing equation in the dimensionless form are
obtained as
∂V
∂y
= 0 (10)
𝜕𝑢
𝜕𝑡
+ 𝑉
𝜕𝑢
𝜕𝑦
=
𝜕2 𝑢
∂y2 −
𝑀
1+𝑚2 u + mw + 𝐺𝑟 𝜃 + 𝐺𝑐 𝐶 (11)
𝜕𝑤
𝜕𝑡
+ 𝑉
𝜕𝑤
𝜕𝑦
=
𝜕2 𝑤
∂y2 −
𝑀
1+𝑚2 w − mu (12)
𝜕𝜃
𝜕𝑡
+ 𝑉
𝜕𝜃
𝜕𝑦
=
1
Pr
𝜕2 𝜃
∂y2 + Ec
∂u
∂y
2
(13)
𝜕𝐶
𝜕𝑡
+ 𝑉
𝜕𝐶
𝜕𝑦
=
1
Sc
𝜕2 𝐶
∂y2 − 𝐾1 𝑐 (14)
Where
𝑀 =
σ𝜇 𝑒
2 𝐵0
2 𝑣
𝜌 𝑈2 𝑖𝑠 Magnetic field parameter, 𝑃𝑟 =
μ Cp
k
is Prandtl number
Sc =
v
D
is Schmidt number, Ec =
U0
2
Cp(Tw −T∞)
is Eckert number
𝐺𝑟 =
vgβ(Tw −T∞)
U0
3 is Modified Grashof number for heat transfer
𝐺𝑐 =
vgβ∗
Cw −C∞
U0
3 is Modified Grashof number for mass transfer
And the boundary conditions (equation (8)) in the non-dimensional form are
𝑡 ≤ 0: 𝑢 = 0, 𝑤 = 0, 𝑇 = 0, 𝐶 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑦
𝑡 > 0:
u = 0, w = 0, θ = 1, C = 1 at y = 0
u = 0, w = 0, θ = 0, C = 0 as y → ∞
(15)
From equation (10), it is seen that 𝑣 is either constant or a function of time t. Similarly, solutions of equations
(11) to (14) with the boundary conditions equation (15) exist only if
𝑉 = −𝜆𝑡−
1
2 (16)
Where λ is a non-dimensional transpiration parameter. For suction 𝜆 > 0 and for blowing λ < 0. From equation
(16), it can be observed that the assumption is valid only for small values of time variable.
III. METHOD OF SOLUTION
By applying the Galerkin finite element method for equation (11) over a typical two-noded linear
element 𝑒 ( 𝑦𝑗 ≤ 𝑦 ≤ 𝑦 𝑘 ) is
𝑁 𝑇 𝜕2 𝑢 𝑒
𝜕𝑦2 −
𝜕𝑢 𝑒
𝜕𝑡
− 𝑉
𝜕𝑢 𝑒
𝜕𝑦
−
𝑀
1+𝑚2 𝑚𝜔 + 𝑢 𝑒
+ 𝐺𝑟 𝜃 + 𝐺𝑐 𝐶 𝑑𝑦
𝑦 𝑘
𝑦 𝑗
(17)
𝜕𝑁
𝜕𝑦
.
𝜕𝑢 𝑒
𝜕𝑦
− 𝑁 𝑇
−𝑉
𝜕𝑢 𝑒
𝜕𝑦
−
𝜕𝑢 𝑒
𝜕𝑡
−
𝑀
1+𝑚2 𝑢 𝑒
+ 𝑅∗
𝑑𝑦
𝑦 𝑘
𝑦 𝑗
=0 (18)
𝑤ℎ𝑒𝑟𝑒 𝑅∗
= −
𝑀𝑚
1+𝑚2 ω + 𝐺𝑟 𝜃 + 𝐺𝑐 𝐶 , 𝑁 = 𝑁𝑗 , 𝑁𝑘 , ∅ 𝑒
=
𝑢𝑗
𝑢 𝑘
,
𝑢 𝑒
= 𝑁. ∅ 𝑒
, 𝑁𝑗 =
𝑦 𝑘 −𝑦
𝑙(𝑒) , 𝑁𝑘 =
𝑦−𝑦 𝑗
𝑙(𝑒) , 𝑙(𝑒)
= 𝑦 𝑘 − 𝑦𝑗 = ℎ
The element equation given by
𝑁𝑗
′
𝑁𝑗
′
𝑁𝑗
′
𝑁𝑘
′
𝑁𝑘
′
𝑁𝑗
′
𝑁𝑘
′
𝑁𝑘
′
𝑢𝑗
𝑢 𝑘
𝑑𝑦 − 𝑉
𝑁𝑗 𝑁𝑗
′
𝑁𝑗 𝑁𝑘
′
𝑁𝑘 𝑁𝑗
′
𝑁𝑘 𝑁𝑘
′
𝑢𝑗
𝑢 𝑘
𝑑𝑦 +
𝑦 𝑘
𝑦 𝑗
𝑁𝑗 𝑁𝑗 𝑁𝑗 𝑁𝑘
𝑁𝑘 𝑁𝑗 𝑁𝑘 𝑁𝑘
𝑢𝑗
𝑢 𝑘
𝑑𝑦 +
𝑀
1+𝑚2
𝑁𝑗 𝑁𝑗 𝑁𝑗 𝑁𝑘
𝑁𝑘 𝑁𝑗 𝑁𝑘 𝑁𝑘
𝑢𝑗
𝑢 𝑘
𝑑𝑦 − 𝑅∗
𝑁𝑗
𝑁𝑘
𝑑𝑦 = 0 (19)
𝑆 𝑒
+ 𝐴 𝑒
+ 𝑅 𝑑𝑦 = 0
𝑦 𝑘
𝑦 𝑗
(20)
where 𝑆 𝑒
=
𝑁𝑗
′
𝑁𝑗
′
𝑁𝑗
′
𝑁𝑘
′
𝑁𝑘
′
𝑁𝑗
′
𝑁𝑘
′
𝑁𝑘
′
𝑢𝑗
𝑢 𝑘
− 𝑉
𝑁𝑗 𝑁𝑗
′
𝑁𝑗 𝑁𝑘
′
𝑁𝑘 𝑁𝑗
′
𝑁𝑘 𝑁𝑘
′
𝑢𝑗
𝑢 𝑘
+
𝑀
1+𝑚2
𝑁𝑗 𝑁𝑗 𝑁𝑗 𝑁𝑘
𝑁𝑘 𝑁𝑗 𝑁𝑘 𝑁𝑘
𝑢𝑗
𝑢 𝑘
,
𝐴 𝑒
=
𝑁𝑗 𝑁𝑗 𝑁𝑗 𝑁𝑘
𝑁𝑘 𝑁𝑗 𝑁𝑘 𝑁𝑘
𝑢𝑗
𝑢 𝑘
𝑎𝑛𝑑 𝑅 = 𝑅∗
𝑁𝑗
𝑁𝑘
Here the prime and dot denote differentiation with respect to y and t. we obtain
4. Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a…
www.ijeijournal.com P a g e | 25
𝑆 𝑒
=
1
𝑙(𝑒)
1 −1
−1 1
𝑢𝑗
𝑢 𝑘
−
𝑉
2
−1 1
−1 1
𝑢𝑗
𝑢 𝑘
+
𝑀
1 + 𝑚2
𝑙(𝑒)
6
2 1
1 2
𝑢𝑗
𝑢 𝑘
𝐴 𝑒
=
𝑙(𝑒)
6
2 1
1 2
𝑢𝑗
𝑢 𝑘
𝑎𝑛𝑑 𝑅 = 𝑅∗
𝑙(𝑒)
2
1
1
We write the element equation for the elements 𝑦𝑖−1 ≤ 𝑦 ≤ 𝑦𝑖 and 𝑦𝑖 ≤ 𝑦 ≤ 𝑦𝑖+1. Assembling these element
equations, we get
1
𝑙 𝑒
1 −1 0
−1 2 −1
0 −1 1
𝑢𝑖−1
𝑢𝑖
𝑢𝑖+1
−
𝑉
2
−1 1 0
−1 0 1
0 −1 1
𝑢𝑖−1
𝑢𝑖
𝑢𝑖+1
+
𝑀
1+𝑚2
𝑙 𝑒
6
2 1 0
1 4 1
0 1 2
𝑢𝑖−1
𝑢𝑖
𝑢𝑖+1
+
𝑙 𝑒
6
2 1 0
1 4 1
0 1 2
𝑢𝑖+1
𝑢𝑖
𝑢𝑖+1
= 𝑅∗ 𝑙 𝑒
2
1
2
1
(21)
Now put row corresponding to the node i to zero, from equation (21) the difference schemes with 𝑙 𝑒
= ℎ is
ℎ
6
𝑢𝑖−1 + 4𝑢𝑖 + 𝑢𝑖+1 +
𝑀
1 + 𝑚2
ℎ
6
−
𝑉
2
−
1
ℎ
𝑢𝑖−1 +
2
ℎ
+
2𝑀
1 + 𝑚2
ℎ
3
𝑢𝑖 +
𝑀
1+𝑚2
ℎ
6
+
𝑉
2
−
1
ℎ
𝑢𝑖+1 = R and here R = 𝑅∗ 𝑙 𝑒
2
1
2
1
(22)
Using the Cranck-Nicolson method to the equation (22), we obtain:
𝐴1 𝑢𝑖−1
𝑗+1
+ 𝐴2 𝑢𝑖
𝑗+1
+ 𝐴3 𝑢𝑖+1
𝑗+1
+ 𝐴4 𝑢𝑖−1
𝑗
+ 𝐴5 𝑢𝑖
𝑗
+ 𝐴6 𝑢𝑖+1
𝑗
= 𝑅 (23)
Similarly, the equations (12), (13), (14) are becoming as follows:
𝐵1 𝑤𝑖−1
𝑗 +1
+ 𝐵2 𝑤𝑖
𝑗 +1
+ 𝐵3 𝑤𝑖+1
𝑗 +1
+ 𝐵4 𝑤𝑖−1
𝑗
+ 𝐵5 𝑤𝑖
𝑗
+ 𝐵6 𝑤𝑖+1
𝑗
= 𝑅 𝐴 (24)
𝐶1 𝜃𝑖−1
𝑗+1
+ 𝐶2 𝜃𝑖
𝑗 +1
+ 𝐶3 𝜃𝑖+1
𝑗+1
+ 𝐶4 𝜃𝑖−1
𝑗
+ 𝐶5 𝜃𝑖
𝑗
+ 𝐶6 𝜃𝑖+1
𝑗
= 𝑅 𝐵 (25)
𝐷1 𝑐𝑖−1
𝑗 +1
+ 𝐷2 𝑐𝑖
𝑗+1
+ 𝐷3 𝑐𝑖+1
𝑗 +1
+ 𝐷4 𝑐𝑖−1
𝑗
+ 𝐷5 𝑐𝑖
𝑗
+ 𝐷6 𝑐𝑖+1
𝑗
= 0 (26)
The initial and boundary conditions (15) reduce to
𝑢 𝑖, 0 = 0, 𝑤 𝑖, 0 = 0, 𝜃 𝑖, 0 = 0, 𝐶 𝑖, 0 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖
𝑢 0, 𝑗 = 0, 𝑤 0, 𝑗 = 0, 𝜃 0, 𝑗 = 1, 𝐶 0, 𝑗 = 1
𝑢 100, 𝑗 = 0, 𝑤 100, 𝑗 = 0, 𝜃 100, 𝑗 = 1, 𝐶 100, 𝑗 = 0
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗
(27)
𝐴1 =
1
6
− +
𝑘
6
𝑀
1+𝑚2 − 𝑟 +
𝑃𝑉
2
, 𝐴2 =
4
6
+ 2𝑟 +
2𝑘
3
𝑀
1+𝑚2 ,
𝐴3 =
1
6
+
𝑘
6
𝑀
1+𝑚2 −
𝑃𝑉
2
− 𝑟 , 𝐴4 = −
𝑘
6
𝑀
1+𝑚2 + 𝑟 −
𝑃𝑉
2
−
1
6
,
𝐴5 = −2𝑟 −
𝑀
1+𝑚2
2𝑘
3
−
4
6
, 𝐴6 = −
𝑘
6
𝑀
1+𝑚2 + 𝑟 −
𝑃𝑉
2
−
1
6
,
𝐵1 =
1
6
+
𝑘
6
𝑀
1+𝑚2 − 𝑟 +
𝑃𝑉
2
, 𝐵2 =
4
6
+ 2𝑟 +
2𝑘
3
𝑀
1+𝑚2 ,
𝐵3 =
1
6
+
𝑘
6
𝑀
1+𝑚2 −
𝑃𝑉
2
− 𝑟 , 𝐵4 = −
𝑘
6
𝑀
1+𝑚2 + 𝑟 −
𝑃𝑉
2
−
1
6
𝐵5 = −2𝑟 −
𝑀
1+𝑚2
2𝑘
3
−
4
6
, 𝐵6 = −
𝑘
6
𝑀
1+𝑚2 + 𝑟 −
𝑃𝑉
2
−
1
6
𝐶1 =
1
6
−
𝑟
𝑆𝑐
+
𝑃𝑉
2
, 𝐶2 =
4
6
+
2𝑟
𝑆𝑐
, 𝐶3 =
1
6
−
𝑟
𝑆𝑐
−
𝑃𝑉
2
, 𝐶4 =
1
6
+
𝑟
𝑆𝑐
−
𝑃𝑉
2
,
𝐶5 =
4
6
−
2𝑟
𝑆𝑐
, 𝐶6 =
1
6
+
𝑟
𝑆𝑐
+
𝑃𝑉
2
,
𝐷1 = 1 − 6𝑟
1
𝑆 𝑐
+ 3𝑝𝑉 + 𝐾1 𝑘 , 𝐷2 = 4 + 12𝑟
1
𝑆 𝑐
+ 4𝐾1 𝑘 ,
𝐷3 = 1 − 6𝑟
1
𝑆 𝑐
− 3𝑝𝑉 + 𝐾1 𝑘 , 𝐷4 = 1 + 6𝑟
1
𝑆 𝑐
− 3𝑝𝑉 − 𝐾1 𝑘 ,
𝐷5 = 4 − 12𝑟
1
𝑆 𝑐
− 4𝐾1 𝑘 , 𝐷6 = 1 + 6𝑟
1
𝑆 𝑐
+ 3𝑝𝑉 − 𝐾1 𝑘 ,
𝑅 𝐴 =
𝑀𝑚
1+𝑚2 𝑢, 𝑅 𝐵 = 𝐸𝑐
𝑈 𝑖+1−𝑈 𝑖−1
2ℎ
2
Here 𝑟 =
𝑘
ℎ2 where k, h is mesh sizes along y direction and time direction respectively. Index i refers to space
and j refers to time. The mesh system consists of h=0.1 for velocity profiles and concentration profiles and
k=0.05 has been considered for computations. In equation (23)-(26), taking i=1(1) n and using initial and
boundary conditions (15), the following system of equation are obtained.
𝐴𝑖 𝑋𝑖 = 𝐵𝑖 , 𝑖 = 1,2,3 …. (28)
Where 𝐴𝑖’s are matrices of order n and 𝑋𝑖 𝑎𝑛𝑑 𝐵𝑖’ 𝑠 are column matrices having n-components. The
solution of above system of equations are obtained using Thomas algorithm for velocity, temperature and
concentration. Also, numerical solutions for these equations are obtained by MATLAB-Program, Galerkin finite
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element method. The same MATLAB-Program was run with slightly changed values of h and k and no
significant change was observed in the values of 𝑢, 𝑤, 𝜃, 𝑐.
IV. RESULTS AND DISCUSSION
To get the physical insight of the problem, the results are discussed through graphs. The numerical
results are obtained for the velocity profiles of main (primary) flow and cross (secondary) flow and the
temperature profiles, concentration profiles have been shown graphically for different flow parameters, such as
magnetic field parameter M, Hall current m, Eckert number Ec, Chemical reaction parameter 𝐾1 Modified
Grashof number Gr for heat transfer, Modified Grashof number Gc for mass transfer, Schmidt number Sc,
transpiration parameter 𝜆, and time t. The primary velocity profiles are shown in fig.1 (a), fig.2 (a) and fig.3 (a).
The Hall current effect is to increases the primary velocity component u, and this is because of the fact that Hall
Effect, in general reduces the level of Lorentz forces effect on the fluid, whose tendency is to suppress the flow.
The primary velocity flow is decreases with the increase of magnetic field parameter M and the transpiration
parameter λ. It is also observed that the primary velocity u increases with the time t. It can be seen that the effect
of Gr, and Gc is to increase the primary velocity profiles u, which is due to with an increase in buoyancy force
due to temperature differences, the flow is accelerated in the boundary layer. Here the positive of Gr (>0)
correspond to cooling of the plate by natural convection. The effect of Prandtl number Pr is reduces velocity
profiles, the reason is that the smaller values of Pr are equivalent to increasing the thermal conductivities, and
therefore heat is able to diffuse away from the heated surface more rapidly than for higher values of Pr. The
influence of Schmidt number Sc is to reduce the velocity profiles. It is also noticed that the influence of Eckert
number Ec, on the velocity profiles is increases the primary velocity profiles u, and designates the ratio of
kinetic energy of the flow into the boundary layer enthalpy differences. It embodies the conversion of kinetic
energy to internal energy by work done against the viscous fluid stresses. Hence, greater viscous dissipative heat
causes a rise in the velocity profiles.
The secondary velocity profiles w are shown graphically in fig.1 (b), fig.2 (b) and fig.3 (b). The
secondary velocity profiles w increase with time t, Hall parameter m and the magnetic field parameter M. But
decreases owing to increase of transpiration parameter λ, it is observed from figures. It also can be observed that
the secondary velocity profiles w increases as the Grashof number Gr (for heat transfer) and Gc (for mass
transfer) increases, but it decreases as Prandtl number Pr and Schmidt number Sc increases. It also seen that
effects of viscous dissipation increases the secondary velocity profiles w.
From the fig.5 it is evident that the temperature of the fluid increases with the increase of time t. But it decreases
with the increase of transpiration parameter λ and Prandtl number Pr. The viscous dissipation effect on
temperature profiles shown in fig.3(c). The viscous dissipation heat causes a rise in the temperature which is
shown from the figure.
Fig.6 is drawn for the concentration profiles. From the figure it can be seen that the concentrations
profiles are increases with the increase of time t, while it is decreases with the increase of transpiration
parameter λ and Schmidt number Sc.
The effect of chemical reaction parameter 𝐾1 on primary velocity and secondary velocity profiles,
temperature and concentration profiles are shown in fig.4 (a)-(d) respectively. From the fig.4 (a) & (b) it can be
observed that the effect of 𝐾1, leads to decreases the primary velocity profiles as well as secondary velocity
profiles. Fig4(c) reviles that an increase of chemical reaction parameter 𝐾1 leads to decrease the temperature
profiles. Fig.4 (d) shows that there is fall in concentration profiles due to the increasing the value of chemical
reaction parameter 𝐾1 is noticed.
Fig.1(a): The Primary velocity profiles
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Fig.1 (b): The Secondary velocity profiles
Fig.2 (a): The Primary velocity profiles
Fig.2 (b): The Secondary velocity profiles
Fig.3 (a): The Primary velocity profiles for different values of Ec
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Fig.3 (b): The Secondary velocity profiles for different values of Ec
Fig.3 (c): Temperature Profiles for different values of Ec
Fig.4 (a): The Primary Velocity Profile for different values of 𝐾1
Fig.4 (b): The Secondary Velocity Profile for different values of 𝐾1
t=1.0, λ=0.5, m=0.2, M=1
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Fig.4(c): The Temperature Profile for different values of 𝐾1
Fig.4 (d): Concentration Profile for different values of 𝐾1
Fig.5: The Temperature Profiles
Fig.6: The Concentration Profiles
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