International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Effect of 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.
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.
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.
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.
Effects of Variable Viscosity and Thermal Conductivity on MHD free Convection...theijes
This document summarizes a study that numerically investigates the effects of variable viscosity and thermal conductivity on magnetohydrodynamic (MHD) free convection and mass transfer flow over an inclined vertical surface in a porous medium with heat generation. The governing equations are reduced to ordinary differential equations using similarity transformations and then solved numerically using a shooting method. The results show that increasing the viscosity variation parameter, thermal conductivity parameter, magnetic parameter, permeability parameter, or Schmidt number decreases the fluid velocity, while increasing the heat generation parameter, local Grashof number, or mass Grashof number increases the fluid velocity. Skin friction, Nusselt number, and Sherwood number are also computed and presented in tabular form.
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.
Investigation of the Effect of Nanoparticles Mean Diameter on Turbulent Mixed...A Behzadmehr
Abstract
Turbulent mixed convection of a nanofluid (water/Al2O3, Φ=.02) has been studied numerically. Two-phase
mixture model has been used to investigate the effects of nanoparticles mean diameter on the flow parameters. Nanoparticles distribution at the tube cross section shows that the particles are uniformly dispersed. The non-uniformity of the particles distribution occurs in the case of large nanoparticles and/or high value of the Grashof numbers. The study of particle size effect showed that the effective Nusselt number and turbulent intensity increases with the decreased of particle size.
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 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.
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.
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.
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.
Effects of Variable Viscosity and Thermal Conductivity on MHD free Convection...theijes
This document summarizes a study that numerically investigates the effects of variable viscosity and thermal conductivity on magnetohydrodynamic (MHD) free convection and mass transfer flow over an inclined vertical surface in a porous medium with heat generation. The governing equations are reduced to ordinary differential equations using similarity transformations and then solved numerically using a shooting method. The results show that increasing the viscosity variation parameter, thermal conductivity parameter, magnetic parameter, permeability parameter, or Schmidt number decreases the fluid velocity, while increasing the heat generation parameter, local Grashof number, or mass Grashof number increases the fluid velocity. Skin friction, Nusselt number, and Sherwood number are also computed and presented in tabular form.
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.
Investigation of the Effect of Nanoparticles Mean Diameter on Turbulent Mixed...A Behzadmehr
Abstract
Turbulent mixed convection of a nanofluid (water/Al2O3, Φ=.02) has been studied numerically. Two-phase
mixture model has been used to investigate the effects of nanoparticles mean diameter on the flow parameters. Nanoparticles distribution at the tube cross section shows that the particles are uniformly dispersed. The non-uniformity of the particles distribution occurs in the case of large nanoparticles and/or high value of the Grashof numbers. The study of particle size effect showed that the effective Nusselt number and turbulent intensity increases with the decreased of particle size.
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
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.
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.
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.
Moving Lids Direction Effects on MHD Mixed Convection in a Two-Sided Lid-Driv...A Behzadmehr
Magnetohydrodynamic (MHD) mixed convection flow of Cu–water nanofluid inside a two-sided lid-driven square enclosure with adiabatic horizontal walls and differentially heated sidewalls has been investigated numerically. The effects of moving lids direction, variations of Richardson number, Hartmann number, and volume fraction of nanoparticles on flow and temperature fields have been studied. The obtained results show that for a constant Grashof number (), the rate of heat transfer increases with a decrease in the Richardson and Hartmann numbers. Furthermore, an increase of the volume fraction of nanoparticles may result in enhancement or deterioration of the heat transfer performance depending on the value of the Hartmann and Richardson numbers and the configuration of the moving lids. Also, it is found that in the presence of magnetic field, the nanoparticles have their maximum positive effect when the top lid moves rightward and the bottom one moves leftward.
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.
This document summarizes a study that examines heat and mass transfer over a vertical plate in a porous medium with Soret and Dufour effects, a convective surface boundary condition, chemical reaction, and magnetic field. The governing equations for the fluid flow, heat transfer, and mass transfer are presented. Similarity solutions are used to transform the governing partial differential equations into ordinary differential equations, which are then solved numerically. The results are presented graphically to show the influence of various parameters on velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number.
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.
Chemical reaction and radiation effect on mhd flow past an exponentially acce...Alexander Decker
This document describes a mathematical analysis of MHD fluid flow past an exponentially accelerated vertical plate embedded in a porous medium. The analysis considers the effects of variable temperature, mass diffusion, radiation, and a heat source on the flow characteristics. The governing equations for this problem are derived and non-dimensionalized. The non-dimensional equations are then solved using the Laplace transform technique. The effects of various physical parameters like the magnetic field, radiation, heat generation, and chemical reaction on the velocity, temperature, and concentration profiles are determined.
Boundary layer flow and heat transfer of a dusty fluid over a vertical permea...eSAT Journals
This document discusses boundary layer flow and heat transfer of a dusty fluid over a vertical permeable stretching surface. The governing equations for steady, two-dimensional flow are presented and non-dimensionalized. The equations are then solved numerically using the Runge-Kutta method. Results are presented graphically showing the effects of various parameters like fluid-particle interaction, local Grashof number, radiation parameter, and Eckert number on flow and heat transfer characteristics. Comparisons are made to previous studies to validate the numerical method.
Effect of Thermo-Diffusion and Chemical Reaction on Mixed Convective Heat And...IJERA Editor
A finite element study of combined heat and mass transfer flow through a porous medium in a circular cylindrical annulus with Soret and Dufour effects in the presence of heat sources has been analyzed. The coupled velocity, energy, and diffusion equations are solved numerically by using Galerkin- finite element technique. Shear stress, Nusslet number and Sherwood number are evaluated numerically for different values of the governing parameters under consideration and are shown in tabular form.
The results show that, with proper selection of physical parameters, significant heat transfer
enhancements and pressure drop reductions can be achieved simultaneously with porous pin fins and
the overall heat transfer performances in porous pin fin channels are much better than those in
traditional solid pin fin channels. The effects of pore density are significant. As PPI increases, the
pressure drops and heat fluxes in porous pin fin channels increase while the overall heat transfer
efficiencies decrease and the maximal overall heat transfer efficiencies are obtained at PPI 20.
Furthermore, the effects of pin fin form are also remarkable. With the same physical parameters, the
overall heat transfer efficiencies in the long elliptic porous pin fin channels are the highest while they
are the lowest in the short elliptic porous pin fin channels
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.
NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFE...IAEME Publication
In the present paper an analysis of mixed convection of an unsteady magneto hydrodynamic (MHD) flow of an incompressible viscous fluid through porous media due to a vertical porous stretching sheet in the presence of viscous dissipation, thermal radiation and heat source /sink has been carried out. The fluid considered is viscous and incompressible. The governing partial differential equations of the flow, mass and heat transfer are highly non linear hence are converted into a system of ordinary differential equations using suitable similarity transformations. These ordinary differential equations are further converted into 7 first order ordinary differential equations and are solved numerically by Matlab ode-45 solver via shooting method.
This document presents an analysis of free convective flow and heat transfer of a viscous incompressible fluid over a linearly moving vertical porous plate with suction and viscous dissipation. The fluid viscosity is assumed to vary linearly with temperature. Governing boundary layer equations are non-dimensionalized using similarity transformations and reduced to a boundary value problem. The problem is solved numerically using a shooting method along with the 4th order Runge-Kutta method. Results for dimensionless velocity and temperature distributions are presented for various parameter values, and skin friction and Nusselt numbers are reported in tables.
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.
Study of Forced Convection Heat Transfer with Single phase and mixture phase ...IOSRJMCE
In this study, forced convection heat transfer of nanoliquids is done using both single-phase and mixture-phase models and the results are compared with experimental results. The governing equations of the study here are discretized using the finite volume method. Hybrid differencing scheme is used to calculate the face values of the control volumes. A code is written using SIMPLER algorithm and then solved using the MATLAB engine. The mixture-phase model studied here, considers two slip mechanisms between nanoparticle and base-fluid, namely Brownian diffusion and thermophoresis. Al2O3-water nanofluid is used for the study of nanofluid and the study shows significant increase in convective heat transfer coefficient while the mixturephase model demonstrates slightly lower values than the single-phase model. The study is done with various nanoparticle concentrations and Reynolds numbers. With increasing particle concentration and Reynolds number, the convective heat transfer coefficient increases and as well as the shear stress. For low concentrations of the nanoparticle, Nusselt number is slightly lower than the base fluid and as the concentration increases, the Nusselt number also rises higher than the base fluid
Laplace Analysis of Periodic Heat and Mass Transport on a Parabolic Started S...IJERA Editor
An analysis of periodic heat and mass transport of unsteady hydromagnetic flow past a parabolic started motion of the infinite vertical plate immersed in Darcian porous regime in presence of a first order chemical reaction has been presented. Here the plate temperature as well as concentration level near the plate are increased linearly with time. The boundary layer conservation equations have been solved by Laplace transforms technique. It has been observed that both the velocity and concentration are decreased with increasing values of chemical reaction parameter. But the opposite behavior has been found for the flow velocity when the values of free convection as well as porosity parameter are increased. Application of magnetic fields to medical science is growing rapidly, with the development of novel magnetic pumps, hydromagnetic separation devices with chemical engineering and geophysical energy systems.
The Talking point is one of the many initiatives from Alzheimer’s society to support people with dementia and their carers through online forum. They alone don’t consider this has as innovation. Through this report I showcase, how the online forum has all the potential to create awareness for all the people who are affected by dementia in all the countries.
Antigua and Barbuda has a small, open economy dominated by tourism which accounts for over half of GDP. The document provides information on Antigua and Barbuda's economy, trade, market access conditions, investment profile, and business establishment requirements. It details the country's tariff rates and free trade agreements, distribution channels, transportation infrastructure, and cultural customs. The document is a comprehensive guide for those looking to do business in Antigua and Barbuda.
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.
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.
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.
Moving Lids Direction Effects on MHD Mixed Convection in a Two-Sided Lid-Driv...A Behzadmehr
Magnetohydrodynamic (MHD) mixed convection flow of Cu–water nanofluid inside a two-sided lid-driven square enclosure with adiabatic horizontal walls and differentially heated sidewalls has been investigated numerically. The effects of moving lids direction, variations of Richardson number, Hartmann number, and volume fraction of nanoparticles on flow and temperature fields have been studied. The obtained results show that for a constant Grashof number (), the rate of heat transfer increases with a decrease in the Richardson and Hartmann numbers. Furthermore, an increase of the volume fraction of nanoparticles may result in enhancement or deterioration of the heat transfer performance depending on the value of the Hartmann and Richardson numbers and the configuration of the moving lids. Also, it is found that in the presence of magnetic field, the nanoparticles have their maximum positive effect when the top lid moves rightward and the bottom one moves leftward.
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.
This document summarizes a study that examines heat and mass transfer over a vertical plate in a porous medium with Soret and Dufour effects, a convective surface boundary condition, chemical reaction, and magnetic field. The governing equations for the fluid flow, heat transfer, and mass transfer are presented. Similarity solutions are used to transform the governing partial differential equations into ordinary differential equations, which are then solved numerically. The results are presented graphically to show the influence of various parameters on velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number.
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.
Chemical reaction and radiation effect on mhd flow past an exponentially acce...Alexander Decker
This document describes a mathematical analysis of MHD fluid flow past an exponentially accelerated vertical plate embedded in a porous medium. The analysis considers the effects of variable temperature, mass diffusion, radiation, and a heat source on the flow characteristics. The governing equations for this problem are derived and non-dimensionalized. The non-dimensional equations are then solved using the Laplace transform technique. The effects of various physical parameters like the magnetic field, radiation, heat generation, and chemical reaction on the velocity, temperature, and concentration profiles are determined.
Boundary layer flow and heat transfer of a dusty fluid over a vertical permea...eSAT Journals
This document discusses boundary layer flow and heat transfer of a dusty fluid over a vertical permeable stretching surface. The governing equations for steady, two-dimensional flow are presented and non-dimensionalized. The equations are then solved numerically using the Runge-Kutta method. Results are presented graphically showing the effects of various parameters like fluid-particle interaction, local Grashof number, radiation parameter, and Eckert number on flow and heat transfer characteristics. Comparisons are made to previous studies to validate the numerical method.
Effect of Thermo-Diffusion and Chemical Reaction on Mixed Convective Heat And...IJERA Editor
A finite element study of combined heat and mass transfer flow through a porous medium in a circular cylindrical annulus with Soret and Dufour effects in the presence of heat sources has been analyzed. The coupled velocity, energy, and diffusion equations are solved numerically by using Galerkin- finite element technique. Shear stress, Nusslet number and Sherwood number are evaluated numerically for different values of the governing parameters under consideration and are shown in tabular form.
The results show that, with proper selection of physical parameters, significant heat transfer
enhancements and pressure drop reductions can be achieved simultaneously with porous pin fins and
the overall heat transfer performances in porous pin fin channels are much better than those in
traditional solid pin fin channels. The effects of pore density are significant. As PPI increases, the
pressure drops and heat fluxes in porous pin fin channels increase while the overall heat transfer
efficiencies decrease and the maximal overall heat transfer efficiencies are obtained at PPI 20.
Furthermore, the effects of pin fin form are also remarkable. With the same physical parameters, the
overall heat transfer efficiencies in the long elliptic porous pin fin channels are the highest while they
are the lowest in the short elliptic porous pin fin channels
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.
NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFE...IAEME Publication
In the present paper an analysis of mixed convection of an unsteady magneto hydrodynamic (MHD) flow of an incompressible viscous fluid through porous media due to a vertical porous stretching sheet in the presence of viscous dissipation, thermal radiation and heat source /sink has been carried out. The fluid considered is viscous and incompressible. The governing partial differential equations of the flow, mass and heat transfer are highly non linear hence are converted into a system of ordinary differential equations using suitable similarity transformations. These ordinary differential equations are further converted into 7 first order ordinary differential equations and are solved numerically by Matlab ode-45 solver via shooting method.
This document presents an analysis of free convective flow and heat transfer of a viscous incompressible fluid over a linearly moving vertical porous plate with suction and viscous dissipation. The fluid viscosity is assumed to vary linearly with temperature. Governing boundary layer equations are non-dimensionalized using similarity transformations and reduced to a boundary value problem. The problem is solved numerically using a shooting method along with the 4th order Runge-Kutta method. Results for dimensionless velocity and temperature distributions are presented for various parameter values, and skin friction and Nusselt numbers are reported in tables.
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.
Study of Forced Convection Heat Transfer with Single phase and mixture phase ...IOSRJMCE
In this study, forced convection heat transfer of nanoliquids is done using both single-phase and mixture-phase models and the results are compared with experimental results. The governing equations of the study here are discretized using the finite volume method. Hybrid differencing scheme is used to calculate the face values of the control volumes. A code is written using SIMPLER algorithm and then solved using the MATLAB engine. The mixture-phase model studied here, considers two slip mechanisms between nanoparticle and base-fluid, namely Brownian diffusion and thermophoresis. Al2O3-water nanofluid is used for the study of nanofluid and the study shows significant increase in convective heat transfer coefficient while the mixturephase model demonstrates slightly lower values than the single-phase model. The study is done with various nanoparticle concentrations and Reynolds numbers. With increasing particle concentration and Reynolds number, the convective heat transfer coefficient increases and as well as the shear stress. For low concentrations of the nanoparticle, Nusselt number is slightly lower than the base fluid and as the concentration increases, the Nusselt number also rises higher than the base fluid
Laplace Analysis of Periodic Heat and Mass Transport on a Parabolic Started S...IJERA Editor
An analysis of periodic heat and mass transport of unsteady hydromagnetic flow past a parabolic started motion of the infinite vertical plate immersed in Darcian porous regime in presence of a first order chemical reaction has been presented. Here the plate temperature as well as concentration level near the plate are increased linearly with time. The boundary layer conservation equations have been solved by Laplace transforms technique. It has been observed that both the velocity and concentration are decreased with increasing values of chemical reaction parameter. But the opposite behavior has been found for the flow velocity when the values of free convection as well as porosity parameter are increased. Application of magnetic fields to medical science is growing rapidly, with the development of novel magnetic pumps, hydromagnetic separation devices with chemical engineering and geophysical energy systems.
The Talking point is one of the many initiatives from Alzheimer’s society to support people with dementia and their carers through online forum. They alone don’t consider this has as innovation. Through this report I showcase, how the online forum has all the potential to create awareness for all the people who are affected by dementia in all the countries.
Antigua and Barbuda has a small, open economy dominated by tourism which accounts for over half of GDP. The document provides information on Antigua and Barbuda's economy, trade, market access conditions, investment profile, and business establishment requirements. It details the country's tariff rates and free trade agreements, distribution channels, transportation infrastructure, and cultural customs. The document is a comprehensive guide for those looking to do business in Antigua and Barbuda.
The Spherical Rubber Bearingst product catalogue from GMT Rubber-Metal Technology. This product brochure features technical specifications and diagrams of a wide range of Spherical Rubber Bearings available within the GMT range.
The document is the 2009 end of year report from the Student Government Association at the University of Alabama. It summarizes the accomplishments of President Cason Kirby's administration, including expanding library hours, improving academic advising, increasing emergency loans for students, and advocating for lower tuition and textbook costs. It also provides budgets and funding allocations to various student organizations.
Artificial intelligence (AI) is everywhere, promising self-driving cars, medical breakthroughs, and new ways of working. But how do you separate hype from reality? How can your company apply AI to solve real business problems?
Here’s what AI learnings your business should keep in mind for 2017.
Magneto-Convection of Immiscible Fluids in a Vertical Channel Using Robin Bou...IJERA Editor
The effects of viscous dissipation on fully developed two fluid magnetohydrodynamic flow in the presence of
constant electric field in a vertical channel is investigated using Robin boundary conditions. The fluids in both
the regions are incompressible, electrically conducting and the transport properties are assumed to be constant.
The plate exchanges heat with an external fluid. Both conditions of equal and different reference temperatures of
the external fluid are considered. First, the simple cases of the negligible Brinkman number or the negligible
Grashof number are solved analytically. Then, the combined effects of buoyancy forces and viscous dissipation
are analyzed by a perturbation series method valid for small values of perturbation parameter. To relax the
condition on the perturbation parameter, the flow fields are solved by using the differential transform method.
The results are presented graphically for different values of the mixed convection parameter, Hartman number,
perturbation parameter, viscosity ratio, width ratio, conductivity ratio and Biot numbers for both open and short
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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.
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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
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
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The unsteady hydromagnetic boundary layer flow of an incompressible and electrically conducting
fluid through a porous medium bounded by a moving surface has been considered. It is assumed that the moving
surface has a velocity profile with respect to time and fluid flow is taken under the influence of a transverse
magnetic field. The similarity solution is used to transform the system of partial differential equations,
describing the problem under consideration, into a boundary value problem of coupled ordinary differential
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technoloTwo dimensional numerical simulation of the combined heat transfer in...ijmech
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Rayleigh number, blowing Reynoldsnumber, radiation parameter, Prandtl number, the ratio of length to width and also ratio of opening thickness to width of the channel. In addition, effects of variation in any of the above non-dimensional numbers on parameters of the flow are clearly illustrated. At the end resultants had been compared with experimental data which demonstrated that in the present study, results have a great accuracy, relative errors are very small and the curve portraits are in a great
agreement with real experiments.
Effects of conduction on magneto hydrodynamics mixed convection flow in trian...Alexander Decker
This document summarizes research on magnetohydrodynamic (MHD) mixed convection flow in triangular enclosures. Key points:
1) The study investigates the effects of conduction on MHD mixed convection flow in triangular enclosures using a finite element method.
2) Parameters like the Hartmann number, Prandtl number, Reynolds number, and Rayleigh number are found to strongly influence the flow and thermal fields.
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Parametric studies on heat transfer by natural convection in vertical channel...IRJET Journal
This document provides an overview of parametric studies on heat transfer by natural convection in a vertical channel using an inclined V-slot plate. It discusses previous related research that investigated natural convection in various channel geometries experimentally and numerically. The proposed work aims to experimentally study heat transfer characteristics for variations in heat input, aspect ratio of the inclined V-slot plate, and its elevation. The goal is to develop a correlation between the Nusselt number and Rayleigh number to aid in the design of systems involving natural convection heat transfer.
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International Journal of Computational Engineering Research(IJCER)
1. International Journal of Computational Engineering Research||Vol, 03||Issue, 10||
Perturbation Technique and Differential Transform Method for
Mixed Convection Flow on Heat and Mass Transfer with
Chemical Reaction
1,
J. Prathap Kumar And 2,J.C. Umavathi
Department of Mathematics, Gulbarga University, Gulbarga-585 106, Karnataka, India
ABSTRACT
A new analytical solution is introduced for the effect of chemical reaction on mixed convective
heat and mass transfer in a vertical double passage channel. The vertical channel is divided into two
passages (by means of a baffle) for two separate flow streams. Each stream has its own individual
velocity, temperature and concentration fields. After placing the baffle the fluid is concentrated in one
of the passage. Approximate analytical solutions are found for the coupled nonlinear ordinary
differential equations using regular perturbation method (PM) and Differential Transform method
(DTM). The validity of the Differential Transform series solutions are verified with the regular
perturbation method. The velocity, temperature and concentration solutions are obtained and
discussed for various physical parameters such as thermal Grashoff number, mass Grashoff number,
Brinkman number and chemical reaction parameter at different positions of the baffle. It is found that
the thermal Grashoff number, mass Grashoff number, Brinkman number enhances the flow whereas
chemical reaction parameter reduces the flow at all baffle positions. It is also found that as Brinkman
number increases the DTM and PM show more error.
KEYWORDS: Baffle, first order chemical reaction, mixed convection, perturbation method,
Differential Transform method.
I.
INTRODUCTION
Study of mixed convection in the channel has been to the focus of lot of investigation during the last
three decades because of the multiple applications in which it is involved. These includes cooling of electronic
equipment, heat exchangers, chemical processing equipment, gas-cooled nuclear reactors and others. Tao [1]
analyzed the laminar fully developed mixed convection flow in a vertical parallel-plate channel with uniform
wall temperatures. Aung and Worku [2, 3] discussed the theory of combined free and forced convection in a
vertical channel with flow reversal conditions for both developing and fully developed flows. The case of
developing mixed convection flow in ducts with asymmetric wall heat fluxes was analyzed by the same authors
[4]. Recently, Prathap Kumar et al. [5] and Umavathi et al. [6, 7] studied the mixed convective flow and heat
transfer in a vertical channel for immiscible viscous fluids.
The rate of heat transfer in a vertical channel could be enhanced by using special inserts. Heat transfer
in such partially divided enclosures has received attention previously due to its applications to design energy
efficient buildings and reduction of heat loss from flat plate solar collectors. When the channel is divided into
several passages by means of plane baffles, as usually occurs in heat exchangers or electronic equipment, it is
quite possible to enhance the heat transfer performance between the walls and fluid by the adjustments of each
baffle position and strengths of the separate flow streams. In such configurations, perfectly conductive and thin
baffles may be used to avoid significant increase of the transverse thermal resistance. For a number of fluids, the
density-temperature relation exhibits an extreme. Because the coefficient of thermal expansion changes signs at
this extremum. Simple linear relations for density as a function of temperature are inadequate near the
extremum. Dutta and Dutta [8] first reported the enhancement of heat transfer with inclined solid and perforated
baffles. Later Dutta and Hossian [9] did the experimental study to analyze the local heat transfer characteristics
in a rectangular channel with inclined solid and perforated baffles. Salah El-Din [10, 11] published a series of
papers on mixed convection in a vertical channel by introducing a perfectly conducting baffle.
||Issn 2250-3005 ||
||October||2013||
Page 26
2. Perturbation Technique And Differential…
Mousavi and Hooman [12] studied numerically the fluid flow and heat transfer in the entrance region
of a two dimensional horizontal channel with isothermal walls and with staggered baffles. Heat transfer
enhancement in a heat exchanger tube by installing a baffle was reported by Nasiruddin and Siddiqui [13]. They
found that the average Nusselt number for the two baffles case is 20% higher than the one baffle case and 82%
higher than the no baffle case. Recently, Prathap Kumar et al. [14, 15] studied the flow characteristics of fully
developed free convection flow of a Walters fluid (Model B’) in a vertical channel divided into two passages.
Umavathi [16] analyzed the effect of the presence of a thin perfectly conductive baffle on the fully developed
laminar mixed convection in a vertical channel containing micropolar fluid.
Combining heat and mass transfer problems with a chemical reaction are of importance in many
processes and have, therefore, received a considerable amount of attention in recent years. In such processes as
drying, energy transfer in a wet cooling tower, and the flow in a desert cooler, heat and mass transfer occurs
simultaneously. Mixed convection processes involving the combined mechanisms are also encountered in many
natural processes, such as evaporation, condensation, and agricultural drying, and in many industrial
applications, such as the curing of plastics and the manufacture of pulp-insulated cables [17]. In many chemical
engineering processes, chemical reactions take place between a foreign mass and the working fluid which
moves due to the stretch of a surface.
The order of the chemical reactions depends on several factors. One of the simplest chemical reactions
is the first-order reaction in which the rate of the reaction is directly proportional to the species concentration.
Chamkha [18] studied the analytical solutions for heat and mass transfer by the laminar flow of a Newtonian,
viscous, electrically conducting and heat generating/absorbing fluid on a continuously moving vertical
permeable surface in the presence of a magnetic field and the first-order chemical reaction. Muthucumaraswamy
and Ganesan [19] studied the numerical solution for the transient natural convection flow of an incompressible
viscous fluid past an impulsively started semi-infinite isothermal vertical plate with the mass diffusion, taking
into account a homogeneous chemical reaction of the first order.
The coupled nonlinear ordinary differential equations governing the flow are solved using regular
perturbation method which is the oldest method used by many researchers. In this paper a new method known as
Differential Transform method is applied to find the analytical solution. The main advantage of DTM is that it
can be applied directly to nonlinear differential equations without requiring linearization, discritization, or
perturbation. This method is well addressed in [20-24]. Recently Umavathi et al. [25] solved the coupled
nonlinear equations governing the flow for magnetoconvection in a vertical channel for open and short circuits
usng Differential Transform method. The aim of this paper is to investigate effect of first order chemical
reaction of viscous fluid in a vertical channel in the presence of a thin conducting baffle. After inserting the
baffle, the fluid in stream-1 is concentrated. Analytical solutions are found using PM and using DTM.
II.
MATHEMATICAL FORMULATION
Consider a steady, two-dimensional laminar fully developed free convection flow in an open ended
vertical channel filled with purely viscous fluid. The X-axis is taken vertically upward, and parallel to the
direction of buoyancy, and the Y-axis is normal to it. Walls are maintained at a constant temperature and the
fluid properties are assumed to be constant. The channel is divided into two passages by means of thin, perfectly
conducting plane baffle and each stream will have its own pressure gradient and hence the velocity will be
individual in each stream.
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3. Perturbation Technique And Differential…
X
Stream - I Stream - II
Y
Y h
Y h
Y h
*
Figure 1. Physical configuration.
The governing equations for velocity, temperature and concentrations are
Stream-I
g T T1 T W
dY
2
X
2
d U1
dY
2
0
(1)
2
2
d T1
2
P
g C (C 1 C 0 )
dU 1
0
dY
CP
(2)
2
d C
D
dY
kC 0
2
(3)
Stream-II
g T T 2 TW
2
dY
P
X
2
d U
dY
2
2
0
(4)
2
2
d T2
2
dU 2
0
dY
CP
(5)
subject to the boundary and interface conditions on velocity, temperature and concentration as
U 1 0 , T1 TW , C C 1 , at Y h
1
U
2
0 , T 2 T W , at Y h
2
U 1 0 ,U
2
0 , T1 T 2 ,
d T1
d T2
dY
dY
, C C 2 ,at Y h
*
(6)
Introducing the following non-dimensional variables,
ui
p
U
i
U1
h
2
, i
T i TW
TW TW
1
p
U 1 X
,Gr
2
g T Th
2
2
3
,GC
g CCh
2
, T TW TW , C C 1 C 0 , Y *
2
3
1
y*
h
,
C1 C 0
C1 C 0
,Y
y
,Re
U 1h
U1
2
, Br
kT
,
(7)
h
where i 1, 2 .
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4. Perturbation Technique And Differential…
The momentum, energy and concentration equations corresponding to stream-I and stream-II become
Stream-I
2
d u1
G RT 1 G R c p 0
2
dy
d 1
d u1
Br
dy
2
2
dy
d
(8)
2
0
(9)
2
dy
0
2
2
(10)
Stream-II
2
d u2
dy
2
G RT 2 p 0
d 2
2
dy
2
du2
Br
dy
(11)
2
0
(12)
subject to the boundary conditions,
u 1 0 , 1 1 , 1 , at y 1
u 2 0 , 2 0 , at y 1
u1 0 , u 2 0 , 1 2 ,
where
kh
2
, n
D
d1
d 2
dy
C2 C0
C1 C 0
, n ,at y y *
(13)
dy
.
III.
SOLUTIONS
The exact solution for concentration distribution is found using Eq. (10) and is given by
B 1 C o s h y B 2 S in h y
(14)
3.1 Perturbation Method
Equations (8), (9), (11) and (12) are coupled non-linear ordinary differential equations. Approximate
solutions can be found by using the regular perturbation method and Differential Transform method. The
perturbation parameter is considered as Brinkman number B r . Adopting this method, solutions for velocity and
temperature are assumed in the form
ui
y
u i0
y
B r u i1
y
B r ui2
2
y
...
(15)
i
y
i0
y
B r i1
y
B r i2
y
...
(16)
2
where the subscript i 1 and 2 represents stream-I and stream-II respectively.
Substituting Eqs. (15) and (16) into Eqs. (8), (9), (11) and (12) and equating the coefficients of like
power of B r to zero and one, we obtain the zeroth and first order equations as
Stream-I
Zeroth order equations
d 10
2
dy
2
0
(17)
G RT 10 G R c p 0
(18)
2
d u10
dy
2
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5. Perturbation Technique And Differential…
First order equations
d 11
2
dy
2
2
d u10
0
dy
(19)
G RT 11 0
(20)
2
d u11
dy
2
Stream-II
Zeroth order equations
d 20
2
dy
2
0
(21)
G RT 20 p 0
(22)
2
d u 20
dy
2
First order equations
d 21
2
dy
2
2
d u 20
0
dy
(23)
G RT 21 0
(24)
2
d u 21
dy
2
The corresponding zeroth order boundary conditions reduces to
u 1 0 0 , 1 0 1 , at y 1
u 2 0 0 , 2 0 0 , at y 1
u10 0 , u 20 0 , 10 20 ,
d 10
dy
d 20
, at y y *
(25)
dy
The corresponding first order boundary conditions reduces to
u 1 1 0 , 1 1 0 at y 1
u 2 1 0 , 2 1 0 at y 1
u11 0 , u 21 0 , 11 21 ,
d 11
dy
d 21
at y y *
(26)
dy
The solutions of zeroth and first order equations (17) to (24) using the boundary conditions as given in
Eqs. (25) and (26) are
Zeroth-order solutions
Stream-I
10 C 1 y C 2
u 1 0 A 2 A1 y r1 y
r2 y r4 C o s h y r5 S in h y
2
3
Stream-II
20 C 3 y C 4
(29)
u 2 0 A 4 A 3 y r5 y r6 y
2
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(28)
3
(30)
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6. Perturbation Technique And Differential…
First order solutions
Stream-I
11 G 2 G 1 y p1 y
2
p2 y
3
p3 y
4
p4 y
5
p5 y
6
p 6 C o s h 2 y
p 7 S in h 2 y p 8 C o s h y p 9 S in h y p 1 0 y C o s h y
(31)
p 1 1 y S in h y p 1 2 y C o s h y p 1 3 y S in h y
2
u 11 G 6 G 5 y R1 y
R2 y
2
2
3
R3 y
4
R4 y
5
R5 y
6
R6 y
7
R7 y
8
R 8 C o s h 2 y R 9 S in h 2 y R 1 0 C o s h y R 1 1 S in h y
(32)
R 1 2 y C o s h y R 1 3 y S in h y R 1 4 y C o s h y R 1 5 y S in h y
2
2
Stream-II
21 G 4 G 3 y q1 y q 2 y q 3 y q 4 y q 5 y
2
u 21 G 8 G 7 y S1 y
3
S2 y
2
4
3
S3 y
4
5
S4 y
5
6
S5 y
(33)
6
S6 y
7
S7 y
8
(34)
3.2 Basic concepts of the differential transform method
The analytical solutions obtained in Section 3.1 are valid only for small values of Brinkman number
B r . In many practical problems mentioned earlier, the values of B r are usually large. In that case analytical
solutions are difficult, and hence we resort to semi-numerical-analytical method known as Differential
Transform method (DTM). The general concept of DTM is explained here: The kth differential transformation of
an analytical function F k is defined as (Zhou [20])
1 d f
k
k ! d
k
F
k
,
0
(35)
and the inverse differential transformation is given by
f
F
k
0
,
k
(36)
k 0
Combining Eqs. (35) and (36), we obtain
f
0
k!
k 0
k
d
k
f
d
,
k
(37)
0
From Eqs. (35)–(37), it can be seen that the differential transformation method is derived from Taylor’s
series expansion. In real applications the sum
F
k
0
is very small and can be neglected when
k
n
kn
is sufficiently large. So f can be expressed by a finite series, and Eqn. (36) may be written as
n
f
F
k
0
,
k
(38)
k 0
where the value of n depends on the convergence requirement in real applications and F k is the
differential transform of f . Table 1 lists the basic mathematics operations frequently used in the following
analysis.
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7. Perturbation Technique And Differential…
Table 1 The operations for the one-dimensional differential transform method.
Original function
Transformed function
y ( ) g ( ) h ( )
Y (k ) G (k ) H (k )
y ( ) g ( )
Y (k ) G (k )
y ( )
d g ( )
Y ( k ) ( k 1) G ( k 1)
d
d g ( )
2
y ( )
d
Y ( k ) ( k 1 )( k 2 ) G ( k 2 )
2
k
y ( ) g ( ) h ( )
Y (k )
G (l ) H ( k l )
l0
y ( )
1, if k m
Y (k ) (k m )
0 , if k m
m
Taking differential transform of Eqs. (8), (9), (11) and (12), one can obtain the transformed equations as
Stream-I
U1 k 2
1 k 2
1
k
1 k 2
1 k 2
k
2
k 2
k
T
1 k
G Rc k
p
k
(39)
k
Br
k
G R
k
r 1 r 1U 1 k r 1U 1 r 1
(40)
r0
(41)
1 k 2
Stream-II
U
2
k
2
2 k 2
1
k
1 k 2
T
2 k
p
k
(42)
k
Br
k
G R
1 k 2
k
r 1 r 1U
2
k
r 1U
2
r
1
(43)
r0
where, U 1 k , U 2 k , 1 k , 2 k and k are the transformed notations of u 1 y , u 2 y ,
1
y , 2 y and 1 y respectively. ( k )
1, if k 0
.
0 , if k 0
The following are the transformed initial conditions
U 1 0 c1 , U 1 1 c 2 , U
2
0
c3 , U
2
1 0 d 1 , 1 1 d 2 , 2 0 d 3 ,
1
2
c4 ,
1
d4,
0 e1 , 1 e 2
(44)
Using the boundary condition (13), we can evaluate c1 , c 2 , c 3 , c 4 , d 1 , d 2 , d 3 , d 4 , e1 and e 2 .
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8. Perturbation Technique And Differential…
IV.
RESULTS AND DISCUSSIONS
The objective of the present study is to understand the characteristics of mixed convection of a viscous
fluid in a vertical double passage channel in the presence of chemical reaction. The solutions are found using
perturbation method and Differential Transformation method. The physical parameters such thermal Grashoff
number G R T , mass Grashoff number G R C , Brinkman number B r (or perturbation parameter) and chemical
reaction parameter , are fixed as 5, 5, 0.1, and 0.5 respectively, for all the graphs except the varying one. The
effect of these parameters on velocity, temperature and concentration are shown in Figs. 2 – 10.
The effect of
thermal Grashoff number G R T (ratio of Grashoff number to Reynolds number) on the velocity and temperature
is shown in Figs. 2a,b,c and Figs. 3a,b,c at all three different baffle positions (i.e. y * 0 .8 , 0.0 and 0.8). As
the thermal Grashoff number increases, the velocity and temperature increases at all the baffle position whereas
the maximum velocity field is observed in the wider stream. It is also observed form Figs. 3a,b,c that the
temperature distribution is more effective near the left wall when compared to right wall. Further it is wellknown that since Grashoff number is the ratio of buoyancy force to viscous force, increase in Grashoff number
is to increase the buoyancy force and hence increases the concentration also.
Therefore as the thermal
Grashoff number increases velocity and temperature increases at all baffle position in both the streams. The
effect of mass Gerashof number G R C (ratio of modified Grashoff number to Reynolds number) is shown in
Figs. 4a,b,c for velocity field and in Figs. 5a,b,c for the temperature field. Here also the effect of G R C is to
increase the velocity and temperature field in both the streams. It is seen from Figs. 4a and 5a ( y * 0 .8 ) that
the effect of G R C on the velocity and temperature fields is not effective whereas when the baffle position is at
y * 0 .0 and 0.8 the flow field is enhanced as G R C increases. The similar result is also observed by
Fasogbon [26] for irregular channel.
The effect of Brinkman number B r on the velocity and temperature fields are shown in Figs. 6a,b,c
and Figs. 7a,b,c respectively. As the Brinkman number increases, both the velocity and temperature increases in
both the streams at all baffle positions. One can see from temperature equation that increase in Brinkman
number increases the viscous dissipation and hence the temperature increases, which intern influences the
velocity and temperature. The effect of first order chemical reaction parameter , on the velocity,
temperature and concentration fields is shown in Figs. 8a,b,c, Figs. 9a,b,c and Figs. 10a,b,c respectively. As
increases the velocity and temperature decreases in stream-I, and remains invariant in stream-II when the baffle
position y * 0 .8 . But when the baffle position is at y * 0 & 0 .8 the effect of is more effective in
stream –I and less effective in stream –II. This is because the fluid is concentrated in stream-I only. The effect of
chemical reaction parameter is to decrease the concentration distribution as seen in Figs. 10a,b,c, which is
the similar result obtained by Srinivas and Muturajan [27] for mixed convective flow in a vertical channel. It is
observed from Tables 2a, 3a and 4a that results of DTM and PM agree well in the absence of Brinkman number
at all the baffle positions. For large values of Brinkman number B r 0 , DTM and PM solutions show
difference as seen in Tables 2(b,c) to 4(b,c). It is also observed from these tables that the error of DTM and PM
is very less in smaller stream when compared to bigger stream at all baffle position for B r 0 .
V.
CONCLUSION
The effect of first order chemical reaction in a vertical double passage channel filled with purely
viscous fluid was investigated. The solutions of the governing equations and the associated boundary conditions
have been obtained by using regular perturbation method and differential transform method. Main findings are
summarized as follows:
[1] Increasing thermal Grashoff number, mass Grashoff number and Brinkman number increases the velocity
and temperature in both the streams at all different baffle position.
[2] Increase in the chemical reaction parameter suppresses the velocity and temperate in stream-I and remains
invariant in stream-II.
[3] The use of baffle in the flow channel resulted in the heat transfer enhancement as high as compared to the
heat transfer in a channel without baffle.
[4] Chemical reaction parameter was to decrease the flow field.
[5] An excellent agreement was observed with the results of DTM and PM for small values of Brinkman
number.
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9. Perturbation Technique And Differential…
REFERENCES
L.N. Tao, “On Combined Free and Forced Convection in Channel”, ASME Journal of Heat Transfer 82 (1960) 233–238.
W. Aung, and G. Worku, “Theory of Fully Developed Combined Convection Including Flow Reversal”, ASME Journal of Heat
Transfer 108 (1986) 485–488.
W. Aung, and G. Worku, “Developing Flow and Flow Reversal in a Vertical Channel with Symmetric wall Temperatures”, ASME
Journal of Heat Transfer 108 (1986) 299–304.
W. Aung and G. Worku, “Mixed Convection in Ducts with Asymmetric wall Heat Fluxes”, ASME Journal of Heat Transfer 109
(1987) 947-951.
J. Prathap Kumar, J.C. Umavathi and Basavaraj M Biradar, “Mixed Convective Flow of Immiscible Viscous Fluids in a Vertical
Channel”, Heat transfer Asian Research 40 (2011) 1-25.
J.C. Umavathi and M. Shekar, “Mixed Convective Flow of two Immiscible Viscous Fluids in a Vertical Wavy Channel with
Traveling Thermal Waves”, Heat Transfer Asian Research 40 (2011)721-743.
J.C. Umavathi, I.C. Liu, and M. Shekar, “Unsteady Mixed Convective Heat Transfer of two Immiscible Fluids Confined between a
long Vertical Wavy wall and a Parallel Flat wall”, Appl. Math. Mech.-Engl. Ed. 33 (2012) 931–950.
P. Dutta, and S. Dutta, “Effect of Baffle size Perforation and Orientation on Internal heat Transfer Enhancement”, International
Journal of Heat Mass Transfer 41(1998) 3005-3013.
P. Dutta, and A. Hossain, “Internal Cooling Augmentation in Rectangular Channel using two Inclined Baffles”, International
Journal of Heat Fluid Flow 26 (2005) 223-232.
M.M. Salah El-Din, “Developing Laminar Convection in a Vertical Double-Passage channel”, Heat Mass Transfer 41 (1998) 35013513.
M.M. Salah El-Din, “Effect of Viscous Dissipation on Fully Developed on Laminar Mixed Convection in a Vertical DoublePassage channel”, International Journal of Thermal Science 41 (2002) 253-259.
S.S. Mousavi, and K. Hooman, “Heat and Fluid flow in Entrance Region of a channel with Staggered Baffles”, Energy
Conservation and Management 47 (2006) 2011-2019.
M.H. Nasiruddin, and K. Siddiqui, “Heat Transfer Augmentation in a Heat Exchanger Tube using a Baffle”, International Journal of
Heat and Fluid Flow 28 (2007) 318-328.
J. Prathap Kumar, J.C. Umavathi, Ali J. Chamkha and H. Prema, “Free Convection in a Vertical Double Passage Wavy channel
Filled with a Walters Fluid (model B’)”, International Journal of Energy and Technology 3 (2011) 1–13.
J. Prathap Kumar, J.C. Umavathi, and H. Prema, “Free Convection of Walter’s Fluid Flow in a Vertical Double-Passage Wavy
channel with Heat Source”, International Journal of Engineering Science and Technology 3 (2011) 136-165.
J. C. Umavathi, “Mixed Convection of Micropolar Fluid in a Vertical Double-Passage channel”, International Journal of
Engineering Science and Technology 3 (2011) 197-209.
R. Kandasamy, and S.P. Anjalidevi, “Effects of Chemical Reaction, Heat and Mass Transfer on Nonlinear Laminar Boundary-Layer
Flow over a Wedge with Suction or Injection”, Computer Application in Mechanics 5 (2004) 21–31.
A.J. Chamkha, “MHD Flow of a Uniformly Stretched Vertical Permeable Surface in the Presence of Heat Generation/Absorption
and Chemical Reaction”, International Communications in Heat Mass Transfer 30 (2003) 413–422
R. Muthucumaraswamy, and P. Ganesan, “Natural Convection on a Moving Isothermal Vertical Plate with Chemical Reaction”,
Engineering Physics and Thermophysics 75 (2002) 113–119
J.K. Zhou, “Differential Transformation and its Applications for Electrical Circuits”, Huarjung University Press; 1986. (in Chinese)
A.S.V. Ravi Kanth, and K. Aruna, “Solution of Singular Two-Point Boundary Value Problems using Differential Transformation
Method”, Physics Letter A 372 (2008) 4671–4673.
M.M. Rashidi, “The Modified Differential Transform Method for Solving MHD Boundary-Layer Equations”, Computer Physics in
Communication 180 (2009) 2210–2217.
Ming-Jyi Jang, Yen-Liang Yeh, Chieh-Li Chen, Wei-Chih Yeh. “Differential Transformation Approach to Thermal Conductive
Problems with Discontinuous Boundary Condition”, Applied Mathematics and Computers 216 (2010) 2339–2350.
D.D. Ganji, M. Rahimi, M. Rahgoshay, M. Jafari, “Analytical and Numerical Investigation of Fin Efficiency and Temperature
Distribution of Conductive, Convective, and Radiative Straight Fins”, Heat Trans Asian Research 40(3) (2011) 233–245.
J.C. Umavathi, A.S.V. Ravi Kanth and M. Shekar, “Comparison study of Differential Transform Method with Finite Difference
Method for Magnetoconvection in a Vertical channel”, Heat Transfer Asian Research 42(3) (2013) 243–258.
P.F Fasogbon, “Analytical Study of Heat and Mass Transfer by Free Convection in a Two-Dimensional Irregular channel”,
International Journal of Applied Mathematics and Mechanics 6(4) (2010) 17-37.
S. Srinivas, and R. Muthuraj, “Effect of Chemical Reaction and Space Porosity on MHD Mixed Convective flow in a Vertical
Asymmetric channel with Peristalsis”, Mathematical and computer Modeling 1213-1227 (2011).
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
NOMENCLATURE
u 2
1
1
K T
Br
Brinkman number
C1
the Concentration in Stream-I
C0
reference concentration
C
specific heat at constant pressure
cp
D
g
p
dimensionless specific heat at constant pressure
diffusion coefficients
acceleration due to gravity
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10. Perturbation Technique And Differential…
h g T
2
3
Gr
Grashoff number
Gc
modified Grashoff Number
G RT
thermal Grashoff number Gr / Re
G RC
mass Grashof number Gc / Re
h
channel width
g c C h
2
3
*
k
width of passage
thermal conductivity of fluid
p
non-dimensional Pressure Gradient (
Re
Reynolds number (
T1 , T 2
dimensional temperature distributions
h
U 1h
h
p
2
U 1 X
)
)
T w , T w temperatures of the boundaries
1
U1
2
reference velocity
U 1 , U 2 dimensional velocity distributions
u1 , u 2
y
*
non dimensional Velocities in Stream-I, Stream-II
baffle position
GREEK SYMBOLS
chemical reaction parameters
T
coefficients of thermal expansion
C
coefficients of concentration expansion
T , C difference in Temperatures & Concentration
perturbation Parameter
i
non-dimensional temperature
T i TW
2
T T
W2
W1
kinematics viscosity
non-dimensional concentrations
density
viscosity
SUBSCRIPTS
i refer quantities for the fluids in stream-I and stream-II, respectively.
Acknowledgment
The authors thank UGC-New Delhi for the financial support under UGC-Major Research Project.
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11. Perturbation Technique And Differential…
5
Br=0.1
GRc=5
12
15
p=-5
n=1
15
4
20
15
10
16
8
3
10
10
10
u
u
u
12
6
2
5
8
4
5
5
GRT=1
GRT=1
1
GRT=1
4
2
0
-1.0-0.5 0.0 0.5 1.0
y
(a)
0
-1.0 -0.5 0.0 0.5 1.0
y
0
-1.0
-0.50.0 0.5 1.0
y
(c)
(b)
Fig.2: velocity distribution for different values of thermal Grashof number GRT
at (a) y*=-0.8 (b)y*=0.0 (c) y*=0.8
1.8
2.1
Br=0.1
p=-5
n=1
GRc=5
15
1.6
15
1.8
1.4
2.5
10
10
1.5
1.2
5
3.0
5
5
GRT=1
10
2.0
GRT=1
1.0
1.2
15
0.8
1.5
GRT=1
0.9
0.6
1.0
0.6
0.4
0.5
0.3
0.0
-1.0
(a)
0.2
-0.5
0.0
y
0.5
1.0
0.0
-1.0
(b)
-0.5
0.0
y
0.5
1.0
0.0
-1.0
(c)
-0.5
0.0
y
0.5
1.0
Figu.3: Temperature profile for different values of ratio of Grashof number to
Reynolds number GRT at (a)y*=0.8 (b)y*=0 (c)y*=0.8
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12. Perturbation Technique And Differential…
4.0
18
4.0
GRc=1, 5, 10, 15
3.5
16
15
3.0
2.5
15
3.5
3.0
Br=0.1
p=-5
GRT=5
n=1
2.5
14
10
12
10
10
u
2.0
u 2.0
u
5
8
1.5
5
1.5
GRc=1
1.0
6
GRc=1
1.0
4
0.5
0.5
0.0
-1.0 -0.5
(a)
0.0
y
0.5
1.0
2
0.0
-1.0 -0.5
(b)
0.0
y
0.5
1.0
0
-1.0 -0.5
(c)
0.0
y
0.5
1.0
Fig.4: Velocity profile for different values ofratio of modified Grashoff number to
Reynolds number GRC at (a)y*=-0.8 (b)y*=0 (c)y*=0.8
1.4
2.0
GRc=1, 5, 10, 15
4.5
Br=0.1
GRT=5
p=-5
n=1
1.8
1.2
15
1.6
1.0
1.4
3.0
10
5
1.0
2.5
GRc=1
2.0
0.6
15
3.5
10
1.2
0.8
4.0
0.8
5
1.5
0.6
0.4
GRc=1
1.0
0.4
0.2
0.5
0.2
0.0
-1.0
-0.5
(a)
0.0
y
0.5
1.0
0.0
-1.0
(b)
-0.5
0.0
y
0.5
1.0
0.0
-1.0
(c)
-0.5
0.0
0.5
1.0
y
Fig.5: Temperature profile for different values of ratio of modified Grashof number to
Reynolds number GRC at (a)y*=-0.8 (b)y*=0 (c)y*=0.8
||Issn 2250-3005 ||
||October||2013||
Page 37
13. Perturbation Technique And Differential…
4.0
12
p=-5
GRc=5
GRT=5
n=1
1
1
3.5
10
1
28
24
3.0
0.5
8
20
2.5
0.5
0.5
u
u 2.0
6
u
12
1.5
0.1
4
16
0.1
Br=0
Br=0
0.1
8
1.0
Br=0
2
4
0.5
0
-1.0 -0.5 0.0 0.5 1.0
y
(a)
0.0
-1.0 -0.5 0.0
y
(b)
0.5
0
-1.0 -0.5
(c)
1.0
0.0
y
0.5
1.0
Fig.6: Velocity for different values of Brinkman number Br
(a)y*=-0.8 (b)y*=0 (c)y*=0.8
6
6
14
GRc=5
GRT=5
p=-5
n=1
1
5
5
1
1
12
10
4
4
8
0.5
3
3
0.5
0.5
6
2
2
4
0.1
0.1
1
1
Br=0
0.1
2
Br=0
Br=0
0
-1.0
(a)
-0.5
0.0
y
0.5
1.0
0
-1.0
(b)
-0.5
0.0
y
0.5
1.0
0
-1.0
(c)
-0.5
0.0
y
0.5
1.0
Fig.7: Temperature profile for different values of Brinkman number Br
at (a) y*=-0.8 (b)y*=0 (c) y*=0.8
||Issn 2250-3005 ||
||October||2013||
Page 38
14. Perturbation Technique And Differential…
4.0
8
3.0
GRc=5
GRT=5
p=-5
n=1
3.5
2.5
7
3.0
6
2.0
5
2.5
=0.1, 0.5, 1, 1.5
u 1.5
2.0
u
u
4
3
1.5
1.0
2
1.0
0.5
1
0.5
0.0
-1.0 -0.5 0.0 0.5
y
(a)
1.0
0.0
-1.0 -0.5 0.0
(b)
y
0.5
0
-1.0 -0.5 0.0
(c)
y
1.0
0.5
1.0
Fig.8: Velocity profile for different values of chemical reaction parameter
at (a) y*=-0.8 (b)y*=0 (c) y*=0.8
1.4
1.2
1.2
1.0 =0.1, 0.5, 1, 1.5
GRc=5
GRT=5
p=-5
n=1
1.0
2.0
1.8
1.6
1.4
0.8
1.2
0.8
=0.1, 0.5, 1, 1.5
0.6
1.0
0.6
0.8
0.4
0.6
0.4
0.4
0.2
0.2
0.2
0.0
-1.0
(a)
-0.5
0.0
y
0.5
1.0
0.0
-1.0
(b)
-0.5
0.0
y
0.5
1.0
0.0
-1.0
(c)
-0.5
0.0
y
0.5
1.0
Fig.9: Temperature profile for different values of chemical reaction parameter
at (a) y*=-0.8 (b)y*=0 (c) y*=0.8
||Issn 2250-3005 ||
||October||2013||
Page 39
15. Perturbation Technique And Differential…
0.1
1.000
0.1
1.00
0.1
1.0
0.5
0.5
0.5
0.998
0.96
0.996
0.9
0.92
1
0.8
1
0.994
0.992
1
0.88
0.84
0.7
0.6
1.5
1.5
GRc=5
GRT=5
p= -5
n1=1
n2=1
0.990
0.988
0.80
0.5
1.5
0.76
0.4
-1.00-0.95
-0.90-0.85
-0.80
y
(a)
-0.8
(b)
-0.4
-0.8 -0.4 0.0 0.4 0.8
y
(c)
0.0
y
Figure10.Concentration profile for different values of chemical reaction parameter
at (a) y*=-0.8 (b)y*=0 (c) y*=0.8
Table 2a Comparison of velocity and temperature with
Velocity
DTM
y
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
PM
0
1.266461
1.659656
1.227398
0
0.605469
0.781250
0.566406
0
0
1.266461
1.659656
1.227398
0
0.605469
0.781250
0.566406
0
Error
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Table 2b Comparison of velocity and temperature with
y
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
||Issn 2250-3005 ||
Velocity
DTM
PM
0
1.339968
1.771965
1.321337
0
0.682521
0.870491
0.622964
0
0
1.329565
1.755951
1.307845
0
0.670711
0.856765
0.614236
0
Error
0.0000
0.0104
0.0160
0.0135
0.0000
0.0118
0.0137
0.0087
0.0000
Br 0
,
G RT 5
,
G RC 5
Temperature
DTM
p 5
PM
1.000000
0.875000
0.750000
0.625000
0.500000
0.375000
0.250000
0.125000
0
,
1.000000
0.875000
0.750000
0.625000
0.500000
0.375000
0.250000
0.125000
0
Br 0 . 05
,
G RT 5
,
G RC 5
Temperature
DTM
PM
1.000000
0.989529
0.933166
0.870308
0.761836
0.583393
0.393647
0.202259
0
1.000000
0.973754
0.907116
0.834778
0.722594
0.551573
0.371510
0.190149
0
||October||2013||
and
y * 0 .0
.
Error
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
,
p 5
and
y * 0 .0
.
Error
0.0000
0.0158
0.0261
0.0355
0.0392
0.0318
0.0221
0.0121
0.0000
Page 40
17. Perturbation Technique And Differential…
Table 4a Comparison of velocity and temperature with Br 0 , G R T 5 , G R C 5 , p 5 and y * 0 .8 .
Velocity
Temperature
y
DTM
PM
DTM
PM
Error
Error
-1
0
0
0.0000
1.000000
0.850000
0.0000
-0.7
2.720194
2.720194
0.0000
0.700000
0.550000
0.0000
-0.4
4.232842
4.232842
0.0000
0.400000
0.250000
0.0000
-0.1
4.649777
4.649777
0.0000
0.100000
0.100000
0.0000
0.2
4.052842
4.052842
0.0000
0.075000
0.050000
0.0000
0.5
2.495194
2.495194
0.0000
0.025000
0
0.0000
0.8
0
0
0.0000
1.000000
0.850000
0.0000
0.85
0.019844
0.019844
0.0000
0.400000
0.250000
0.0000
0.9
0.026250
0.026250
0.0000
0.100000
0.100000
0.0000
0.95
0.019531
0.019531
0.0000
0.075000
0.050000
0.0000
1
0
0
0.0000
0.025000
0
0.0000
Table 4b Comparison of velocity and temperature with Br 0 . 01 , G R T 5 , G R C 5 , p 5 and y * 0 .8 .
y
-1
-0.7
-0.4
-0.1
0.2
0.5
0.8
0.8
0.85
0.9
0.95
1
Velocity
DTM
0
2.825637
4.412157
4.859428
4.243934
2.615207
0
0
0.020506
0.027007
0.020005
0
PM
0
2.816608
4.396750
4.841378
4.227453
2.604832
0
0
0.020448
0.026941
0.019963
0
Error
0.0000
0.0090
0.0154
0.0181
0.0165
0.0104
0.0000
0.0000
0.0001
0.0001
0.0000
0.0000
Table 4c Comparison of velocity and temperature with
y
-1
-0.7
-0.4
-0.1
0.2
0.5
0.8
0.8
0.85
0.9
0.95
1
||Issn 2250-3005 ||
Velocity
DTM
0
3.707230
5.914596
6.618204
5.848729
3.624481
0
0
0.026132
0.033437
0.024023
0
PM
0
3.202265
5.052384
5.607785
4.925897
3.043386
0
0
0.022866
0.029704
0.021690
0
Temperature
DTM
1.000000
0.924245
0.797661
0.658724
0.517827
0.362559
0.160579
0.160579
0.120437
0.080292
0.040147
0
Br 0 . 05
Error
0.0000
0.5050
0.8622
1.0104
0.9228
0.5811
0.0000
0.0000
0.0033
0.0037
0.0023
0.0000
,
G RT 5
Temperature
DTM
1.000000
1.536854
1.616169
1.571084
1.507863
1.314728
0.674938
0.674938
0.506220
0.337487
0.168751
0
||October||2013||
PM
1.000000
0.918020
0.789269
0.649363
0.507653
0.352732
0.155260
0.155260
0.116447
0.077632
0.038817
0
,
G RC 5
PM
1.000000
1.190102
1.146343
1.046814
0.938267
0.763661
0.376298
0.376298
0.282234
0.188161
0.094085
0
Error
0.0000
0.0062
0.0084
0.0094
0.0102
0.0098
0.0053
0.0053
0.0040
0.0027
0.0013
0.0000
,
p 5
and
y * 0 .8
.
Error
0.0000
0.3468
0.4698
0.5243
0.5696
0.5511
0.2986
0.2986
0.2240
0.1493
0.0747
0.0000
Page 42
18. Perturbation Technique And Differential…
Appendix
1
C1
1
C2
,
2
2
n C o sh
B2
S in h y * C o s h
G R c B1
r3
A1
2
r y *
1 r2
2
1
y*
3
p6
p9
S in h y * n S in h
B1
,
2
S in h y * C o s h
C o sh y *
S in h C o s h y *
r4
,
1
C4
,
2
G Rc B2
r1
,
r5
,
2
p
p G RT
S in h C o s h y *
C2
r6
,
2
G RT C 1
r2
,
2
G RT C 4
1 r3 C o s h y * C o s h
2 A
2
1
r4
2
r3
2
2
2
4
r
r4
2
2
3
, p7
8
2 A
r3
1
2
p1 2
r3 r4
4
3
2 A
1
, p8
3
, p1 3
2 A1 r1
, p2
8 r1 r4 3 6 r2 r3
6 r2 r4
r 4 S in h ,
A3
2
r5 1 y *
G RT C 3
, q1
A3
4 r
2
1
r 1
y*
6
6 A1 r 2
3
, A 4 A 3 r5 r6
, p4
12
2 4 r2 r3
2 A 3 r5
3
2 r
2
5
, q3
3 r2
, p5
5
4 r1 r3
, p1 1
2
2
3 r1 r2
,
10
,
3
4 r1 r4
, q2
2
2
8 r1 r3 3 6 r2 r4
2
6 r2 r3
,
6
r4 S in h y * S in h
y * 1
, p1 0
,
,
, p3
r4
,
6
1 y*
A 2 A1 r1 r2 r3 C o s h
p1
1
,C3
2 4 r2 r4
3 A 3 r6
6
,
2
, q4
3 r5 r6
,
5
2
q5
3 r6
o
p 1 p 2 p 3 p p 5 p C 6 s h
4
T1
p S in h p C o s h p
9
1 0
,
10
T 2 q1 q 2 q 3 q 4 q 5
T3 q1 y *
2
q2 y *
p6 C osh
q3 y *
3
2
4
2
S in h
7
2
p
C o sh
1 2
p
S in h
1 1
q4 y *
5
q5 y *
2
p1 y *
6
2
p2 y *
3
p3 y *
4
p4 y *
2
2 p 6 S in h
p10
p12
G1
G3
R8
4 q3 y * 5q4 y *
3
2 y * C o sh
y * T4
6 q 5 y * 2 p1 y * 3 p 2 y *
5
2
y * y *
T1 T 2 T 3 T 4
2
T 2 T3 T 4 1 y *
,
12
G RT p 6
4
2
,
||Issn 2250-3005 ||
R4
,G 2
T
,
6
,
1
2 y * S in h
T 2 T3 T 4 1 y *
R5
,
20
G RT p 7
2
S in h y *
2
4 p3 y * 5 p4 y *
2
3
4
6 p5 y *
5
,
C o s h y * S in h y *
G 4 T2 G 3 ,
4
p5 y *
5
y * y * C o sh y *
2
,
2
,
G RT p 2
R9
y *
S in h y * p13
2
G R T p1
S in h
1 3
y * p 8 S i n h y * p 9 C o s h y *
y * C o s h y * p11
2
1
4
y * 2 p 7 C o s h
y * S in h
T
R3
2
p
y * p 8 C o s h y * p 9 S in h y *
p10 y * C o s h y * p11 y * S in h y * p12 y * C o s h y * p 13 y *
T 4 2 q1 y * 3 q 2 y *
,
y * p 7 S in h
2
p C o8 h
s
G RT p3
30
,
||October||2013||
R1
,
G RT G 2
R6
R1 0
R2
,
2
G RT p 4
42
p
8
2
,
R7
G RT G1
G RT p5
56
2 p 1 1 6 p 1 2 G R T
4
,
6
,
,
Page 43
19. Perturbation Technique And Differential…
R1 1
R1 4
S4
p
2
9
2 p 1 0 6 p 1 3 G R T
G R T p12
2
G RT q 2
4
G R T p13
R1 5
,
20
p1 0
S1
,
2
S5
,
R1 2
,
G RT q 3
4 p1 3 G R T
3
G RT G 4
S6
30
R1 3
G RT G 3
S2
,
2
,
,
G RT q 4
42
6
,
p 1 1
4 p1 2 G R T
,
S3
S7
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R8 C o s h 2 R9 S in h 2 R1 0 C o s h
T5
R S in h R C o s h R S in h R C o s h R S in h
11
12
13
14
15
T7
G R T q1
G5
T7 T5
1 y*
3
, G7
||Issn 2250-3005 ||
4
T 6 T8
1 y*
5
6
7
8
,
12
G RT q 5
,
56
R1 y * R 2 y * R 3 y * R 4 y * R 5 y * R 6 y * R 7 y * R 8 C o s h 2 y * R 9 S i n h
R10 C o s h y * R11 S in h y * R12 y * C o s h y * R13 y * S in h y *
R y * 2 C o s h y * R y * 2 S in h y *
14
15
2
,
3
2
y *
, G 6 T5 G 5 , G 8 T6 G 7 .
||October||2013||
Page 44