IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Mhd effects on non newtonian micro polar fluid with uniform suctioneSAT Journals
Abstract In this present work, the problem of heat and mass transfer on the flow of Non-newtonian micropolar fluid with uniform suction/blowing with heat generation, radiation, thermophoresis and chemical reaction effects are studied. The non-linear partial differential equation arising from the flow modeling were transformed into coupled non linear ordinary differential equations and subsequently solved by using the implicit finite difference method. The effect of relevant thermo physical parameters such as material parameter K, suction parameter A, heat generation/absorption parameter B, Prandtl number Pr, radiation parameter R, thermophoretic parameter τ, chemical reaction parameter δ , magnetic parameter Ha and Schmidt number Sc are also being numerically investigated and analyzed. Keywords: MHD, micro polar fluid, heat generation, thermophoresis, radiation and finite difference scheme.
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.
MIXED CONVECTIVE HEAT AND MASS TRANSFER MHD FLOW PAST AN UNSTEADY STRETCHING ...IAEME Publication
Mixed two dimensional convection heat and mass transfer flow with suction, viscous
dissipation, heat source/sink effect, mass diffusion including Soret and Dufour effects due to an
unsteady porous stretching sheet is studied in the present analysis. The flow is subjected to
magnetic field normal to the vertical plate in a saturated porous medium. The governing non-linear
partial differential equations have been reduced to ordinary differential equations using suitable
similarity transformation variables. The resultant equations which are coupled and highly nonlinear
are solved by standard Runge-Kutta fourth order numerical technique via shooting method.
The momentum, temperature and concentration field distributions are analyzed and discussed
numerically and presented pictorially through graphs. Numerical values for skin friction
coefficient, local Nusselt number and Sherwood number at the plate in the presence of magnetic
field and porous medium are derived and discussed for various values of physical parameters and
are presented in table. Finally the present results are compared with previously published results
and found to be well in agreement.
MHD Chemically Reacting and Radiating Nanofluid Flow over a Vertical Cone Emb...IJLT EMAS
In this study, we examine the combined effects of
thermal radiation, chemical reaction on MHD hydromagnetic
boundary layer flow over a vertical cone filled with nanofluid
saturated porous medium under variable properties. The
governing flow, heat and mass transfer equations are
transformed into ordinary differential equations using similarity
variables and are solved numerically by a Galerkin Finite
element method. Numerical results are obtained for
dimensionless velocity, temperature, nanoparticle volume
fraction, as well as the skin friction, local Nusselt and Sherwood
number for the different values of the pertinent parameters
entered into the problem. The effects of various controlling
parameters on these quantities are investigated. Pertinent
results are presented graphically and discussed quantitatively.
The present results are compared with existing results and found
to be good agreement. It is found that the temperature of the
fluid remarkably enhances with the rising values of Brownian
motion parameter (Nb).
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.
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 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 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 effects on non newtonian micro polar fluid with uniform suctioneSAT Journals
Abstract In this present work, the problem of heat and mass transfer on the flow of Non-newtonian micropolar fluid with uniform suction/blowing with heat generation, radiation, thermophoresis and chemical reaction effects are studied. The non-linear partial differential equation arising from the flow modeling were transformed into coupled non linear ordinary differential equations and subsequently solved by using the implicit finite difference method. The effect of relevant thermo physical parameters such as material parameter K, suction parameter A, heat generation/absorption parameter B, Prandtl number Pr, radiation parameter R, thermophoretic parameter τ, chemical reaction parameter δ , magnetic parameter Ha and Schmidt number Sc are also being numerically investigated and analyzed. Keywords: MHD, micro polar fluid, heat generation, thermophoresis, radiation and finite difference scheme.
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.
MIXED CONVECTIVE HEAT AND MASS TRANSFER MHD FLOW PAST AN UNSTEADY STRETCHING ...IAEME Publication
Mixed two dimensional convection heat and mass transfer flow with suction, viscous
dissipation, heat source/sink effect, mass diffusion including Soret and Dufour effects due to an
unsteady porous stretching sheet is studied in the present analysis. The flow is subjected to
magnetic field normal to the vertical plate in a saturated porous medium. The governing non-linear
partial differential equations have been reduced to ordinary differential equations using suitable
similarity transformation variables. The resultant equations which are coupled and highly nonlinear
are solved by standard Runge-Kutta fourth order numerical technique via shooting method.
The momentum, temperature and concentration field distributions are analyzed and discussed
numerically and presented pictorially through graphs. Numerical values for skin friction
coefficient, local Nusselt number and Sherwood number at the plate in the presence of magnetic
field and porous medium are derived and discussed for various values of physical parameters and
are presented in table. Finally the present results are compared with previously published results
and found to be well in agreement.
MHD Chemically Reacting and Radiating Nanofluid Flow over a Vertical Cone Emb...IJLT EMAS
In this study, we examine the combined effects of
thermal radiation, chemical reaction on MHD hydromagnetic
boundary layer flow over a vertical cone filled with nanofluid
saturated porous medium under variable properties. The
governing flow, heat and mass transfer equations are
transformed into ordinary differential equations using similarity
variables and are solved numerically by a Galerkin Finite
element method. Numerical results are obtained for
dimensionless velocity, temperature, nanoparticle volume
fraction, as well as the skin friction, local Nusselt and Sherwood
number for the different values of the pertinent parameters
entered into the problem. The effects of various controlling
parameters on these quantities are investigated. Pertinent
results are presented graphically and discussed quantitatively.
The present results are compared with existing results and found
to be good agreement. It is found that the temperature of the
fluid remarkably enhances with the rising values of Brownian
motion parameter (Nb).
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.
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 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 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 Higher Order Chemical Reactions and Slip Boundary Conditions on Na...IJERA Editor
We investigate the effects of higher order chemical reactions on coupled heat and mass transfer from a permeable stretching sheet in nanofluid flow with slip boundary conditions. Nanofluids play an important role in emerging nanotechnology and biotechnology industries where they find application in heat transfer processes. We study the effects of the order of the chemical reaction and chemical reaction parameter. The highly nonlinear and coupled governing differential equations were solved numerically. The solution depends on Brownian motion number, thermophoresis number, thermal convective parameter, mass convective parameter and double diffusive buoyancy parameter. The effects of various engineering parameters are investigated.
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.
Chemical Reaction Effects on Free Convective Flow of a Polar Fluid from a Ver...IOSR Journals
This article deals with a study of two dimensional free convective flow of a polar fluid through a porous medium due to combined effects of thermal and mass diffusion in presence of a chemical reaction of first order. The objective of the present investigation is to analyze the free convective flow in the presence of prescribed wall heat flux and mass flux condition. The governing partial differential equations are non-dimensionalized and transformed into a system of non-similar equations. The resulting coupled nonlinear partial differential equations are solved numerically under appropriate transformed boundary conditions using an implicit finite difference scheme in combination with quasilinearisation technique. Computations are performed for a wide range of values of the various governing flow parameters of the velocity, angular velocity, temperature and species concentration profiles and results are presented graphically. The numerical results for local skin friction coefficient, couple stress coefficient, local Nusselt number and local Sherwood number are also presented. The obtained results are compared with previously published work and were to be in excellent agreement. The study reveals that the flow characteristics are profoundly influenced by the polar effects
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.
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
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.
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.
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.
International journal of engineering and mathematical modelling vol2 no3_2015_1IJEMM
A weak nonlinear stability analysis has been performed for an oscillatory mode of convection, heat and mass transports in terms of
Nusselt, Sherwood numbers are derived and evaluated by a non$-$autonomous complex Ginzburg-Landau equation. The porous layer boundaries are heated sinusoidally with time. The basic state temperature is defined in terms of study and oscillatory parts, where study part show nonlinear throughflow effect on the system and time dependant part show modulation effect. The generalized Darcy model is employed for the momentum equation. The disturbances of the flow are expanded in power series of amplitude of modulation, which is assumed to be small and employed using normal mode technics. The effect of vertical throughflow is found to stabilize or destabilize the system depending on its direction. The time relaxation parameter $\lambda_1$ has destabilizing effect, while time retardation parameter $\lambda_2$ has stabilizing effect on the system. Three types of modulations have been analyzed, and found that, OPM, LBMO cases are effective on heat and mass transfer than IPM case. The effects of amplitude and frequency of modulation on heat and mass transports have been analyzed and depicted graphically. The study establishes that the heat and mass transports can be controlled effectively by a mechanism that is external to the system.
Enhancement of power quality by unified power quality conditioner with fuzzy ...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Hydrostatic transmission as an alternative to conventional gearboxeSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Effects Of Higher Order Chemical Reactions and Slip Boundary Conditions on Na...IJERA Editor
We investigate the effects of higher order chemical reactions on coupled heat and mass transfer from a permeable stretching sheet in nanofluid flow with slip boundary conditions. Nanofluids play an important role in emerging nanotechnology and biotechnology industries where they find application in heat transfer processes. We study the effects of the order of the chemical reaction and chemical reaction parameter. The highly nonlinear and coupled governing differential equations were solved numerically. The solution depends on Brownian motion number, thermophoresis number, thermal convective parameter, mass convective parameter and double diffusive buoyancy parameter. The effects of various engineering parameters are investigated.
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.
Chemical Reaction Effects on Free Convective Flow of a Polar Fluid from a Ver...IOSR Journals
This article deals with a study of two dimensional free convective flow of a polar fluid through a porous medium due to combined effects of thermal and mass diffusion in presence of a chemical reaction of first order. The objective of the present investigation is to analyze the free convective flow in the presence of prescribed wall heat flux and mass flux condition. The governing partial differential equations are non-dimensionalized and transformed into a system of non-similar equations. The resulting coupled nonlinear partial differential equations are solved numerically under appropriate transformed boundary conditions using an implicit finite difference scheme in combination with quasilinearisation technique. Computations are performed for a wide range of values of the various governing flow parameters of the velocity, angular velocity, temperature and species concentration profiles and results are presented graphically. The numerical results for local skin friction coefficient, couple stress coefficient, local Nusselt number and local Sherwood number are also presented. The obtained results are compared with previously published work and were to be in excellent agreement. The study reveals that the flow characteristics are profoundly influenced by the polar effects
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.
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
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.
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.
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.
International journal of engineering and mathematical modelling vol2 no3_2015_1IJEMM
A weak nonlinear stability analysis has been performed for an oscillatory mode of convection, heat and mass transports in terms of
Nusselt, Sherwood numbers are derived and evaluated by a non$-$autonomous complex Ginzburg-Landau equation. The porous layer boundaries are heated sinusoidally with time. The basic state temperature is defined in terms of study and oscillatory parts, where study part show nonlinear throughflow effect on the system and time dependant part show modulation effect. The generalized Darcy model is employed for the momentum equation. The disturbances of the flow are expanded in power series of amplitude of modulation, which is assumed to be small and employed using normal mode technics. The effect of vertical throughflow is found to stabilize or destabilize the system depending on its direction. The time relaxation parameter $\lambda_1$ has destabilizing effect, while time retardation parameter $\lambda_2$ has stabilizing effect on the system. Three types of modulations have been analyzed, and found that, OPM, LBMO cases are effective on heat and mass transfer than IPM case. The effects of amplitude and frequency of modulation on heat and mass transports have been analyzed and depicted graphically. The study establishes that the heat and mass transports can be controlled effectively by a mechanism that is external to the system.
Enhancement of power quality by unified power quality conditioner with fuzzy ...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Hydrostatic transmission as an alternative to conventional gearboxeSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Localization based range map stitching in wireless sensor network under non l...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Mining elevated service itemsets on transactional recordsets using slicingeSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Study of shape of intermediate sill on the design of stilling basin modeleSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
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.
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 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.
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.
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.
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.
Thermal Energy on Water and Oil placed Squeezed Carreau Nanofluids FlowMOHAMMED FAYYADH
this research work is focused on the numerical study regarding Carreau nanofluids’ squeezed flow via a permeable sensor surface. The nanofluids’ thermal conductivity is considered to be dependent on temperature. A convenient transformation is employed to reorganize governing equations into ordinary differential equations. The Runge–Kutta method and shooting technique are employed to accurately solve the boundary layer momentum as well as heat equations. Graphical and tabular aids are used to evaluate the solutions of applicable parameter with regards to temperature as well as the rate of heat transfer. In this work, a comparison is done from three nanofluids, i.e. copper, oxide aluminum and SWCNTs (nanoparticles) based fluids (water, crude oil and ethylene glycol) to improve heat transfer. It is found that the temperature dimensionless was dropped and dominated with the squeezed flow parameter and nanoparticle volume fraction parameter. That is for all nanomaterials. When compared with water and ethylene glycol, crude oil is cooler and a thinner thermal boundary layer is presented. For the rate of heat transfer (Nusselt number) was higher in: Ethylene glycol- SWCNT with high permeable velocity parameter 0.2, Ethylene glycol- SWCNT with low squeeze flow parameter 0.1 and Ethylene glycol- oxide aluminum with low nanoparticle volume fraction 0.05
Thermal Radiation and Viscous Dissipation Effects on an Oscillatory Heat and ...rahulmonikasharma
An anticipated outcome that is intended chapter is to investigate effects of magnetic field on an oscillatory flow of a viscoelastic fluid with thermal radiation, viscous dissipation with Ohmic heating which bounded by a vertical plane surface, have been studied. Analytical solutions for the quasi – linear hyperbolic partial differential equations are obtained by perturbation technique. Solutions for velocity and temperature distributions are discussed for various values of physical parameters involving in the problem. The effects of cooling and heating of a viscoelastic fluid compared to the Newtonian fluid have been discussed.
Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow...iosrjce
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
equations and an efficient numerical technique is implemented to solve the reduced system. The effects of the
parameters such as Magnetic parameter, Prandtl number and Eckert number are discussed graphically on
velocity and temperature distributions
Soret Effect And Effect Of Radiation On Transient Mhd Free Convective Flow Ov...inventionjournals
The present paper is concerned to analyze the radiation, Magneto hydrodynamic and soret effects on unsteady flow heat and mass transfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinite vertical porous plate through porous media the porous plate is subjected to a transverse variable suction velocity. The transient, non linear and coupled dimensionless governing equations for this investigation are solved analytically using perturbation technique about a small parameter . the effects of governing parameters on the flow variables are discussed graphically.
Soret Effect And Effect Of Radiation On Transient Mhd Free Convective Flow Ov...inventionjournals
The present paper is concerned to analyze the radiation, Magneto hydrodynamic and soret effects on unsteady flow heat and mass transfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinite vertical porous plate through porous media the porous plate is subjected to a transverse variable suction velocity. The transient, non linear and coupled dimensionless governing equations for this investigation are solved analytically using perturbation technique about a small parameter . the effects of governing parameters on the flow variables are discussed graphically.
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.
The Effect of Radiation on the Convective Heat and Mass Transfer Flow of a Vi...inventionjournals
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Minimal M-gs Open and Maximal M-gs Closed Sets In Interior Minimal Spaceinventionjournals
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Mhd effects on non newtonian micro polar fluid with
1. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 350
MHD EFFECTS ON NON-NEWTONIAN MICRO POLAR FLUID WITH
UNIFORM SUCTION/BLOWING AND HEAT GENERATION IN THE
PRESENCE OF CHEMICAL REACTION AND THERMOPHORESIS
N. Kishan1
and S. Jagadha2
1
Department of Mathematics, University College of Science, Osmania University, Hyderabad, A.P., India
2
Department of Mathematics, Hyderabad Institute of technology and Management, Medchal, Hyderabad, A.P, .India,
kishan_n@rediffmail.com, jagadhasaravanan@gmail.com
Abstract
In this present work, the problem of heat and mass transfer on the flow of Non-newtonian micropolar fluid with uniform
suction/blowing with heat generation, radiation, thermophoresis and chemical reaction effects are studied. The non-linear partial
differential equation arising from the flow modeling were transformed into coupled non linear ordinary differential equations and
subsequently solved by using the implicit finite difference method. The effect of relevant thermo physical parameters such as material
parameter K, suction parameter A, heat generation/absorption parameter B, Prandtl number Pr, radiation parameter R,
thermophoretic parameter τ, chemical reaction parameter δ , magnetic parameter Ha and Schmidt number Sc are also being
numerically investigated and analyzed.
Keywords: MHD, micro polar fluid, heat generation, thermophoresis, radiation and finite difference scheme.
----------------------------------------------------------------------***-----------------------------------------------------------------------
1. INTRODUCTION
The dynamics of micropolar fluids has attracted considerable
attention during the last few decades because traditional
Newtonian fluids cannot precisely describe the characteristics
of fluid flow with suspended particles. Eringen [1] developed
the theory that the local effects arising from the microstructure
and the intrinsic motion of the fluid elements should be taken
into account. The theory is expected to provide a mathematical
model for the Non-Newtonian fluid behavior observed in
certain man-made liquids such as polymers, lubricants, fluids
with additives, paints, animal blood and colloidal and
suspension solutions, etc. The presence of dust or smoke,
particularly in a gas, may also be modeled using micropolar
fluid dynamics. Later, Eringen [2] extended the theory of
thermo-micropolar fluids and derived the constitutive laws for
fluids with microstructures. An excellent review of micropolar
fluids and their applications was given by Ariman et al. [3],
Gorla [4] discussed the steady state heat transfer in a
micropolar fluid flow over a semi-infinite plate and the
analysis is based on similarity variables. Rees and Pop[5]
studied the free convection boundary layer flow of micropolar
fluid from a vertical flat plate. Singh [6] has studied the free
convection flow of a micropolar fluid past an infinite vertical
plate using the finite difference method
Chemical reactions 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 if the rate of reaction is directly proportional to the
concentration itself. In many chemical engineering processes,
a chemical reaction between a foreign mass and the fluid does
occur. These processes take place in numerous industrial
applications, such as the polymer production, the
manufacturing of ceramics or glassware, the food processing
[7] and so on. Das et al.[8] considered the effects of a first
order chemical reaction on the flow past an impulsively started
infinite vertical plate with constant heat flux and mass
transfer. Muthucumarswamy and Ganesan [9] and
Muthucumarswamy [10] studied the first order homogeneous
chemical reaction on the flow past an infinite vertical plate.
Thermophoresis is a mechanism of migration of small
particles in direction of decreasing thermal gradient by
Hinds.W.C [11]. It is an effective method for particle
collection by Sai et al.[12]. The velocity acquired by the
particle is called thermophoretic velocity and the force
experienced by the suspended particle is called thermophoretic
force by Bakier et al. [13]. Thermophoresis causes small
particles to deposit on the cold surfaces. It has many
applications in aerosol technology, deposition of silicon thin
films, and radioactive particle deposition in nuclear reactor
safety simulations.
2. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 351
Also, the study of micropolar fluids is very significant due to
their potential application in many industrial processes; for
example, in continuous casting glass-fiber production, paper
production, metal extrusion, hot rolling, wire drawing,
drawing of plastic films, metal and polymer extrusion and
metal spinning. Balaram and Sastry [14] solved the problem of
a fully developed free convection flow in a micropolar flow.
Agarwal and Dhanapal [15] obtained a numerical solution to
study the fully developed free convection micropolar fluid
flow between two parallel with constant suction (or injection).
Srinivasacharya et al. [16] studied the effects of microrotation
and frequency parameters on an unsteady flow of micropolar
fluid between two parallel porous plates with a periodic
suction.
The interaction of magnetic field and microrotation plays a
vital role in several engineering applications such as in MHD
electrical power generation, designing cooling system for
nuclear reactors, etc., where microrotation provides an
important parameter for deciding the rate of heat flow. Gorla
et al. [17] developed a numerical scheme to solve the steady
free convection from a vertical isothermal plate in a strong
cross magnetic field immersed in a micropolar fluid.
El-Hakiem et al. [18] analyzed the effect of viscous and Joule
heating on the flow of an electrically conducting and
micropolar fluid past a plate whose temperature varies linearly
with the distance from the leading edge in the presence of a
uniform transverse magnetic field. Helmy et al. [19] studied
the unsteady flow MHD of a conducting micropolar fluid,
through a porous medium, over an infinite plate that is set in
motion in its own plane by an impulse. Bhargava et al. [20]
obtained a numerical solution of a free convection MHD
micropolar fluid flow between two parallel porous vertical
plates by means of the quasi-linearization method.
The study of heat generation/absorption effects in moving
fluids is important in view of several physical problems, such
as fluids undergoing exothermic or endothermic chemical
reaction. Due to the fast growth of electronic technology,
effective cooling of electronic equipment has become
warranted. The cooling of electronic equipment ranges from
individual transistors to main frame computers, from energy
suppliers to telephone switch boards and thermal diffusion
effects has been utilized for isotopes separation in the mixture
between gases with very light molecule weight (hydrogen and
helium) and medium molecular weight. Recently R. A.
Mohamed1 et al. and Ziabakhsh et al. studied [21.22] Heat and
mass transfer analysis on the flow of non-Newtonian
micropolar fluid with uniform suction/blowing, heat
generation, chemical reaction and Thermophoresis .
The purpose of study is to investigate the effects of MHD on
non-Newtonian micro polar fluid flow with uniform
suction/blowing heat generation, radiation, thermophoresis
and chemical reaction. We solved the non-linear boundary
value problem arising from the non dimensionalization and
local non similarity method using an implicit finite difference
scheme along with Gauss-Seidal method. The C-
programming code is used to solve the system equations.
2. MATHEMATICAL FORMULATION
Consider the two-dimensional stagnation point flow of an
incompressible non-Newtonian micropolar fluid implinging
perpendicular on a permeable wall and flowing away along the
x-axis. A uniform magnetic field βo is applied normal to the
walls. And using the boundary layer approximation and
neglecting the dissipation, the equation of energy for
temperature T with heat generation or absorption and thermal
radiation, the equation of mass for concentration C with
thermophoresis and chemical reaction. The simplified two-
dimensional equations governing the flow in the boundary
layer of a steady, laminar and incompressible micropolar fluid
are governed by:
+ (1)
ρ u + v = μ + k + k − u (2)
ρ u + v =
²
²
− 2N + (3)
ρc u
!
+ v
!
= k
²!
²
+ Q T − T$ −
%&
(4)
u
'
+ v
'
= D
²'
²
− v)c − R∗
c (5)
Where N is the microrotation or angular velocity whose
direction of rotation is in the x-y-plane, µ is the viscosity of
the fluid, � is the density, cp is the specific heat capacity at
constant pressure of the fluid, kf is the thermal conductivity
of the fluid, Q is the heat generation/absorption coefficient and
j,g and k are the microinertia per unit mass, spin gradient
viscosity and vortex viscosity respectively which are assumed
to be constant.
The appropriate physical boundary conditions of Eqs (1)-(5)
are
u(x,0)=0; v(x,0) = - vo , N(X, 0) = -n
,-
,.
y ∞: u(x,y) U(x) = ax , v(x,y) 0, N(x,y) 0 (6)
y = 0; T = Tw , c = cw
y ∞: T T∞ , c c∞
Where n is a constant and 0≤ n≤ 1. The case n = ½ indicates
the vanishing of the antisymmetric part of the stress tensor and
3. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 352
denotes weak concentration of microelements, which will be
considered here. Using the transform we have
1 = 2
3
4
5 , u = a x f’(η), v = -√89 f(η)
N = a x 2
3
4
: 1 , : 1 = −
;
<
="
1 , ? =
@A@∞
@BA@∞
C =
DA DB
DBAD∞
(7)
After using the transformation (7), for micropolar fluid, there
are two equations in which one is for angular velocity or
microrotation and physically it is important in micropolar
fluid. In this study, we have two equations f(η) and :(η)
which :(η) equals to
A;
<
f”(η) [22]
So Eqns.(2) and (3) reduce to the single equation as Eq. (8a)
1 +
E
<
= 1 + = 1 =" 1 − =′ 1 ² −F8<
=′
+ 1 = 0
(8a)
1 +
H
I
J ?" 1 + KL= 1 ?′ 1 + KLM? 1 = 0 (8b)
C"(η) + Sc ( f - τ θ') C' - Sc τ C θ" - δ Sc C = 0 (8c)
Subject to the boundary conditions
f (0) = A, f'(0) =0, f'(∞) = 1
θ(0) = 1, θ(∞) = 0
(0) = 1, C (∞) = 0 (8d)
Where K =
E
N
> 0 is the material parameter, A =
4P
√3Q
is the
suction parameter and primes denote differentiation with
respect to η. Pr =
NDR
E
is the Prandtl number and B =
S
TUDR
is
the heat Generation/absorption parameter, Sc =
N
UV
is the
Schmidt number, W =
.X∗
Y
is the chemical reaction parameter,
Z =
AE[
@
]^ − ]∞ is the thermophoretic parameter , R=
H_∗@∞
`
a∗a
, is the radiation parameter, F8 =
_bP
UT
is the magnetic
parameter. For micropolar boundary layer flow, the wall skin
friction ]^
]^ = h i + j
,-
,k
+ jlm y=0 (9)
Using U(x) = ax as a characteristic velocity, the skin friction
Coefficient, Cf can be defined as
no =
@B
UY
(10)
By using this definition we have
Jp.
;/<
= 1 +
E
<
) f "(0) (11)
Where Jp.
;/<
=
.Y
Q
is the local Reynolds Number
The heat transfer from the surface to the fluid is computed by
application of Fourier’s law
q = h
;r_ @∞
`
IE∗
,@
,k
m y=0 (12)
Introducing the transformed variables, the expression for q
becomes and the heat transfer coefficient, in terms of the
Nusselt number Nu can be expressed as
ls =
t
E @BA@∞ 2
u
v
(13)
Then we have Nu = −?′
0
The definition of the local mass flux and the local Sherwood
number are respectively given by
w^ = −x
,D
,k y=0 (14)
yℎ.=
{B
V DBAD∞ 2
u
v
(15)
3. NUMERICAL SOLUTIONS
Applying the Quasi-linearization technique [ 23] to the non-
linear equation (8a) we obtain as
(1+K/2) f“’ +(F f"
+ (-2 F’-Ha2
)f‘+F”f = FF” – F’F’ -1
(16)
Where assumed F is the value of f at nth
iteration and f is at
(n+1)th
iteration. The convergence criterion is fixed as ⃒F- f⃒
< 10-5
.
Using an implicit finite difference scheme for the equation
(16),(8b) and (8c), we obtain
a[i] f[i+2] + b[i] f[i+1] + c[i] f[i] + d[i] f[i-1] =e[I ]
(17)
a1[i]θ[i+1] + b1[i]θ[i] +c1[i]θ[i-1] = 0 (18)
a2[i]C[i+1] + b2 C[i] + c2 C[i-1] = 0 (19)
4. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 353
Where
a[i] = A[i], b[i] = -3A[i] + h B[i] + 0.5*h2
C[i]
c[i] = 3A[i] -2*hB[i] + h3
D[i],
d[i] = -A[i] + h B[i] – 0.5*h2
C[i]
e[i] = h3
E[i]
a1 = A1[i] + 0.5*hB1[i], b1 = -2A1[i] + h2
C1[i]
c1 = A1[i] –0.5*h B1[i], a2 = A2[i]+0.5*hB2[i]
b2 = -2A2[i] + h*hC2[i], c2 = A2[i] – 0.5 *hB2[i]
A[i] = 1+k/2, B[i] = F , C[i] = -2F’ – Ha2
,
D[i] = F” E[i] = FF” – F’F’ -1, A1[i] = 1+4/3R,
B1[i] = Prf[i] C1[i] = PrB
4. RESULTS AND DISCUSSION
Numerical calculations are carried out for different values of
dimensionless parameters such as material parameter K,
suction parameter A, heat generation/absorption parameter B,
Prandtl number Pr, radiation parameter R, thermophoretic
parameter τ, chemical reaction parameter δ, Schmidt number
Sc and magnetic number Ha and a representative set of results
is reported in graphs from figures 1 to 18. These results are
obtained to show that the flow field is influenced appreciably
by A, K, B, Pr, Sc, R,τ,δ, Ha.
Figures (1)– (3) represent the dimensionless velocity,
temperature and concentration profiles for the values of A
respectively. From these figures it is clear that f‘(η) increase
with the increase of suction parameter A, whereas the
temperature and concentration profiles decrease with the
increase of suction parameter A. The effect of material
parameter K on velocity, temperature and concentration
profiles is displayed in figs. (4) - (6). It is seen that the
velocity profile, temperature and concentration profiles
decrease with the increase of material parameter K.
The effect of magnetic parameter Ha is shown in figs. (7) - (9)
for velocity, temperature and concentration profiles
respectively. The velocity profile f‘ decrease with the increase
of magnetic parameter Ha, whereas the temperature and
concentration profiles increase with the increase of magnetic
parameter Ha.
Figures (10 )and( 11) show the effect of radiation parameter R
on temperature and concentration profiles respectively. It is
noticed that the temperature distribution increase with the
increase value of radiation parameter while the concentration
profile decrease with the increase of radiation parameter R.
The effect of dimensionless Prandtl number Pr on temperature
distribution θ is placed in fig.(12) It can be seen that the
temperature profile decrease with the increase of Pr value.
The effect of Prandtl number Pr is to decrease the
concentration profile is observed from fig.(13).
Fig.(1) Effects of Suction parameter A on velocity profile
B=0.1,K=1.0,Pr=0.72,Sc=0.66,R=2.0,δ=0.4,τ=1.0and Ha=0
Fig.(2) Effects of Suction parameter A on temperature profile
B=1,K=1.0,Pr=0.72,Sc=0.66,R=2.0,δ=0.4,τ=1.0 and Ha=0.5
Fig.(3) Effects of Suction parameter A on concentration
profile Sc = 0.66,K=1.0,Pr=0.72,δ=0.4,B = 0.1,τ=1.0,R = 2
and Ha=0.5
0
0.2
0.4
0.6
0.8
1
0 2 4 6
f'
η
A = -2
A = -1
A = 0
A = 1
A = 2
0
0.5
1
1.5
2
0 2 4 6
θ
η
A = -2
A = -1
A = 0
A = 1
0
0.2
0.4
0.6
0.8
1
0 2 4 6
CCCC
η
A = -2
A = -1
A = 0
A = 1
A = 2
5. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 354
Fig.(4) Effects of material parameter K on velocity profile
B=0.1,A = 0 ,Pr=0.72,Sc=0.66,R=2.0,δ=0.4,τ=1.0 and Ha = 0
Fig.(5)Effects of material parameter K on temperature profile
B=0.1,A = 0 ,Pr=0.72,Sc=0.66,R=2.0,δ=0.4,τ=1.0 and Ha = 0
Fig.(6)Effects of material parameter K on concentration
profile B=0.1,A = 0 ,Pr=0.72,Sc=0.66,R=2.0,δ=0.4,τ=1.0 and
Ha = 0
Fig.(7)Effects of magnetic parameter Ha on velocity profile
B=0.1,K=1.0,Pr=0.72,Sc=0.66,R=2.0,δ=0.4,τ=1.0 and A =0
Fig.(8) Effects of Magnetic parameter Ha on temperature
profile A=0,K=1.0,Pr=0.72,Sc=0.66,B = 0.1,δ=0.4,τ=1.0 and
R = 2
Fig.(9) Effects of Magnetic parameter Ha on concentration
profile Sc = 0.66,K=1.0,Pr=0.72,δ=0.4,B = 0.1,τ=1.0,R = 2
and A = 0
0
0.2
0.4
0.6
0.8
1
0 2 4 6
f'
η
K = 0
K = 0.5
K = 1.0
K = 1.5
K = 2.0
0
0.5
1
1.5
2
0 2 4 6
θ
η
K = 0
K = 1
K = 2
K = 3
0
0.2
0.4
0.6
0.8
1
0 2 4 6
CCCC
η
K = 0
K = 1
K = 2
0
0.2
0.4
0.6
0.8
1
0 2 4 6
f'
η
Ha = 1
Ha = 2
Ha = 5
0
0.2
0.4
0.6
0.8
1
0 2 4 6
θ
η
Ha = 0
Ha = 1
Ha = 2
Ha = 5
0
0.2
0.4
0.6
0.8
1
0 2 4 6
CCCC
η
Ha = 0
Ha = 1
Ha = 2
Ha = 5
6. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 355
Fig.(10)Effects of Radiation parameter R on temperature
profile A=0,K=1.0,Pr=0.72,Sc=0.66,B = 0.1,δ=0.4,τ=1.0 and
Ha=0.5
Fig.(11)Effects of Radiation parameter R on concentration
profile A=0,K=1.0,Pr=0.72,Sc=0.66,
B = 1.0,δ=0.4,τ=1.0 and Ha=0.5
Fig.(12) Effects of Prandtl number Pr on temperature profile
A=0,K=1.0,Ha = 0.5 ,Sc=0.66,B = 0.1,δ=0.4,τ=1.0 and R = 2
Fig.(13) Effects of Prandtl number Pr on concentration profile
A=0,K=1.0,Ha = 0.5 ,Sc=0.66,B = 0.1,δ=0.4,τ=1.0 and R = 2
Fig.(14)Effects of heat generation/absorption parameter B
temperature profile A=0, K=1.0, Pr=0.72, Sc=0.66,
R=2.0,δ=0.4, τ=1.0 and Ha=0.5
Fig.(15)Effects of heat generation/absorption parameter B on
concentration profile A=0,K=1.0,Pr=0.72,Sc=0.66,
R=2.0,δ=0.4,τ=1.0 and Ha=0.5
0
0.2
0.4
0.6
0.8
1
0 2 4 6
θ
η
R = 0
R = 0.5
R = 1.0
R = 1.5
R = 2.0
0
0.2
0.4
0.6
0.8
1
0 2 4
CCCC
η
R = 0
R = 1
R = 2
R = 5
R = 7
0
0.2
0.4
0.6
0.8
1
0 2 4 6
θ
η
Pr = 0.1
Pr = 0.5
Pr = 0.71
Pr = 1.0
Pr = 7.0
0
0.5
1
0 2 4
CCCC
η
Pr = 0.1
Pr = 1.0
Pr = 7.0
Pr = 9.0
0
0.2
0.4
0.6
0.8
1
0 2 4 6
θ
η
B = -0.5
B = -0.1
B = 0
B = 0.1
B = 0.5
0
0.2
0.4
0.6
0.8
1
0 2 4 6
CCCC
η
B = -2
B = -1
B = 0
B = 1
B = 2
7. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 356
Fig.(16) Effects of thermophoretic parameter τ on
concentration profile A=0,K=1.0,Pr=0.72,Sc=0.66,
B = 0.1,δ=0.4,R = 2 and Ha=0.5
Fig (17)Effects of chemical reaction parameter δ on
concentration profile A=0,K=1.0,Pr=0.72,Sc=0.66,B =
0.1,τ=1.0,R = 2 and Ha=0.5
Fig.(18) Effects of Schmidt number Sc on concentration
profile A=0,K=1.0,Pr=0.72,δ=0.4,B = 0.1,τ=1.0,R = 2 and
Ha=0.5
Table (1) Results of f“(0) and - θ’0) for different values of
K and A when B = 0.1 and Pr = 0.7
K A f”(0) -θ’(0)
0 1 1.419836 0.403682
0.5 1 -1.201999 0.398784
1.0 1 -1.055735 0.39472
1.5 1 -0.9498 0.391236
2.0 1 -0.86894 0.388179
1 -2 0.378311 0.030677
1 -1 0.519675 0.108716
1 0 0.737433 0.234122
1 1 1.055735 0.39472
1 2 1.494145 0.573393
Table (2) Results of - θ’0) for different values of Pr and B
when A = 0 and B = 0
Pr B -θ’(0)
0.05 0.1 0.171646
0.1 0.1 0.17674
0.5 0.1 0.219637
1.0 0.1 0.271005
1.5 0.1 0.314981
0.7 -0.1 0.303411
0.7 0 0.273324
0.7 0.1 0.240921
In fig.(14), the effect of heat source/sink parameter B is on the
temperature exhibited. From the figure it is noticed that the
dimensionless temperature θ decreases for increasing strength
of the heat sink and due to increase of heat source strength the
temperature increases. So, the thickness of the boundary layer
reduces for increase of heat sink parameter, but it increases
with heat source parameter. This result is very much
significant for the flow where heat transfer is given prime
importance. The concentration profile C decrease with the
increase of heat generation/absorption parameter B is seen
from fig. (15).
Figure(16) is displaced the effect of thermophoretic parameter
τ on concentration profile C . From the figure it is observed
that the influence of thermophoretic parameter τ is to reduce
the concentration profile.
Figure (17) illustrate the variation of concentration profile fpr
difference values of δ. The effect of increase of chemical
reaction parameter δ is to decrease concentration profile is
noticed from the figure.
Figure (18) shows the effect of Schmidt number Sc on
concentration profile. From this figure it is clear that an
increase in the Schmidt number results in decrease in the
concentration profile.
0
0.2
0.4
0.6
0.8
1
0 2 4
CCCC
η
τ = 0.0
τ = 0.5
τ = 1.5
τ = 3.0
τ = 5.0
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
CCCC
η
δ = -0.4
δ = -0.2
δ = 0
δ = 0.2
δ = 0.4
0
0.2
0.4
0.6
0.8
1
0 2 4 6
CCCC
η
Sc = 0.66
Sc = 0.78
Sc = 1
Sc = 2
8. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 357
Table (1) shown the value of skin friction coefficient f”(0) and
the rate of heat transfer coefficient –θ’
(0) for different values
of K and A for fixed values of B and Pr. It is observed from
the table the skin friction coefficient f”(0) decrease with the
increase of K whereas the value of f”(0) increase with the
increase of A. It can also be seen that the heat transfer
coefficient –θ’
(0) decrease with the increase of K value. The
heat transfer coefficient –θ’
(0) increase with the increase of A
value from -2 to 1.
Table (2) shown that heat transfer coefficient –θ’
(0) for
different values of Prandtl number Pr and heat
generation/absorption parameter B. It is observed from the
table heat transfer coefficient –θ’
(0) value increase with the
increase of Pr value whereas it decreases with the increase of
B value from -0.1 to 0.1
CONCLUSIONS
The problem of steady, laminar, two dimensional stagnation
point flow of an incompressible flow non Newtonian
micropolar fluid with uniform suction/blowing , heat
generation/absorption, radiation, thermophoresis and chemical
reaction under the influence of magnetic field. The effects of
the various dimensionless parameters are investigated.
1.The velocity profile is decreased with the increase of
material parameter K and magnetic parameter Ha suction
parameter A.
2.The temperature profile increases with the increase of
magnetic parameter Ha , heat generation/absorption B and
radiation parameter R. The temperature profile decreases with
the increase of suction parameter A, material parameter K and
Prandtl number Pr.
3.The concentration profile increase with the increase of
magnetic parameter Ha whereas the concentration profiles
decreases with increase of material parameter K, suction
parameter A, Prandtl number Pr, thermophoretic parameter τ,
chemical reaction parameter δ , heat generation/absorption
parameter B, radiation parameter R and Schmidt number Sc.
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