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.
Chemical Reaction on Heat and Mass TransferFlow through an Infinite Inclined ...iosrjce
The numerical studies are performed to examine the mass transfer flow with thermal diffusion and
diffusion thermo effect past an infinite, inclined vertical plate in a porous medium in the presence of chemical
reaction. First of all, the governing equations are transformed to a system of dimensionless coupled partial
equations. Explicit finite difference method has been used to solve these dimensionless equations for momentum,
concentration and energy equations. During the course of discussion, it is found that various parameters related
to the problem influence the calculated result. Finally, the profiles of velocity, concentration and temperature
are analyzed and illustrated with graphs.
Effects of Variable Viscosity and Thermal Conductivity on MHD free Convection...theijes
A steady two-dimensional MHD free convection and mass transfer flow past an inclined vertical surface in the presence of heat generation and a porous medium have been studied numerically when the fluid viscosity and thermal conductivity are assumed to be vary as inverse linear function of temperature. The governing partial differential equations are reduced to a system of ordinary differential equations by introducing similarity transformations. The non-linear similarity equations are solved numerically by applying the Runge-Kutta method of fourth order with shooting technique. The numerical results are presented graphically to illustrate influence of different values of the parameters on the velocity, temperature and concentration profiles. Skin friction, Nusselt number and Sherwood number are also completed and presented in tabular form.
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability
STEADY FLOW OF A VISCOUS FLUID THROUGH A SATURATED POROUS MEDIUM AT A CONSTAN...Journal For Research
In this paper the Steady flow of a viscous fluid through a porous medium over a fixed horizontal, impermeable and thermally insulated bottom. The flow through the porous medium satisfies the general momentum and energy equations are obtained when the temperature on the fixed bottom and on free surface prescibed. By using Galerkin Method, the expression for Velocity and Drag force are obtained. The Galerkin Method endowed with distinct features that account for its superiority over competing methods. The effect of different parameters on Velocity and Drag force are discussed with the help of graphs.
Convective Heat And Mass Transfer Flow Of A Micropolar Fluid In A Rectangular...IJERA Editor
In this chapter we make an investigation of the convective heat transfer through a porous medium in a Rectangular enclosure with Darcy model. The transport equations of liner momentum, angular momentum and energy are solved by employing Galerkine finite element analysis with linear triangular elements. The computation is carried out for different values of Rayleigh number – Ra micropolar parameter – R, spin gradient parameter, Eckert number Ec and heat source parameter. The rate of heat transfer and couple stress on the side wall is evaluated for different variation of the governing parameters.
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.
Chemical Reaction on Heat and Mass TransferFlow through an Infinite Inclined ...iosrjce
The numerical studies are performed to examine the mass transfer flow with thermal diffusion and
diffusion thermo effect past an infinite, inclined vertical plate in a porous medium in the presence of chemical
reaction. First of all, the governing equations are transformed to a system of dimensionless coupled partial
equations. Explicit finite difference method has been used to solve these dimensionless equations for momentum,
concentration and energy equations. During the course of discussion, it is found that various parameters related
to the problem influence the calculated result. Finally, the profiles of velocity, concentration and temperature
are analyzed and illustrated with graphs.
Effects of Variable Viscosity and Thermal Conductivity on MHD free Convection...theijes
A steady two-dimensional MHD free convection and mass transfer flow past an inclined vertical surface in the presence of heat generation and a porous medium have been studied numerically when the fluid viscosity and thermal conductivity are assumed to be vary as inverse linear function of temperature. The governing partial differential equations are reduced to a system of ordinary differential equations by introducing similarity transformations. The non-linear similarity equations are solved numerically by applying the Runge-Kutta method of fourth order with shooting technique. The numerical results are presented graphically to illustrate influence of different values of the parameters on the velocity, temperature and concentration profiles. Skin friction, Nusselt number and Sherwood number are also completed and presented in tabular form.
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability
STEADY FLOW OF A VISCOUS FLUID THROUGH A SATURATED POROUS MEDIUM AT A CONSTAN...Journal For Research
In this paper the Steady flow of a viscous fluid through a porous medium over a fixed horizontal, impermeable and thermally insulated bottom. The flow through the porous medium satisfies the general momentum and energy equations are obtained when the temperature on the fixed bottom and on free surface prescibed. By using Galerkin Method, the expression for Velocity and Drag force are obtained. The Galerkin Method endowed with distinct features that account for its superiority over competing methods. The effect of different parameters on Velocity and Drag force are discussed with the help of graphs.
Convective Heat And Mass Transfer Flow Of A Micropolar Fluid In A Rectangular...IJERA Editor
In this chapter we make an investigation of the convective heat transfer through a porous medium in a Rectangular enclosure with Darcy model. The transport equations of liner momentum, angular momentum and energy are solved by employing Galerkine finite element analysis with linear triangular elements. The computation is carried out for different values of Rayleigh number – Ra micropolar parameter – R, spin gradient parameter, Eckert number Ec and heat source parameter. The rate of heat transfer and couple stress on the side wall is evaluated for different variation of the governing parameters.
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
Unsteady Mhd free Convective flow in a Rotating System with Dufour and Soret ...IOSRJM
Numerical analysis is used to examine the unsteady MHD free convection and mass transfer fluid flow through a porous medium in a rotating system. Impulsively started plate moving its individual plane is considered. Similarity equations of the corresponding momentum, energy, and concentration equations are derived by introducing a time dependent length scale which infect plays the role of a resemblance parameter. The velocity component is taken to be inversely proportional to this parameter. The effects on the velocity, temperature, concentration, local skin-friction coefficients, Nusselt number, Prandl number, Dufour, Soret number and the Sherwood number of the various important parameters entering into the problem separately are discussed with the help of graphs.
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 Heat Source And Thermal Diffusion On An Unsteady Free Convection F...IOSR Journals
The present analysis is made to investigate the effects of heat source and thermal diffusion on an unsteady free convection flow along a porous vertical plate in a rotating system. The plate is subjected to constant heat and mass flux also. The problem is solved analytically and expressions for velocity. Energy and temperature profiles, skin friction and Nusselt number are obtained. The effects of different parameter entered in the problem are discussed on the primary and secondary velocities, temperature and concentration distributions, primary and secondary skin frictions and Nusselt number with the help of tables and graphs.
Radiation Effects on MHD Free Convective Rotating Flow with Hall EffectsIJERA Editor
In this paper, we have studied the unsteady an incompressible MHD rotating free convection flow of Viscoelastic fluid through a porous medium with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter. The expressions for the velocity, temperature, concentration have been derived analytically and also its behaviour is computationally discussed with reference to different flow parameters with the help of graphs. The skin friction, the Nusselt number and the Sherwood number are also obtained and their behaviour discussed.
Abstract— A scientific investigation is created to examine the nonlinear unfaltering blended convection limit
layer stream and warmth exchange of an incompressible digression hyperbolicnon-Newtonian liquid from a
non-isothermal wedge in the nearness of attractive field. The changed preservation conditions are understood
numerically subject to physically fitting limit conditions utilizing a second-arrange precise verifiable limited
distinction Keller Box method. The numerical code is accepted with past studies. The impact of various rising
non-dimensional parameters, to be specific Weissenberg number (We), power law record (n), blended
convection parameter, weight angle parameter (m), Prandtl number (Pr), Biot number, attractive parameter
(M)and dimensionless extraneous direction on speed and temperature development in the limit layer
administration are inspected in subtle element. Moreover, the impacts of these parameters on surface warmth
exchange rate and nearby skin erosion are additionally examined. Approval with prior Newtonian studies is
introduced and amazing relationship accomplished. It is found that speed is lessened with expanding We,
while, temperature is increased. Expanding n improves speed yet diminishes temperature, a comparable
pattern was seen. An expanding M is found to decline speed however temperature increments.
Keywords— Magnetic parameter, Mixed Convection parameter, Non-Newtonian digression hyperbolic liquid,
power law index, Weissenberg number, Weight inclination parameter.
Non-NewtonianFluid Flow and Heat Transfer over a Non- Linearly Stretching Sur...IJERA Editor
This paper investigates the MHD flow and heat transfer of an electrically conducting non-newtonian power-law
fluid over a non-linearly stretching surface along with porous plate in porous medium. The governing equations
are reduced to non-linear ordinary differential equations by means of similarity transformations. These
equations are then solved numerically with the help ofRunge – Kutta shooting method. The effect of various
flow parameters in the form of dimensionless quantities on the flow field are discussed and presented
graphically.
MHD convection flow of viscous incompressible fluid over a stretched vertical...IJERA Editor
The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs.
Melting Heat Transfer in MHD Boundary Layer Stagnation-Point Flow towards a S...iosrjce
An analysis is carried out to study the MHD steady two-dimensional stagnation-point flow and heat transfer from
a warm, laminar liquid flow to a melting stretching sheet. The governing partial differential equations are transformed into
ordinary differential equations by similarity transformation, before being solved numerically using Runge-Kutta-Fehlberg
method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are
presented for different values of the governing parameters. Effects of the Magnetic parameter, melting parameter, stretching
parameter and prandtl number on the flow and heat transfer characteristics are thoroughly examined.
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.
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.
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 Magnetohydrodynamic Fluid Flows and Their Applicationsijtsrd
In this paper a detailed study is presented on magnetohydrodynamic fluid flows with their basic governing equations. Basic concept of magnetohydrodynamic is discussed in detail. The results of various problems done by researchers are presented and addressed properly. The various applications of magnetohydrodynamic fluid flows have been presented. Dr. Vishal Saxena ""Study of Magnetohydrodynamic Fluid Flows and Their Applications"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020, URL: https://www.ijtsrd.com/papers/ijtsrd29931.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/applied-mathematics/29931/study-of-magnetohydrodynamic-fluid-flows-and-their-applications/dr-vishal-saxena
Effect of Mass Transfer and Hall Current on Unsteady MHD Flow with Thermal Di...IJERA Editor
The paper investigated the effect of mass transfer and Hall current on unsteady MHD flow with Thermal Diffusivity of a viscoelastic fluid in a porous medium. The resultant equations have been solved analytically. The velocity, temperature and concentration distributions are derived, and their profiles for various physical parameters are shown through graphs. The coefficient of Skin friction, Nusselt number and Sherwood number at the plate are derived and their numerical values for various physical parameters are presented through tables. The influence of various parameters such as the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter, Hartmann number, Hall parameter, and the frequency of oscillation on the flow field are discussed. It is seen that, the velocity increases with the increase in Gc, Gr, m and K, and it decreases with increase in Sc,M, n and Pr, temperature decreases with increase in Pr and n, Also, the concentration decreases with the increase in Sc and n.
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
Unsteady Mhd free Convective flow in a Rotating System with Dufour and Soret ...IOSRJM
Numerical analysis is used to examine the unsteady MHD free convection and mass transfer fluid flow through a porous medium in a rotating system. Impulsively started plate moving its individual plane is considered. Similarity equations of the corresponding momentum, energy, and concentration equations are derived by introducing a time dependent length scale which infect plays the role of a resemblance parameter. The velocity component is taken to be inversely proportional to this parameter. The effects on the velocity, temperature, concentration, local skin-friction coefficients, Nusselt number, Prandl number, Dufour, Soret number and the Sherwood number of the various important parameters entering into the problem separately are discussed with the help of graphs.
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 Heat Source And Thermal Diffusion On An Unsteady Free Convection F...IOSR Journals
The present analysis is made to investigate the effects of heat source and thermal diffusion on an unsteady free convection flow along a porous vertical plate in a rotating system. The plate is subjected to constant heat and mass flux also. The problem is solved analytically and expressions for velocity. Energy and temperature profiles, skin friction and Nusselt number are obtained. The effects of different parameter entered in the problem are discussed on the primary and secondary velocities, temperature and concentration distributions, primary and secondary skin frictions and Nusselt number with the help of tables and graphs.
Radiation Effects on MHD Free Convective Rotating Flow with Hall EffectsIJERA Editor
In this paper, we have studied the unsteady an incompressible MHD rotating free convection flow of Viscoelastic fluid through a porous medium with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter. The expressions for the velocity, temperature, concentration have been derived analytically and also its behaviour is computationally discussed with reference to different flow parameters with the help of graphs. The skin friction, the Nusselt number and the Sherwood number are also obtained and their behaviour discussed.
Abstract— A scientific investigation is created to examine the nonlinear unfaltering blended convection limit
layer stream and warmth exchange of an incompressible digression hyperbolicnon-Newtonian liquid from a
non-isothermal wedge in the nearness of attractive field. The changed preservation conditions are understood
numerically subject to physically fitting limit conditions utilizing a second-arrange precise verifiable limited
distinction Keller Box method. The numerical code is accepted with past studies. The impact of various rising
non-dimensional parameters, to be specific Weissenberg number (We), power law record (n), blended
convection parameter, weight angle parameter (m), Prandtl number (Pr), Biot number, attractive parameter
(M)and dimensionless extraneous direction on speed and temperature development in the limit layer
administration are inspected in subtle element. Moreover, the impacts of these parameters on surface warmth
exchange rate and nearby skin erosion are additionally examined. Approval with prior Newtonian studies is
introduced and amazing relationship accomplished. It is found that speed is lessened with expanding We,
while, temperature is increased. Expanding n improves speed yet diminishes temperature, a comparable
pattern was seen. An expanding M is found to decline speed however temperature increments.
Keywords— Magnetic parameter, Mixed Convection parameter, Non-Newtonian digression hyperbolic liquid,
power law index, Weissenberg number, Weight inclination parameter.
Non-NewtonianFluid Flow and Heat Transfer over a Non- Linearly Stretching Sur...IJERA Editor
This paper investigates the MHD flow and heat transfer of an electrically conducting non-newtonian power-law
fluid over a non-linearly stretching surface along with porous plate in porous medium. The governing equations
are reduced to non-linear ordinary differential equations by means of similarity transformations. These
equations are then solved numerically with the help ofRunge – Kutta shooting method. The effect of various
flow parameters in the form of dimensionless quantities on the flow field are discussed and presented
graphically.
MHD convection flow of viscous incompressible fluid over a stretched vertical...IJERA Editor
The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs.
Melting Heat Transfer in MHD Boundary Layer Stagnation-Point Flow towards a S...iosrjce
An analysis is carried out to study the MHD steady two-dimensional stagnation-point flow and heat transfer from
a warm, laminar liquid flow to a melting stretching sheet. The governing partial differential equations are transformed into
ordinary differential equations by similarity transformation, before being solved numerically using Runge-Kutta-Fehlberg
method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are
presented for different values of the governing parameters. Effects of the Magnetic parameter, melting parameter, stretching
parameter and prandtl number on the flow and heat transfer characteristics are thoroughly examined.
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.
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.
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 Magnetohydrodynamic Fluid Flows and Their Applicationsijtsrd
In this paper a detailed study is presented on magnetohydrodynamic fluid flows with their basic governing equations. Basic concept of magnetohydrodynamic is discussed in detail. The results of various problems done by researchers are presented and addressed properly. The various applications of magnetohydrodynamic fluid flows have been presented. Dr. Vishal Saxena ""Study of Magnetohydrodynamic Fluid Flows and Their Applications"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020, URL: https://www.ijtsrd.com/papers/ijtsrd29931.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/applied-mathematics/29931/study-of-magnetohydrodynamic-fluid-flows-and-their-applications/dr-vishal-saxena
Effect of Mass Transfer and Hall Current on Unsteady MHD Flow with Thermal Di...IJERA Editor
The paper investigated the effect of mass transfer and Hall current on unsteady MHD flow with Thermal Diffusivity of a viscoelastic fluid in a porous medium. The resultant equations have been solved analytically. The velocity, temperature and concentration distributions are derived, and their profiles for various physical parameters are shown through graphs. The coefficient of Skin friction, Nusselt number and Sherwood number at the plate are derived and their numerical values for various physical parameters are presented through tables. The influence of various parameters such as the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter, Hartmann number, Hall parameter, and the frequency of oscillation on the flow field are discussed. It is seen that, the velocity increases with the increase in Gc, Gr, m and K, and it decreases with increase in Sc,M, n and Pr, temperature decreases with increase in Pr and n, Also, the concentration decreases with the increase in Sc and n.
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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.
Thermal Effects in Stokes’ Second Problem for Unsteady Second Grade Fluid Flo...IOSR Journals
In this paper, we investigated the effects of magnetic field and thermal in Stokes’ second problem for unsteady second grade fluid flow through a porous medium. The expressions for the velocity field and the temperature field are obtained analytically. The effects of various pertinent parameters on the velocity field and temperature field are studied through graphs in detail.
Natural convection heat transfer oscillatory flow of an elastico viscous flui...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.
Heat and Mass Transfer Hydromagnetic Radiative Casson Fluid Flow over an Expo...inventionjournals
The present paper examines the hydromagnetic two-dimensional boundary layer flow of a nonNewtonian fluid accompanied by heat and mass transfer towards an exponentially stretching sheet in the presence of chemical reaction and thermal radiation. Casson model is used to characterize non-Newtonian fluid behavior. Using the similarity transformations, the governing partial differential equations are transformed into self-similar ordinary differential equations is solved by using Matlab bvp4c package. The effects of pertinent parameters are presented in both graphical and tabular form.
Heat and Mass Transfer Hydromagnetic Radiative Casson Fluid Flow over an Expo...inventionjournals
The present paper examines the hydromagnetic two-dimensional boundary layer flow of a nonNewtonian fluid accompanied by heat and mass transfer towards an exponentially stretching sheet in the presence of chemical reaction and thermal radiation. Casson model is used to characterize non-Newtonian fluid behavior. Using the similarity transformations, the governing partial differential equations are transformed into self-similar ordinary differential equations is solved by using Matlab bvp4c package. The effects of pertinent parameters are presented in both graphical and tabular form.
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Journals
Abstract This paper examines the problem of two dimensional unsteady free convection oscillatory flow of an Elastico-viscous fluid past an impulsively started infinite vertical plate with constant heat flux and heat generation. The flow is subjected to a constant suction. Using perturbation techniquethe expressions are obtained for velocity, temperature distribution and skin friction. The results obtained are discussed for various numerical values of the parameters entered into the equations governing the flow, with help of graphs.
Soret Effect And Effect Of Radiation On Transient Mhd Free Convective Flow Ov...inventionjournals
The present paper is concerned to analyze the radiation, Magneto hydrodynamic and soret effects on unsteady flow heat and mass transfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinite vertical porous plate through porous media the porous plate is subjected to a transverse variable suction velocity. The transient, non linear and coupled dimensionless governing equations for this investigation are solved analytically using perturbation technique about a small parameter . the effects of governing parameters on the flow variables are discussed graphically.
Soret Effect And Effect Of Radiation On Transient Mhd Free Convective Flow Ov...inventionjournals
The present paper is concerned to analyze the radiation, Magneto hydrodynamic and soret effects on unsteady flow heat and mass transfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinite vertical porous plate through porous media the porous plate is subjected to a transverse variable suction velocity. The transient, non linear and coupled dimensionless governing equations for this investigation are solved analytically using perturbation technique about a small parameter . the effects of governing parameters on the flow variables are discussed graphically.
The International Journal of Engineering & 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.
Non-Darcy Convective Heat and Mass Transfer Flow in a Vertical Channel with C...IJERA Editor
In this paper, We made an attempt to study thermo-diffusion and dissipation effect on non-Darcy convective
heat and Mass transfer flow of a viscous fluid through a porous medium in a vertical channel with Radiation and
heat sources. The governing equations of flow, heat and mass transfer are solved by using regular perturbation
method with δ, the porosity parameter as a perturbation parameter. The velocity, temperature, concentration,
shear stress and rate of Heat and Mass transfer are evaluated numerically for different variations of parameter.
MHD CASSON FLUID FLOW AND HEAT TRANSFER WITH PST AND PHF HEATING CONDITIONS D...IAEME Publication
In the present paper two-dimensional flow of non-Newtonian MHD flow of Casson fluid heat transfer with PST & PHF is considered. Using Nevier Stoke’s Equations of Motion the momentum and energy equations of Casson fluid are derived. These governing equations of motion and temperature are non-linear partial differential equations which are tedious to solve as they are, hence these partial differential equations are converted into Ordinary differential equations using suitable similarity transformations. These ODE’s are then solved numerically by the efficient Runge Kutta Fehlberg method. Effects of various governing parameters on the flow and heat transfer profiles are analyzed. Further the numerical Values of Wall temperature and Wall temperature Profiles are calculated and discussed in detail through graphs in both the cases of PST and PHF.
nternational Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
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Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 04, Issue 04 (April. 2014), ||V1|| PP 46-57
International organization of Scientific Research 46 |
P a g e
Unsteady MHD flow of a visco-elastic fluid along vertical porous
surface with fluctuating temperature and concentration
Arpita Mohanty, Pravat Kumar Rath, G. C. Dash
1
Department of Mathematics, Prananath Autonomous College, Khordha-752057, Odisha, India
2
Dept. of Mathematics, B.R.M. International Institute of Technology, Bhubaneswar-751010, Odisha, India
3
Department of Mathematics, S.O.A. University, Bhubaneswar-751030, Odisha, India
Abstract: - An unsteady magneto-hydrodynamics flow of an incompressible visco-elastic fluid [Walters’
fluid (Model B)] along an infinite hot vertical porous surface with fluctuating temperature and concentration
has been studied. The governing equations of motion, energy and concentration are solved by the successive
perturbation techniques. The expressions for the skin friction, Nusselt number and Sherwood number are also
derived. The variations in the fluid velocity, temperature and concentration are shown graphically whereas
numerical values of skin friction, Nusselt number and Sherwood number are presented in a tabular form for
various values of pertinent flow parameters.
Keywords: - MHD flow; porous medium; chemical reaction/heat source
I. INTRODUCTION
Flow between porous boundaries is of practical, as well as theoretical interest. The practical interest
includes problems of gaseous diffusion, transpiration cooling, lubrication of porous bearing etc. Flow through
porous medium under the influence of temperature and concentration differences find great applications in
Geothermy, Geophysics and Technology.
Several researches [1-6] have studied the two dimensional free convection MHD flow with heat and mass
transfer. Walters [7] have analyzed second order effects in elasticity plasticity and fluid dynamics. Sharma and
Pareek [8] later examined the unsteady flow and heat transfer through an elastico-viscous liquid along an infinite hot
vertical porous moving plate with variable free stream and suction. Flow and heat transfer of an electrically
conducting visco-elastic fluid between two horizontal squeezing/stretching plates have been studied by Rath et al.[9].
Alhabi et al.[10] have analyzed heat and mass transfer in MHD visco-elastic fluid flow through a
porous medium over a stretching sheet with chemical reaction. Kumar and Sivraj [11] have studied MHD mixed
convective visco-elastic fluid flow in a permeable vertical channel with Dufour effect and chemical reaction
parameter. Damseh and Sharnaik [12] have also studied visco-elastic fluid flow past an infinite vertical porous
plate in the presence of first order chemical reaction. MHD flow through a porous medium past a stretched
vertical permeable surface in the presence of heat source/sink and a chemical reaction has been studied by Dash
et al.[13]. Rath et al. [14] have studied the three dimensional free convection flow through porous medium in a
vertical channel with heat source and chemical reaction. Sharma and Sharma [15] have studied the unsteady two
dimensional flow and heat transfer through an elastico-viscous liquid along an infinite hot vertical porous
surface bounded by porous medium.
The main objective of the present study is to investigate the combined effect of magnetic field, heat
source and chemical reaction on two-dimensional visco-elastic flow through porous media in a vertical surface
with fluctuating temperature and concentration. The governing equations of flow under the Boussinesq
approximation are solved analytically using successive perturbation method.
II. FORMULATION OF THE PROBLEM
An unsteady two dimensional MHD flow past a vertical infinite surface embedded in a porous medium has
been considered with the visco-elastic fluid model (Walters’ fluid (Model B) [7]). A uniform transverse magnetic
field B0 is applied normal to the direction of the fluid flow. The x*
-axis is taken along the surface in upwards direction
i.e., opposite to the direction of gravity and y*
-axis is taken normal to the surface. The governing equations of
continuity, motion, energy and species concentration for flow of a visco-elastic fluid through porous medium bounded
by an infinite, vertical porous surface with oscillatory suction velocity are given by :
*
*
v
y
= 0 (1)
2. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
International organization of Scientific Research 47 | P a g e
* * 2 *
* *
* * * *2 *
1u u p u v
v g u
t y x y k
* 2 *3 * 3 *
*0 0
* *2 *3
K B uu u
v
t y y
(2)
* * *2
* * *
* * *2
( )
p
T T k T
v S T T
t y C y
(3)
* * 2 *
* * *
* * *2 c
C C C
v D K C C
t y y
(4)
Where u*
, v*
denote the components of velocity along x*
and y*
directions respectively, the density of
the fluid, g the acceleration due to gravity, the Kinematic viscosity,
*
k the permeability parameter
*
0K the
non-Newtonian parameter, T*
the temperature of the fluid, k the thermal conductivity of fluid, Cp the specific
heat of the fluid at constant pressure, t*
the time, C*
the concentration of fluid,
*
cK the chemical reaction
parameter, D the mass diffusion coefficient and the electrical conductivity.
The boundary conditions are given by :
t>0: y*
= 0; u*
= 0, T*
= *
,i t i t
w w w wT T T e C C C C e
y*
; u*
U*
(t*
), T*
T , C*
C (5)
where wT and Cw are the surface temperature and concentration respectively, U*
(t*
) the free stream velocity,
T the free stream temperature and C is the concentration of the fluid for away from the wall.
In view of Bousinesq approximation, the equation (2) is reduced to
2
* * * 2 *
* * *
* * * * *
u u U u
v U u
t y t k y
3 * 3 *
* * 2 * * *
0 0* *2 *3
u u
K v B U u g T T
t y y
* *
g C C (6)
where and *
are the volumetric coefficient of thermal and concentration expansion respectively and the
coefficient of dynamic viscosity.
From equation of community (1), it is clear that the suction velocity normal to the plate is either a
constant or a function of time. Here it is assumed in the form of
v*
= – v0 (1+ ei*t*
) (7)
where , *
the frequency of vibration, is a small parameter i.e., 0 < < 1 and v0 is a non-zero positive suction
velocity. Here the negative sign indicates that the suction is towards the plate.
The governing equations of the flow are non-dimentionalised using the following transformation.
* * 2 * *
0 0
2
0
4
, , , ,
4
y v t v u v
y t u
U v
** *
( )
( ) , ,
p
r
w
CT TU t
U t T P
U T T k
kp =
* 2 * 2
0 0 0
02 2 2
0
, , w
r
g T Tk v K v
K G
Uv
,
2 *
0
2 2
0 0
B
M , c
c
K
K
v v
3. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
International organization of Scientific Research 48 | P a g e
C =
** *
2 2
0 0
, , w
c
w
g C CC C S
S G
C C v Uv
(8)
In view of equations (7) and (8), equations (3) – (4) and (6) become :
ti
pp
cr
ti
2
2
e1M
k
1
uM
k
1
CGTG
t
u
4
1
y
u
e1
y
u
3
3
iwt
2
3
0
ti
y
u
e1
yt
u
4
1
kei
4
1
(9)
STP
y
T
e1P
y
T
t
T
4
P
r
ti
r2
2
r
(10)
CKS
y
C
e1S
y
C
t
C
4
S
cc
ti
c2
2
c
(11)
where Gr the Grashof number for heat transfer, Pr the Prandtl number, K0 the non-Newtonian parameter, Gc
the Grashof number for mass transfer, Sc the Schmidt number, M the magnetic parameter and kp the porosity
parameter.
The corresponding boundary conditions in non-dimensional form:
y = 0: u = 0, T = 1+eit
, C = 1 + eit
y : u U(t)=1+eit
, T 0, C 0 (12)
Solution procedure
In view of the above assumption it is justified to assume that :
u( y, t) = u0 (y) + u1 (y)eit
(13)
T (y, t) = T0 (y) + T1 (y) eit
(14)
C (y, t) = C0 (y) + C1 (y) eit
(15)
Substituting equations(13) – (15) into equations (9) – (11) and equating the coefficients of 0
and , we get
Zeroth order :
p
0000
k
1
MuuuK u0 = – Gr T0 0c
p
CG
k
1
M
(16)
0r0r0 STPTPT = 0 (17)
0 0 0 0c c cC S C K S C (18)
First order:
0
0 1 1 1 1
1
1
4 4p
k i i
K u u u M u
k
= – Gr T1 – Gc C1 – K0
4
i
k
1
Muu
p
00
(19)
0r1r1r1 TPT
4
i
SPTPT
(20)
0c1cc1c1 CSCSk
4
i
CSC
(21)
where prime denote differentiation with respect to y.
The corresponding boundary conditions (12) are reduced to :
y = 0: u0 = 0, u1 = 0,T0 =1, T1 = 1,C0 = 1, C1 = 1,
y : u0 1, u1 1, T0 0,T1 0, C0 0, C1 0 (22)
Solving equations (17), (18), (20) and (21) under the boundary conditions (22), we get
4. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
International organization of Scientific Research 49 | P a g e
T0 = 1m y
e (23)
C0 = 3m y
e (24)
2 1
1 1 1
4 4
1 m y m yi i
T m e m e
(25)
C1= 34
3 3
4 4
1 m ym yi i
m e m e
(26)
The equations (16) and (19) are still of third order where K0 0 and reduce to second order differential
equation when K0 = 0 i.e., for a Newtonian fluid case. Hence, the presence of a non-Newtonian parameter
increases the order of differential equation. While from physical consideration only two boundary conditions are
available. Since the non-Newtonian parameter (K0) is very small for incompressible fluid (Walters7
), therefore
u0 and u1 can be expanded in powers of K0 which is given by
u0 (y) = u00(y) + K0 u01(y) + O (
2
0K )
u1 (y) = u10 (y) + K0 u11 (y) + O (
2
0K ) (27)
introducing equation (27) into equations (16) and (19), we obtain the following systems of equations.
0c0r
p
00
p
0000 CGTG
k
1
Mu
k
1
Muu
(28)
4
i
k
1
MuCGTGu
4
i
k
1
Muu
p
001c1r10
p
1010
(29)
0001
p
0101 uu
k
1
Muu
(30)
1010010011
p
1111 u
4
i
uuuu
4
i
k
1
Muu
(31)
With the corresponding boundary conditions:
y = 0, u00 = 0, u01 = 0, u10 = 0, u11 = 0
y : u00 1, u01 0, u10 1,u11 0 (32)
Solving equations (28) – (31) under the boundary condition (32), we get
u00 = (A3 + A4 – 1) 5 31
3 4 1m y m ym y
e A e A e (33)
u01 = A9
5 31
7 8
m y m ym y
e A e A e (34)
u10 = A19
6 3 51 2 4
17 18 10 14 12 1m y m y m ym y m y m y
e A e A e A e A e A e (35)
u11 = A38
6 5 34 2 1
35 28 36 26 37
m y m y m ym y m y m y
e A e A e A e A e A e (36)
Finally, the velocity u(y, t) is given by
u(y, t) = (A3 + A4 – 1) 5 3 5 31 1
3 4 0 9 7 81m y m y m y m ym y m y
e A e A e K A e A e A e
+ 6 3 51 2 4
19 17 18 10 14 12 1m y m y m ym y m y m y
A e A e A e A e A e A e
6 5
0 38 35
m y m y
K A e A e + 34 2 1
28 36 26 37
m ym y m y m y i t
A e A e A e A e e
(37)
where A1 – A38, m1 – m6 are given in the Appendix.
Skin friction (Cf)
5. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
International organization of Scientific Research 50 | P a g e
* 2 2
0 2
0 0
1
1
4
i tw
f
y
u u u
C K e
U y t y y
(38)
Nusselt Number (Nu)
Nu
0yw0 y
T
TTkv
q
2
1 1 2 1
4 4
1i t
u
i i
N m e m m m
(39)
Sherwood Number (Sh)
0 0
( )
h
w y
m c
S
v D C C y
=
2
3 3 4 3
4 4
1i t i i
m e m m m
(40)
Here
*
w is the dimensionless shear stress component of the elastico-viscous fluid and the symbols q and m
represent heat and mass flux and they are given by
* *
* *
* 0 * 0
and m
y y
T C
q k D
y y
III. RESULTS AND DISCUSSION
The effects of physical parameters such as magnetic parameter (M), Elastic parameter (K0),
permeability parameter (kp), chemical reaction parameter (Kc), heat source parameter (S), Grashoff number for
heat transfer (Gr), Grashoff number for mass transfer (Gc), Prandtl number (Pr), Schmidt number (Sc) and Phase
angle (wt) on the velocity, temperature, concentration profile have been presented with the help of graphs and
Tables.
Fig. 1 exhibits the velocity distribution for various values of Gr, Gc, K0, kp, Kc, Sc, , t and M. It is
seen that an increase in Gr and Gc increases the velocity near the plate (curves I, II and IX), due to free
convection current which accelerates the velocity. The increase is insignificant in case of thermal buoyancy (Gr),
whereas it is quite significant in case of mass buoyancy (Gc). Further, it is seen that the presence of transverse
magnetic field reduces the velocity due to magnetic interaction, producing Lorentz force which opposes the
motion (curves I and X). From the curves XI, XII and I it is observed that the velocity increases in case of
generating reaction (Kc < 0) but is reversed in case of destructive reaction (Kc > 0). It is also observed that the
fluid velocity increases due to increase in the elastic parameter (K0) and reverse effect is observed in case of
phase angle (t) and Schmidt number (Sc).
Fig. 2 shows the velocity distribution for various values of Pr and S. it is seen that higher Prandtl
number fluid contributes very insignificantly in increasing the velocity (curves I, III). It is further to note that
the heat source parameter reduces the velocity at all points (curve I, II).
Fig.3. depicts the temperature distribution in the flow domain. An increase in frequency parameter ()
and heat source parameter (S) contribute to increase the thickness of thermal boundary layer (curves I, IV and
VI) but with an increase in the value of Prandtl number (Pr) and Phase angle (t) the temperature decreases
(curves I, II, III and V). This may be attributed to the fact that smaller values of Pr lead to increase the thermal
conductivity of the fluid and thereby more amount of heat is diffused which causes the decrease in temperature.
Fig. 4 exhibits the concentration distribution for various values of Schmidt number (Sc), chemical
reaction parameter (Kc), frequency parameter () and phase angle (t). For heavier species i.e., for higher
values of (Sc) concentration decreases at all the layers in the flow domain. Similar effects are also observed in
case of chemical reaction parameter (Kc), frequency parameter () and phase angle (t).
Table 1 shows that the skin friction (Cf) decreases due to higher value of Prandtl number and other
parameters such as permeability parameter, magnetic parameter, elastic parameter, chemical reaction
parameter, Schmidt number, frequency of oscillation but reverse effect is observed in case of Grashoff
numbers, heat source parameter and phase angle (t).
6. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
International organization of Scientific Research 51 | P a g e
From Table 2 it is observed that the rate of heat transfer (Nu) at the surface increases in case of higher
Prandtl number (Pr) and frequency of oscillation () but reverse effect is observed in case of heat source
parameter (S) and phase angle (t).
From Table 3 it is observed that the rate of mass transfer (Sh) increases due to increase in Schmidt
number, chemical reaction parameters and frequency parameter but decreases with an increase in phase angle.
IV. CONCLUSION
i) Flow characteristic in the present study is more dependent on mass buoyancy effect (Gc) rather than thermal
buoyancy effect (Gr).
ii) Magnetic field, chemical reaction parameter and heat source parameter with high Prandtl number flow have
a retarding effect on the velocity.
iii) Thinning of thermal boundary layer occurs for higher Prandtl number fluid.
iv) Higher rate of mass transfer is experienced in case of heavier species and in the presence of destructive
reaction.
REFERENCES
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constant suction and Mass transfer. Journal of Heat and Mass Transfer 20, 1977, 1363-1373.
[2] K.D. Singh, Unsteady free convection flow past a hot vertical porous plate with variable temperature.
Proc. Indian Nat. Set. Acad. 58, 1992, 537-544.
[3] K. Anwar, MHD unsteady free convection flow past a vertical porous plate. ZAMM 78, 1998, 255-270.
[4] H. Poonia, and R.C. Chaudhary, MHD free convection and mass transfer flow over an infinite vertical
porous plate with viscous dissipation. Theo Appl., Mech., 37, 2010, 263-287.
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flow through porous medium with constant suction and constant heat flux. Indian J. Pure Appl. Maths.
31, 2000, 1-18.
[6] M. Acharya, G.C. Dash and L.P. Singh, Thermal and mass diffusion on unsteady hydromagnetic flow
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[7] K. Walters, Non-Newtonian effects in some elastico-viscous liquids whose behavior at small rates of shear is
characterized by a general linear equation of state. Quant. J. Mech. Appl. Math, 15, 1962, 63-76.
[8] P.R. Sharma and D. Pareek, Unsteady flow and heat transfer through an elastico-viscous liquid along an
infinite hot vertical porous moving plate with variable free stream suction. Bull. Cal. Math. Sec. 98, 2006,
97-108.
[9] Pravat Kumar Rath, G.C. Dash, and P.K. Rath, Flow and heat transfer of an electrically conducting visco-
elastic fluid between two horizontal squeezing/stretching plates. AMSE Modelling Measurement and
Control 70, 2001, 45-63.
[10] S.M.B., Alhari, A.A. Mohamed and M. S.E.L. Gerdy, Heat and Mass transfer in MHD viscous elastic
fluid flow through a porous medium over a stretching sheet with chemical reaction. Applied Mathematics
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[11] B. Kumar and R. Sivaraj, MHD mixed convective visco-elastic fluid flow in a permeable vertical
channel with Dufour effect and chemical reaction. Int. J. of Appl. Math. and Mech. 14, 2011, 79-96.
[12] R.A. Damesh and B.A. Shannak, Visco-elastic fluid flow past an infinite vertical porous plates in the
presence of first order chemical reaction. Int. J. of Appl. Math. Mech. Engl. Ed. 31, 2010, 955-962.
[13] S. Dash, G.C. Dash and D.P. Mishra, MHD flow through a porous medium past a stretched vertical
permeable surface in the presence of heat source/sink and a chemical reaction. Proc. Nat. Acad. Sci.
India., 78A, 2008, 49-55.
[14] P.K. Rath, T. Parida and G.C. Dash, Three-Dimension free convective flow through porous medium in a
vertical channel with heat source and chemical reaction. Proc. Nat. Acad. Sci. India, 82A, 2012, 225-232.
[15] P.R. Sharma and S. Sharma, Unsteady two dimensional flow and heat transfer through an elastico-
viscous liquid along an infinite hot vertical porous surface bounded by porous medium. Bull. Cal. Math.
Seoc. 97, 2005; 477-488.
7. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
International organization of Scientific Research 52 | P a g e
APPENDIX
2
1
4
2
r r rP P P S
m
,
2
2
4
4
2
r r r
i
P P P S
m
2
3
4
2
c c c c
S S K S
m
, 1 ,
4
c c
iw
K S
2
1
4 2
4 1
,
2
c c
p
S S
m M
k
,
2
5
1 1 4
2
m
,
2
6
1 1 4
4
2
i
m
A1= 2 2
1 1 2 2 3 3 2, A ,m m m m 2
3 4 5 5 5 2
1 2
A , A , Acr GG
m m
A A
3 3 3
3 4 5 3 1 4 3
6 7 8
1 1 2
1
A , A , A
A A m A m A m
A A A
A9 = - (A7 + A8),
1 1
10 11
2
12 2 2
4 41
A , A ,
44
r r
i iG m G m
ii
Am m
3 3
3 4 5
12 13 14
2
2 54 4 2
4 41
1
A ,A ,A
4 44
c c
i iG m G m A A m
i ii
A Am m
3 1 4 3
15 16
1 2
A ,A ,
4 4
A m A m
i i
A A
A17 = A11 + A15 , A18 = A13 + A16, A19 = (- 1 – A17 – A18 – A10 – A14 – A12 )
A20 =
2 2 2
6 6 2 21 2 2 2 22 4 4 2m , A m ,A mm m m
2 3 2 3
6 6 19 1 1 17
23 24
20 1
i i
m m
4 4
A ,A
i i
4 4
m A m A
A A
2 3 2 3
3 3 18 2 2 10
25 26
2 21
i i
m m
4 4
A ,A
i i
4 4
m A m A
A A
,
8. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
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2 3 2 3
5 5 14 4 4 12
27 28
5 22
i i
m m
4 4
A ,A
i i
4 4
m A m A
A A
,
3 3
3 4 5 3 1
29 30
5 1
A 1 A
A ,A
i i
4 4
A m m
A A
,
3
4 3 9 5
31 32
2 5
A ,A
i i
4 4
A m A m
A A
, 7 1 8 3
33 34
1 2
A ,A
i i
4 4
A m A m
A A
A35 = A27 + A29 +A32, A36 = A25 + A31 +A34, A37 = A22 + A30 + A33, A38 = - (A35 + A28 + A36 + A26 + A37)
TABLE - 1 Skin friction (Cf)
Gr Pr Kp t K0 Gc M Kc Sc S Cf
5 5 0.4 5 0.2 2 2 1 0.22 1 5.906
7 5 0.4 5 0.2 2 2 1 0.22 1 6.236
-5 5 0.4 5 0.2 2 2 1 0.22 1 4.249
-7 5 0.4 5 0.2 2 2 1 0.22 1 3.918
5 7 0.4 5 0.2 2 2 1 0.22 1 -442.203
5 5 100 5 0.2 2 2 1 0.22 1 -1.971
5 5 0.4 15 0.2 2 2 1 0.22 1 -1.328
5 5 0.4 5 0.2 2 2 1 0.22 1 11.626
5 5 0.4 5 0 2 2 1 0.22 1 22.009
5 5 0.4 5 0.2 5 2 1 0.22 1 10.885
5 5 0.4 5 0.2 2 5 1 0.22 1 2.269
5 5 0.4 5 0.2 2 2 -0.04 0.22 1 9.202
5 5 0.4 5 0.2 2 2 0 0.22 1 8.780
5 5 0.4 5 0.2 2 2 1 0.78 1 2.577
5 5 0.4 5 0.2 2 2 1 0.22 0.5 -76.803
TABLE - 2 Nusselt number (Nu)
Pr t S Nu
5 1 6.04581
7 5 1 8.81121
11.4 5 1 14.68664
5 15 1 6.31914
5 5 1 5.77632
5 5 0 7.20342
5 5 -1 8.16158
TABLE - 3 Sherwood number (Sh)
Sc t Kc Sh
1.002 1 1.69103
0.600 5 1 1.17297
0.780 5 1 1.41045
1.002 5 0 1.35615
1.002 5 1.5 1.75429
1.002 5 2 1.78982
1.002 15 1 1.88002
1.002 5 1 1.52351
9. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
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Curve Gr
P
r
K
p
t K
0
G
c
M K
c
S
c
S
-------------------------------------------------
I 5 5 0.4 5 /4 0.2 2 2 1 0.22 1
II 7 5 0.4 5 /4 0.2 2 2 1 0.22 1
III -5 5 0.4 5 /4 0.2 2 2 1 0.22 1
IV -7 5 0.4 5 /4 0.2 2 2 1 0.22 1
V 5 5 100 5 /4 0.2 2 2 1 0.22 1
VI 5 5 0.4 15 /4 0.2 2 2 1 0.22 1
VII 5 5 0.4 5 /3 0.2 2 2 1 0.22 1
VIII 5 5 0.4 5 /4 0.0 2 2 1 0.22 1
IX 5 5 0.4 5 /4 0.2 5 2 1 0.22 1
X 5 5 0.4 5 /4 0.2 2 5 1 0.22 1
XI 5 5 0.4 5 /4 0.2 2 2-.04 0.22 1
XII 5 5 0.4 5 /4 0.2 2 2 0 0.22 1
XIII 5 5 0.4 5 /4 0.2 2 2 1 0.78 1
Fig- 1 Velocity distribution for various values of Gr, kp, , t, K0, Gc, M, Sc and Kc when
= 0.5, Pr
= 5 and S = 1
10. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
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Fig- 2 Velocity distribution for various values of Pr
and S when = 0.5, Gr
= 5,
t = /4 ,Kp
= 0.4, = 5, K0
= 0.2, Gc
= 2, M = 2, Kc
= 1 & Sc
= 0.22.
Curve P
r
S
---------------
I 5 1.0
II 5 0.5
III 7 1.0
11. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
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Fig- 3 Temperature distribution for = 0.5
Curve Pr
t S
-------------------------
I 5 5 /4 1
II 7 5 /4 1
III 11.4 5 /4 1
IV 5 15 /4 1
V 5 5 /3 1
VI 5 5 /4 -1
12. Unsteady MHD flow of a visco-elastic fluid along vertical porous surface with fluctuating
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Fig- 4 Concentration distribution for = 0.5
Curve Sc
t Kc
-------------------------
I 1.002 5 /4 1
II 0.600 5 /4 1
III 0.780 5 /4 1
IV 1.002 5 /4 0
V 1.002 5 /4 1.5
VI 1.002 5 /4 2
VII 1.002 15 /4 1
VIII 1.002 5 /3 1