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.
Effect of Radiation on Mixed Convection Flow of a Non-Newtonian Nan fluid ove...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Computational Engineering Research(IJCER) is 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
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.
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
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.
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.
Effect of Radiation on Mixed Convection Flow of a Non-Newtonian Nan fluid ove...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Computational Engineering Research(IJCER) is 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
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.
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
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.
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.
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.
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.
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.
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.
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).
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
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.
Investigation of the Effect of Nanoparticles Mean Diameter on Turbulent Mixed...A Behzadmehr
Abstract
Turbulent mixed convection of a nanofluid (water/Al2O3, Φ=.02) has been studied numerically. Two-phase
mixture model has been used to investigate the effects of nanoparticles mean diameter on the flow parameters. Nanoparticles distribution at the tube cross section shows that the particles are uniformly dispersed. The non-uniformity of the particles distribution occurs in the case of large nanoparticles and/or high value of the Grashof numbers. The study of particle size effect showed that the effective Nusselt number and turbulent intensity increases with the decreased of particle size.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The Force Convection Heat Transfer of A Nanofluid Over A Flat Plate: Using Th...AEIJjournal2
A drift-flux model is utilized to theoretically analyze the boundary layer flow and heat transfer of a
nanofluid over a flat plate. The concentration of nanoparticles at the plate is obtained using the solution of
the governing equations. Assuming a fixed magnitude of free stream velocity, the results show that the heat
transfer may enhance up to 22% or decrease about -7% by using nanofluids compared to the pure base
fluid.
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.
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.
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.
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.
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.
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.
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.
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).
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
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.
Investigation of the Effect of Nanoparticles Mean Diameter on Turbulent Mixed...A Behzadmehr
Abstract
Turbulent mixed convection of a nanofluid (water/Al2O3, Φ=.02) has been studied numerically. Two-phase
mixture model has been used to investigate the effects of nanoparticles mean diameter on the flow parameters. Nanoparticles distribution at the tube cross section shows that the particles are uniformly dispersed. The non-uniformity of the particles distribution occurs in the case of large nanoparticles and/or high value of the Grashof numbers. The study of particle size effect showed that the effective Nusselt number and turbulent intensity increases with the decreased of particle size.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The Force Convection Heat Transfer of A Nanofluid Over A Flat Plate: Using Th...AEIJjournal2
A drift-flux model is utilized to theoretically analyze the boundary layer flow and heat transfer of a
nanofluid over a flat plate. The concentration of nanoparticles at the plate is obtained using the solution of
the governing equations. Assuming a fixed magnitude of free stream velocity, the results show that the heat
transfer may enhance up to 22% or decrease about -7% by using nanofluids compared to the pure base
fluid.
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.
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.
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.
Evaluation of Tensile and Flexural Properties of Polymer Matrix CompositesIJMER
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.
Influence of Hadoop in Big Data Analysis and Its Aspects IJMER
This paper is an effort to present the basic understanding of BIG DATA and
HADOOP and its usefulness to an organization from the performance perspective. Along-with the
introduction of BIG DATA, the important parameters and attributes that make this emerging concept
attractive to organizations has been highlighted. The paper also evaluates the difference in the
challenges faced by a small organization as compared to a medium or large scale operation and
therefore the differences in their approach and treatment of BIG DATA. As Hadoop is a Substantial
scale, open source programming system committed to adaptable, disseminated, information
concentrated processing. A number of application examples of implementation of BIG DATA across
industries varying in strategy, product and processes have been presented. This paper also deals
with the technology aspects of BIG DATA for its implementation in organizations. Since HADOOP has
emerged as a popular tool for BIG DATA implementation. Map reduce is a programming structure for
effectively composing requisitions which prepare boundless measures of information (multi-terabyte
information sets) in- parallel on extensive bunches of merchandise fittings in a dependable,
shortcoming tolerant way. A Map reduce skeleton comprises of two parts. They are “mapper" and
"reducer" which have been examined in this paper. The paper deals with the overall architecture of
HADOOP along with the details of its various components in Big Data.
Live Streaming With Receiver-Based P2P Multiplexing for Future IPTV NetworkIJMER
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.
Accelerometer and EOG Based Wireless Gesture Controlled Robotic ArmIJMER
- This paper deals with the Robotic arm controlled by arm and eye movements, where data
is acquired through the use of accelerometer and EOG amplifier. The hand movements are captured by
the Accelerometer and the eye movements are captured by EOG amplifier. The Robotic arm will pick
and place the object according to the movement of the hand and Robotic arm will move left and right
according to our eye movements. Signals are wirelessly transmitted through RF Encoder and Decoder.
Robotic arm is based on Arduino Duemilanove controller for signal processing, and interfaced with the
Controller serially. This Embedded system will help disabled persons to do work by their own. The
Robotic arm can be used to pick and place hazardous objects.
The Effects of Vehicle Speeds on Accident Frequency within Settlements along ...IJMER
Literature provides overwhelming evidence that a strong relationship exist between
vehicle speed and accident risk, and an outcome severity in the event of an accident. Excessive speed
is said to be a major causal factor of road accidents on trunk roads; contributing 60% of all vehicular
accidents. However, speed rationalization measures implemented on a number of trunk roads in
Ghana have realized very little success. This study therefore investigated the effects of vehicle speeds
on accident frequency within settlements along trunk roads. Data was collected on accidents, vehicle
speeds and other road and environment-related features for ninety-nine (99) settlements delineated
from four (4) trunk roads. Correlation analysis was employed to establish useful relationships and
provided insight into the contributions of relevant road and environmental-related variables to the
occurrence of road traffic accidents. Using the Negative Binomial error structure within the
Generalized Linear Model framework, core (flow-based) models were formulated based on accident
data and exposure variables (vehicle mileage, daily pedestrian flow and travel speed). Incremental
addition of relevant explanatory variables further expanded the core models into comprehensive
models. Findings indicate the main risk factors are number of accesses, daily pedestrian flow and
total vehicle kilometers driven, as vehicle speed did not appear to influence the occurrence of road
traffic accidents within settlements along trunk roads. In settlement corridors, mitigating accident
risks should not focus only on traffic calming but rather on measures that reduce pedestrian and
vehicular conflict situations as well as improve conspicuity around junctions
Many entrepreneurs don't get to startup in major tech hubs. Being the center of your own startup oasis is a way to keep sane, especially during the prototyping / MVP dev / bootstrapping phase.
Impact and Dynamics of Centralization in Transportation Cost of Cement Bag’s ...IJMER
The goal of many research efforts cognate to supply chain management is to propose
mechanisms to reduce operational costs. Inventory holding and conveyance costs are regarded as the
most paramount operational costs in inventory management. Many researches in supply chain
management only consider the inventory cost as a criterion to decide replenishment policy. In the
replenishment process, in juxtaposition of the inventory cost, the conveyance cost is a major cost factor
which affects the shipment size. Thus in this research work the conveyance cost is additionally considered
to minimize the inventory cost.
Two models are studied: when the retailers make decisions independently i.e. Decentralized decision
model and when the retailers are branches of the same firm i.e. Centralized decision model to determine
the best solution to minimize costs.
Determination of Some Mechanical And Hydraulic Properties Of Biu Clayey Soils...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.
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 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.
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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
MHD Flow past a Vertical Oscillating Plate with Radiation and Chemical Reacti...iosrjce
An analysis is performed to study the effect of Thermal radiation and first order Chemical reaction on unsteady natural Convective flow of a viscous incompressible Conducting fluid past over an infinite isothermal vertical oscillating plate in Porous medium. The dimensionless governing equations are solved using the Laplace transform technique. The velocity, temperature and concentration are studied for different
parameters like the magnetic field parameter, radiation parameter, chemical reaction parameter, thermal
Grashof number, Schmidt number, phase angle and time. It is observed that the velocity increases with decreasing magnetic field parameter or radiation parameter. It is also observed that the velocity increases with decreasing magnetic field parameter, radiation parameter and phase angle
The present study analyzes the steady boundary layer slip flow of magneto-nanofluid due to an exponentially permeable stretching sheet with heat generation/absorption. In this paper, the effects of Brownian motion and thermophoresis on heat transfer and nanoparticle volume friction are considered. Using shooting technique along with fourth-order Runge-Kutta method the transformed equations are solved. The study reveals that the governing parameters, namely, the magnetic parameter, wall mass suction parameter, Prandtl number, the Lewis number, slip parameter, heat generation/absorption parameter, Brownian motion parameter, and thermophoresis parameter, have major effects on the flow field, the heat transfer, and the nanoparticle volume fraction as well as skin friction, local Nusselt number and local Sherwood number has been discussed in detail.
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.
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
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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.
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Radiation and Mass Transfer Effects on MHD Natural Convection Flow over an Inclined Plate
1. Radiation and Mass Transfer Effects on Mhd Natural Convection Flow Over An Inclined Plate
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 2 | Feb. 2014 |90|
Radiation and Mass Transfer Effects on MHD Natural
Convection Flow over an Inclined Plate
R.L.V.Renuka Devi1
, T.Poornima2
, N. Bhaskar Reddy3
, S.Venkataramana4
1,2,3,4
Department of Mathematics, Sri Venkateswara University, Tirupati, India
I. INTRODUCTION
The buoyancy force induced by density differences in a fluid cause’s natural convection. Natural
convection flows are frequently encountered in physics and engineering problems such as chemical catalytic
reactors, nuclear waste material etc. Transient free convection is important in many practical applications such
as thermal regulation process, security of energy systems etc. In literature, extensive research work has been
performed to examine the effect of natural convection on flow past a plate. The first attempt in this direction
was made by Callahan and Marner [1] who solved the non-linear system of equations by explicit finite
difference scheme, which is not always convergent. Soundalgekar and Ganesan [2] studied the same problem by
implicit finite difference scheme which is always stable and convergent. Recently, finite difference solution of
natural convection flow over a heated plate with different inclination was studied by Begum et al. [3].
Two dimensional natural convection heat and mass transfer flow past a semi-infinite flat plate has been
receiving the attention of many researchers because of its wide applications in industry and technological fields.
Natural convection along an inclined plate has received less attention than the case of vertical and horizontal
plates. Finite-difference technique has been used in natural convective flow analysis by many researchers.
Callahan and Marner [4] have presented a paper on transient free convection with mass transfer effects and to
solve the problem by explicit finite difference technique. Soundalgekar and Ganesan [5] solved the same
problem using implicit finite difference technique and compared the result with those of Marner and Callahan
[6] and both the results agree well. Chamkha et al. [7] presented similarity solutions for hydromagnetic
simultaneous heat and mass transfer by natural convection from inclined plate with thermal heat generation or
absorption employing implicit finite difference technique. Ganesan and Palani [8] studied free convection
effects on the flow of water at 4ºC past a semi-infinite inclined flat plate and solved the problem using implicit
finite difference technique.
Magnetohydrodynamic flows have applications in meteorology, solar physics, cosmic fluid dynamics,
astrophysics, geophysics and in the motion of earth’s core. In addition from the technological point of view,
MHD free convection flows have significant applications in the field of stellar and planetary magnetospheres,
aeronautical plasma flows, chemical engineering and electronics. An excellent summary of applications is to be
found in Hughes and Young [9]. Raptis [10] studied mathematically the case of time varying two dimensional
natural convective flow of an incompressible, electrically conducting fluid along an infinite vertical porous
plate embedded in a porous medium. Helmy [11] studied MHD unsteady free convection flow past a vertical
porous plate embedded in a porous medium. Elbashbeshy [12] studied heat and mass transfer along a vertical
plate in the presence of magnetic field.
In the context of space technology and in the processes involving high temperatures, the effects of
radiation are of vital importance. Recent developments in hypersonic flights, missile re-entry, rocket
combustion chambers, power plants for inter planetary flight and gas cooled nuclear reactors, have focused
attention on thermal radiation as a mode of energy transfer, and emphasized the need for improved
understanding of radiative transfer in these processes. Cess [13] presented radiation effects on the boundary
layer flow of an absorbing fluid past a vertical plate, by using the Rosseland diffusion model. Several authors
have also studied thermal radiating MHD boundary layer flows with applications in astrophysical fluid
dynamics. Mosa [14] discussed one of the first models for combined radiative hydromagnetic heat transfer,
ABSTRACT: 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.
Keywords: Unsteady flow, Inclined plate, Finite difference method, Radiation, Mass transfer.
2. Radiation and Mass Transfer Effects on Mhd Natural Convection Flow Over An Inclined Plate
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 2 | Feb. 2014 |91|
considering the case of free convective channel flows with an axial temperature gradient. Nath et al. [15]
obtained a set of similarity solutions for radiative-MHD stellar point explosion dynamics using shooting
method. Takhar et al. [16] studied radiation effects on MHD free convection flow past a semi infinite vertical
plate where the viscosity and thermal conductivity were assumed constant. Azzam [17] studied radiation effects
on the MHD mixed convective flow past a semi infinite moving vertical plate for the case of high temperature
differences.
However, the interaction of natural convection flow of an electrically conducting fluid past an inclined
plate in the presence of radiation and mass transfer has received little attention. Hence, in the present chapter an
attempt is made to analyze the mass transfer effects on MHD natural convection flow along a heated inclined
semi-infinite plate in the presence of radiation. The equations of continuity, linear momentum, energy and
diffusion, which govern the flow field, are solved by an implicit finite difference method of Crank-Nicolson
type. The behavior of the velocity, temperature, concentration, skin-friction, Nusselt number and Sherwood
number has been discussed for variations in the physical parameters.
II. MATHEMATICAL ANALYSIS
An unsteady two-dimensional natural convection flow of a viscous, incompressible, electrically
conducting, radiating fluid past a heated inclined semi infinite plate is considered. The fluid is assumed to be
gray, absorbing-emitting but non-scattering. The x-axis is taken along the plate and the y-axis normal to it.
Initially, it is assumed that the plate and the fluid are at the same temperature 'T and concentration level 'C
everywhere in the fluid. At time t >0, the temperature of the plate and the concentration level near the plate are
raised to 'wT and 'wC respectively and are maintained constantly thereafter. A uniform magnetic field is
applied in the direction perpendicular to the plate and that the induced magnetic field is neglected. The
transverse applied magnetic field and magnetic Reynolds number are assumed to be very small, so that the
induced magnetic field is negligible. It is assumed that the concentration C of the diffusing species in the binary
mixture is very less in comparison to the other chemical species, which are present, and hence the Soret and
Dufour effects are negligible. It is also assumed that there is no chemical reaction between the diffusing species
and the fluid. Then, under the above assumptions, in the absence of an input electric field, the governing
boundary layer equations with Boussinesq’s approximation are
0
u v
x y
(1)
2 2
* 0
2
( )sin ( )sin
'
u u u u B
u v g T T g C C u
t x y y
(2)
2
' ' ' ' 1
'
r
p
T T T T q
u v
t x y y c y
(3)
2
2
' ' ' '
'
C C C C
u v D
t x y y
(4)
The initial and boundary conditions are
' 0: 0, 0, ' ' , 't u v T T C C
' 0: 0, 0, ' ' , ' 0w wt u v T T C C at y
0, ' ' , 'u T T C C
at x 0
0, ' ' , 'u T T C C
as y (5)
Where u , v are the velocity components in x , y directions respectively, 't - the time, g - the
acceleration due to gravity, - the volumetric coefficient of thermal expansion,
*
- the volumetric coefficient
of expansion with concentration, T - the temperature of the fluid in the boundary layer, C - the species
concentration in the boundary layer, - the kinematic viscosity, 'wT - the wall temperature, 'T - the free
stream temperature far away from the plate, 'wC - the concentration at the plate, 'C - the free stream
concentration far away from the plate, - the electrical conductivity, 0B - the magnetic induction, - the
density of the fluid, - the thermal diffusivity, cp - the specific heat at constant pressure, rq - the radiation heat
flux and D - the species diffusion coefficient.
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The second term on the right hand side of equation (2.3) represents the radiative heat flux; Thermal
radiation is assumed to be present in the form of a unidirectional flux in the y-direction i.e., rq (transverse to
the surface).
By using the Rosseland approximation ( Brewster [18] ), the radiative heat flux rq is given by
y
T
k
q
e
s
r
4
3
4 (6)
Where s is the Stefan-Boltzmann constant and ek - the mean absorption coefficient. It should be
noted that by using the Rosseland approximation, the present analysis is limited to optically thick fluids. If the
temperature differences within the flow are sufficiently small, then Equation (2.6) can be linearized by
expanding
4
'T into the Taylor series about 'T , which after neglecting higher order terms takes the form
3 44
' 4 ' ' 3 'T T T T (7)
In view of equations (2.6) and (2.7), equation (2.3) reduces to
2 3 2
2 2
' ' ' ' 16 ' '
' 3
s
e p
T T T T T T
u v
t x y y k c y
(8)
From the technological point of view, for the type of problem under conditions, the coefficient of skin-
friction, heat and mass transfer are important.
Local and average skin-frictions are given respectively by
0
y
x
y
u
(9)
L
y
L dx
y
u
L 0 0
1
(10)
Local and average Nusselt numbers are given respectively by
0
'
' '
y
x
w
T
x
y
Nu
T T
(11)
0 0
'
( )
L
L w
y
T
Nu T T dx
y
(12)
Local and average Sherwood numbers are given respectively by
0
'
y
x
w
C
x
y
Sh
C C
(13)
0 0
'
( )
L
L w
y
C
Sh C C dx
y
(14)
In order to write the governing equations and the boundary conditions in dimensionless form, the
following non-dimensional quantities are introduced.
1 1 1 3 324 2 41
2
2 2
*2
0
2 3
*
, , ' , , , ,
' ' '
, , ,Pr , , ,
' ' 4
w w
w e
w w w s
g L T T g L C Cx yGr L uLGr vLGr
X Y t Gr U V Gr Gc
L L v
C CT T C C B k k
T C M Sc N F
T T C C L D T T T
where L is the characteristic length of the plate and k-the thermal conductivity.
In view of (2.15), the equations (1), (2), (8) and (4) are reduced to the following non-dimensional form
0
Y
V
X
U (16)
2
2
( )sin
U U U U
U V T NC MU
t X Y Y
(17)
(15)
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2
2
1 4
1
Pr 3
T T T T
U V
t X Y F Y
(18)
2
2
1C C C C
U V
t X Y Sc Y
(19)
where, Gr, M, F, Pr, N and Sc are thermal Grashof number, radiation parameter, Prandtl number, buoyancy
ratio parameter and Schmidt number respectively.
The corresponding initial and boundary conditions are
0: 0 0 0 0t U ,V ,T ,C
0 : 0, 0, 1, 1 0t U V T C at Y
0, 0, 0, 0 0U V T C at X
0, 0, 0U T C as Y (20)
Local and average skin-frictions in non-dimensional form are
0
4
3
y
x
y
u
Gr (21)
1
3
4
0 0y
u
Gr dX
y
(22)
Local and average Nusselt numbers in non-dimensional form are
1
4
0
x
y
T
Nu XGr
Y
(23)
1
1
4
0 0y
T
Nu Gr dX
Y
(24)
Local and average Sherwood numbers in non-dimensional form are
1
4
0
x
y
C
Sh XGr
Y
(25)
1
1
4
0 0y
C
Sh Gr dX
Y
(26)
III. METHOD OF SOLUTION
In order to solve the unsteady, non-linear, coupled equations (16) - (19), under the boundary conditions
(20), an implicit finite difference scheme of Crank-Nicholson type has been employed. The region of
integration is considered as a rectangle with sides maxX (=1) and maxY (=14), where maxY corresponds to
Y , which lies very well outside the momentum, energy and concentration boundary layers. The
maximum value of Y was chosen as 14 after some preliminary investigations so that the last two of the
boundary conditions of equation (20) are satisfied with in the tolerance limit 10-5
. The grid system is shown in
the following figure A.
The finite difference equations corresponding to equations (16) - (19) are as follows
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1 1 1 1 1 1
, 1, , 1, , 1 1, 1 , 1 1, 1 , , 1 , , 1
0
4 2
n n n n n n n n n n n n
i j i j i j i j i j i j i j i j i j i j i j i jU U U U U U U U V V V V
X Y
(27)
1 1 1 1 1
, , , 1, , 1, , 1 , 1 , 1 , 1
, ,
2 4
n n n n n n n n n n
i j i j i j i j i j i j i j i j i j i jn n
i j i j
U U U U U U U U U U
U V
t X Y
=
1 1 1 1 1 1
, , , , , 1 , , 1 , 1 , , 1 , ,
2
2 2
sin
2 2 2( ) 2
n n n n n n n n n n n n
i j i j i j i j i j i j i j i j i j i j i j i jT T C C U U U U U U U U
N M
Y
(28)
1 1 1 11
, 1, , 1, , 1 , 1 , 1 , 1, ,
, ,
[ ]
2 4
n n n n n n n nn n
i j i j i j i j i j i j i j i ji j i j n n
i j i j
T T T T T T T TT T
U V
t X Y
1 1 1
, 1 , , 1 , 1 , , 1
2
2 21 4
1
Pr 3 2( )
n n n n n n
i j i j i j i j i j i jT T T T T T
F Y
(29)
1 1 1 1 1
, , , 1, , 1, , 1 , 1 , 1 , 1
, ,
2 4
n n n n n n n n n n
i j i j i j i j i j i j i j i j i j i jn n
i j i j
C C C C C C C C C C
U V
t X Y
1 1 1
, 1 , , 1 , 1 , , 1
2
2 21
2( )
n n n n n n
i j i j i j i j i j i jC C C C C C
Sc Y
(30)
Here, the subscript i-designates the grid point along the X-direction, j-along the Y-direction and the
superscript n along the t-direction. An appropriate mesh size considered for the calculation is X = 0.018, Y
= 0.25,and the time step t 0.01.During any one time step, the coefficients
n
jiU . and
n
jiV. appearing in the
difference equations are treated as constants. The values of C, T, U and V at time level (n+1) using the known
values at previous time level (n) are calculated as follows.
The finite difference equation (30) at every internal nodal point on a particular i-level constitute a
tridiagonal system of equations. Such a system of equations is solved by using Thomas algorithm as discussed
in Carnahan et al. [33]. Thus, the values of C are known at every internal nodal point on a particular i at (n+1)th
time level. Similarly, the values of T are calculated from the equation (29). Using the values of C and T at
(n+1)th
time level in the equation (28), the values of U at (n+1)th
time level are found in similar manner. Thus
the values of C, T and U are known on a particular i-level. Then the values of V are calculated explicitly using
the equation (27) at every nodal point at particular i-level at (n+1)th
time level. This process is repeated for
various i-levels. Thus the values of UTC ,, and V are known, at all grid points in the rectangular region at
(n+1)th
time level.
Computations are carried out until the steady state is reached. The steady-state solution is assumed to
have been reached, when the absolute difference between the values of U as well as temperature T and
concentration C at two consecutive time steps are less than 10-5
at all grid points.
The derivatives involved in the equations (21) - (26) are evaluated using five-point approximation formula and
then the integrals are evaluated using Newton-Cotes closed integration formula.
IV. STABILITY OF THE SCHEME
In this section, the stability of the finite difference equations has been discussed using the well-known
Von-Neumann technique. This method introduces an initial error represented by a finite Fourier series and
examines how this error propagates during the solution. The general terms of the Fourier expansion for U, T and
C at a time arbitrarily called t = 0 are assumed to be of the form
i x i y
e e
(here i = 1 ). At a later time t,
these terms will become
U = F(t)
i x i y
e e
T = G(t)
i x i y
e e
(4.1)
C = H(t)
i x i y
e e
Substituting (4.1) in the equations (27) – (30) under the assumption that the coefficients U and V are
constants over any one time step and denoting the values after on time step by HandGF , , one may get
after simplification.
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1 sin
2 2
i X
F F eF F F F i Y
U V
t X Y
2
cos 1
sin
2 2
G G H H F F Y
N
Y
(4.2)
1 sin
2 2
i X
G G eG G G G i Y
U V
t X Y
2
cos 14
1
3 Pr
G G Y
F Y
(4.3)
1 sin
2 2
i X
H H eH H H H i Y
U V
t X Y
2
cos 1H H Y
Sc Y
(4.4)
Equations (4.2) – (4.4) can be written as
(1 ) ' (1 )
2
t
A F A F G G N H H Sin
(4.5)
(1+B) G = (1-B)G (4.6)
(1+E) H= (1-E)H (4.7)
where
2
1 sin
2 2
cos 1
i XU t V t
A e i Y
X Y
t
Y
Y
2
1 sin
2 2
4
cos 1 1
3 Pr
i XU t V t
B e i Y
X Y
t
Y
F Y
2
1 sin
2 2
cos 1
i XU t V t
E e i Y
X Y
t
Y
Sc Y
After eliminating G and H in the equation (4.5) using the equations (4.6) and (4.7), the resultant
equation and equations (4.6) and (4.7) can be written in matrix form as follows.
H
G
F
=
E
E
B
B
DD
A
A
1
1
00
0
1
1
0
1
1
21
H
G
F
(4.8)
where 1D =
1 1
tsin
A B
and 2D =
1 1
tsin
A E
Now, for the stability of the finite difference scheme, the modulus of each eigen value of the
amplification matrix must not exceed unity. Since this matrix in the equation (4.8) is triangular, the eigen values
are diagonal elements. Hence, the eigen values of the amplification matrix are (1A)/(1+A), (1B)/(1+B) and
(1E)/(1+E). Assuming that U is everywhere non-negative and V is everywhere non-positive we get
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2 2
2 sin 2 sin
2 2
sin sin
X Y
A a c
i a X b Y
2 24
2 sin 2 1 sin
2 3 Pr 2
sin sin
X c Y
B a
F
i a X b Y
2 22
2 sin sin
2 2
sin sin
X Y
A a c
Sc
i a X b Y
where
a =
X
tU
2
, b =
Y
tV
2
and c =
2
Y
t
Since the real part of A is greater than or equal to zero, 11/1 AA always. Similarly,
11/1 BB and 11/1 EE .
Hence, the scheme is unconditionally stable. The local truncation error is O( XYt 22
) and it tends to
zero as t , Y and X tend to zero. Therefore the scheme is compatible. The stability and compatibility
ensures convergence.
V. RESULTS AND DISCUSSION
The aim of present study is to investigate the effect of radiation and mass transfer on an unsteady free
convection flow of a viscous incompressible electrically conducting fluid over an inclined heated plate. A
representative set of numerical results is shown graphically in Figs.1-20, to illustrate the influence of physical
parameters viz., radiation parameter F, buoyancy ratio parameter N, Prandtl number Pr, Schmidt number Sc,
and magnetic parameter M on the velocity, temperature, concentration, skin-friction, Nusselt number and
Sherwood number. Here the value of Pr is chosen as 0.72, which corresponds to air. The values of Sc are
chosen such that they represent Helium (0.24), Ammonia (0.78). The other parameters are arbitrarily chosen.
In order to ascertain the accuracy of the numerical results, the present study is compared with the
previous study. The velocity profiles of the present problem for Sc = 0.0, Pr = 0.72, N = 0.0, M=0.0, F=0.0,
t=0.40, α=600
, X = 1.0 are compared with the available solution of Begum et al [3] in Fig.1. It is observed that
the present results are in good agreement with that of Begum et al. [3] .
The effects of inclined angle α, on the transient velocity are displayed in Fig.2. It is noticed that the
velocity increases with increasing values of the inclined angle. The effects of the magnetic parameter M on the
transient velocity are illustrated in Fig.3. It is seen that, as expected, the velocity decreases with an increase in
the magnetic parameter. The magnetic parameter is found to retard the velocity at all points of the flow field. It
is because that the application of transverse magnetic field will result in a resistive type force (Lorentz force)
similar to drag force which tends to resist the fluid flow and thus reducing its velocity. Also, the boundary layer
thickness decreases with an increase in the magnetic parameter. The effect of N, on the transient velocity is
displayed in Fig.4. It is noticed that the velocity increases with increasing values of the buoyancy ratio
parameter. The effect of F, on the transient velocity is displayed in Fig.5. It can be seen that an increase in the
thermal radiation parameter produces significant decreases in the velocity boundary layer.
The steady state temperature for different values of M is displayed in Fig.6. It can be seen that an
increase in the magnetic parameter produces significant increases in the temperature boundary layer. Fig.7
shows the distribution of steady state temperature against Y for various N values. The profiles in Fig.7 attest
that with an increase in N the thermal boundary layer will be decreased in thickness and there will be a
corresponding uniformity of temperature distributions across the boundary layer. The effect of radiation
parameter on the transient temperature can be observed from Fig.8. It can be seen that an increase in the
radiation parameter leads to a decrease in the temperature.
Fig.9 shows the distribution of T against Y for various Pr values. The profiles in Fig.9 attest that with
an increase in Pr the thermal boundary layer will be decreased in thickness and there will be a corresponding
uniformity of temperature distributions across the boundary layer. It is observed that the maximum temperature
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correspond to lower Pr values. The profiles also steepen and intersect the abscissa faster for higher Pr fluids i.e.
temperatures across the boundary layer (normal to wall) reach zero faster.
The steady state concentration for the different values M is displayed in Fig.10. It can be seen that an
increase in the magnetic parameter produces significant increase in the concentration boundary layer. The effect
of N, on the steady state concentration is displayed in Fig.11. It is noticed that the concentration decreases with
increasing values of the buoyancy ratio parameter. The effect of F, on the steady state concentration is
represented in Fig.12. It is noticed that the concentration decreases with increasing values of the radiation
parameter. Fig.13 shows the distribution of transient concentration against Y for various Sc values. The profiles
in Fig.13 attest that with an increase in Sc the concentration boundary layer will be decreased in thickness and
there will be a corresponding uniformity of concentration distributions across the boundary layer.
Fig.14 illustrates the effects of M, N and F on the local skin-friction. The local skin-friction is found to
decrease due to an increase in the magnetic field strength. An increase in N or F produces an increase in the
local skin-friction. The effects M, N and F on the average skin-friction are shown in Fig.15. It is observed that
the average skin-friction increases as N or F increases, and it decreases as M increases. Figs.16 and 17 show the
effect of Pr and F on the local and average Nusselt numbers respectively. It is observed that the local and
average Nusselt numbers increase as Pr increases, and decrease as F increases. Figs.18, 19 and 20 display the
effect of Sc and F on the local and average Sherwood numbers respectively. It can be observed that as Sc or F
increases the local and average Sherwood numbers increase.
VI. CONCLUTIONS
An Unsteady two dimensional natural convection boundary layer flow of heat and mass transfer over
heated plate with different inclinations in the presence of radiation has been studied. Implicit finite difference
scheme of Crank-Nicolson type was employed to obtain the solution of the governing equations. The present
solutions were validated by comparing with solutions existing in the literature. Our results show a good
agreement with the existing work in the literature. The results are summarized as follows:
1. Magnetic field elevates the temperature and concentration, and reduces the velocity.
2. The angle of inclination enhances the velocity
3. The radiation enhances the velocity and temperature, and reduces the concentration.
4. The radiation enhances the local and average skin-friction, and local and average Sherwood number, and
reduces the local and average Nusselt number.
Fig. 1 Comparison of velocity for t=0.40, Pr=0.72 and α=600
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Fig 2 Transient velocity at X=1.0 for the different values of α.
Fig 3 Transient velocity at X=1.0 for the different values of M.
Fig 4 Transient velocity at X=1.0 for the different values of N.
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Fig 5 Transient velocity for at X=1.0 the different values of F.
Fig 6 Steady state temperature for different values of M at X=1.0.
Fig 7 Steady state temperature for different values of N at X=1.0.
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Fig 8 Transient temperature at X=1.0 for different values of F.
Fig 9 Transient temperature at X=1.0 for different values of Pr.
Fig 10 Steady state concentration for different values of M at X=1.0.
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Fig 11 Steady state concentration for different values of N at X=1.0.
Fig 12 Steady state concentration for different values of F at X=1.0.
Fig 13 Transient concentration at X=1.0 for the different values of Sc.
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Fig 14 The effects of M, N and F on local skin friction.
Fig 15 The effects of M, N and F on average skin friction.
Fig 16 The effects of Pr and F on local Nusselt number.
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Fig 17 The effects of Pr and F on average Nusselt number.
Fig 18 The effects of Sc and F on local Sherwood number.
Fig 19 The effect of Schmidt number on average Sherwood number.
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Fig 20 The effect of R on average Sherwood number.
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