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.
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.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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.
Heat Transfer on Steady MHD rotating flow through porous medium in a parallel...IJERA Editor
We discussed the combined effects of radiative heat transfer and a transverse magnetic field on steady rotating flow of an electrically conducting optically thin fluid through a porous medium in a parallel plate channel and non-uniform temperatures at the walls. The analytical solutions are obtained from coupled nonlinear partial differential equations for the problem. The computational results are discussed quantitatively with the aid of the dimensionless parameters entering in the solution.
International journal of engineering and mathematical modelling vol2 no3_2015_1IJEMM
A weak nonlinear stability analysis has been performed for an oscillatory mode of convection, heat and mass transports in terms of
Nusselt, Sherwood numbers are derived and evaluated by a non$-$autonomous complex Ginzburg-Landau equation. The porous layer boundaries are heated sinusoidally with time. The basic state temperature is defined in terms of study and oscillatory parts, where study part show nonlinear throughflow effect on the system and time dependant part show modulation effect. The generalized Darcy model is employed for the momentum equation. The disturbances of the flow are expanded in power series of amplitude of modulation, which is assumed to be small and employed using normal mode technics. The effect of vertical throughflow is found to stabilize or destabilize the system depending on its direction. The time relaxation parameter $\lambda_1$ has destabilizing effect, while time retardation parameter $\lambda_2$ has stabilizing effect on the system. Three types of modulations have been analyzed, and found that, OPM, LBMO cases are effective on heat and mass transfer than IPM case. The effects of amplitude and frequency of modulation on heat and mass transports have been analyzed and depicted graphically. The study establishes that the heat and mass transports can be controlled effectively by a mechanism that is external to the system.
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.
OSCILLATORY FLOW OF MHD POLAR FLUID WITH HEAT AND MASS TRANSFER PAST A VERTIC...IAEME Publication
The study of unsteady two-dimensional laminar boundary layer flow of a viscous incompressible fluid (polar fluid) through porous medium past a semi-infinite vertical porous stretching plate in the presence of transverse magnetic field is investigated.
The sheet makes with a constant velocity in the longitudinal direction and the free stream velocity follows an exponentially increasing or decreasing small perturbation law. A uniform magnetic field acts perpendicularly to the porous sheet which absorbs the polar fluid with a suction velocity varying with time component. The effects of all parameters encountering in the problem are investigated for velocity and temperature fields across the boundary layer.
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.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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.
Heat Transfer on Steady MHD rotating flow through porous medium in a parallel...IJERA Editor
We discussed the combined effects of radiative heat transfer and a transverse magnetic field on steady rotating flow of an electrically conducting optically thin fluid through a porous medium in a parallel plate channel and non-uniform temperatures at the walls. The analytical solutions are obtained from coupled nonlinear partial differential equations for the problem. The computational results are discussed quantitatively with the aid of the dimensionless parameters entering in the solution.
International journal of engineering and mathematical modelling vol2 no3_2015_1IJEMM
A weak nonlinear stability analysis has been performed for an oscillatory mode of convection, heat and mass transports in terms of
Nusselt, Sherwood numbers are derived and evaluated by a non$-$autonomous complex Ginzburg-Landau equation. The porous layer boundaries are heated sinusoidally with time. The basic state temperature is defined in terms of study and oscillatory parts, where study part show nonlinear throughflow effect on the system and time dependant part show modulation effect. The generalized Darcy model is employed for the momentum equation. The disturbances of the flow are expanded in power series of amplitude of modulation, which is assumed to be small and employed using normal mode technics. The effect of vertical throughflow is found to stabilize or destabilize the system depending on its direction. The time relaxation parameter $\lambda_1$ has destabilizing effect, while time retardation parameter $\lambda_2$ has stabilizing effect on the system. Three types of modulations have been analyzed, and found that, OPM, LBMO cases are effective on heat and mass transfer than IPM case. The effects of amplitude and frequency of modulation on heat and mass transports have been analyzed and depicted graphically. The study establishes that the heat and mass transports can be controlled effectively by a mechanism that is external to the system.
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.
OSCILLATORY FLOW OF MHD POLAR FLUID WITH HEAT AND MASS TRANSFER PAST A VERTIC...IAEME Publication
The study of unsteady two-dimensional laminar boundary layer flow of a viscous incompressible fluid (polar fluid) through porous medium past a semi-infinite vertical porous stretching plate in the presence of transverse magnetic field is investigated.
The sheet makes with a constant velocity in the longitudinal direction and the free stream velocity follows an exponentially increasing or decreasing small perturbation law. A uniform magnetic field acts perpendicularly to the porous sheet which absorbs the polar fluid with a suction velocity varying with time component. The effects of all parameters encountering in the problem are investigated for velocity and temperature fields across the boundary layer.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Effect of Chemical Reaction and Radiation Absorption on Unsteady Convective H...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
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
Effect of Thermo-Diffusion and Chemical Reaction on Mixed Convective Heat And...IJERA Editor
A finite element study of combined heat and mass transfer flow through a porous medium in a circular cylindrical annulus with Soret and Dufour effects in the presence of heat sources has been analyzed. The coupled velocity, energy, and diffusion equations are solved numerically by using Galerkin- finite element technique. Shear stress, Nusslet number and Sherwood number are evaluated numerically for different values of the governing parameters under consideration and are shown in tabular form.
Non-Newtonian Visco-elastic Heat Transfer Flow Past a Stretching Sheet with C...IJERA Editor
In this paper two dimensional flow of a viscoelastic fluid due to stretching surface is considered. Flow analysis is carried out by using closed form solution of fourth order differential equation of motion of viscoelastic fluid. Further (Walters’ liquid B’ model) heat transfer analysis is carried out using convective surface condition. The governing equations of flow and heat transfer are non-linear partial differential equations which are unable to solve analytically hence are solved using Runge-Kutta Numerical Method with efficient shooting technique. The flow and heat transfer characteristics are studied through plots drawn. Numerical values of Wall temperature are calculated and presented in the table and compared with earlier published results which are in good agreement
Double Diffusive Convection and the Improvement of Flow in Square Porous AnnulusIJERA Editor
There has been increased interest shown in recent years to investigate the behavior of heat and mass transfer in a square annulus with a porous medium fixed between the inner and outer walls. This paper aims to evaluate the Soret effect arising in the case of heat and mass transfer in a porous medium bounded by a square annulus and subjected to isothermal heating of the inner surfaces as well as the outer horizontal surfaces. The phenomenon is governed by 3 partial differential equations, the momentum, energy and concentration equations, that are coupled together and result in a situation where change in one variable affects the other equations and vice versa. The partial differential equations are converted into finite element equations with the help of the Galerkin method and then solved to predict solution variables such as temperature, stream function and concentration in the porous medium. It is found that the heat transfer rate at the hot wall decreases with increasing viscous dissipation effect in the porous medium.
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
Fuzzy numbers, Nth - order Fuzzy Initial Value Problems, Runge-Kutta method, ...IOSR Journals
A numerical study is presented of two-dimensional laminar steady-state on megneto-hydrodynamics
(MHD) free convection for heat flow patterns within trapezoidal enclosures. A finite element analysis is
performed to investigate the effects of unifor heating and is also used for solving the Navier-Stokes and
Energybalance equations.In this study, cold bottom walls, uniformly heated left and right (side) walls and
insulated top walls with inclination angles (ф) are considered in a trapezoidal enclosure. The present numerical
procedure adopted in this investigation yields consistent performance over a wide range of parameters, Prandtl
numbers, (Pr = 0.026 - 0.7), and Rayleigh numbers (Ra = 103 – 105), Hartmann number (Ha = 50) with various
tilt angles Ф = 450, 300 and 00(square).Numerical results are presented in terms of streamlines, isotherms, heat
function (total heat flux) and nusselt numbers.for different Ra and Pr. As Ra increases conduction dominant
region changes for different Pr. Complete heat transfer analysis is performed in terms of local and average
nusselt numbers.
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.
Numerical study of natural convection in an enclosed square cavity using cons...eSAT Journals
Abstract In the present study, Constrained Interpolated Profile (CIP) method was used to simulate the natural convection heat transfer and fluid flow in an enclosed square cavity with differentially heated side walls. The fundamental idea of this method is to solve the advection phase equation with CIP method and the non-advection phase equation is calculated with finite difference method. CIPNSE is applied to predict the temperature and velocity profiles in a square cavity for various Rayleigh number: Ra=103, 104 and 105. The streamline and isotherms obtained under these conditions were then compared with those published in literature and found a good agreement. Keywords: Constrained Interpolated Profile (CIP), Finite Difference Method (FDM) and Lattice Boltzmann Method (LBM), Natural Convection, Square Cavity, Stream-Function Vorticity
Transient thermal analysis of phase change material based heat sinkseSAT Journals
Abstract Integrated circuits operate best in a limited range of temperature hence their package must be designed to remove the excessive heat. As an alternative passive cooling technique means, phase change materials or PCMs have been widely investigated for such transient cooling applications considering their advantage such as high latent heat of fusion, high specific heat, controllable temperature stability and small volume change during phase change, etc. This PCM based cooling techniques have great potential application in the devices which are not operated continuously over a long period, but in intermittently using devices like cellular phones, digital cameras, notebook etc. The PCM absorbs heat from the electronic component when it operates and melts, the molten PCM needs to be re-solidified by dissipating heat to the surroundings while the electronic device are idle, such a cooling system is applicable only for intermittent use devices and not those in continuous operation. To achieve effective cooling it is important to ensure that the operating duration of the electronic device does not exceed the time of full melting of PCM. Advanced transient analysis is required for clear understanding of the mechanism behind this method of cooling while practical implementations are considered. Controlled convective cooling techniques can be implemented for continuous operation such kind of systems. The present work is a numerical study consists of thermal analysis of various configurations of finned heat sink with PCM. The configurations considered are finned heat sink without PCM and with PCM, Half-filled PCM towards the fin tip side and cases which includes forced convection for systems with continuous operation. The unsteady analyses were performed to record the transient nature of problem. The characteristic of PCM and the design of operational time of convective cooling are estimated. By analyzing these different configurations a vivid picture of the physics of heat transfer in PCM based heat sink is figured out. Keywords: Phase change materials; Heatsink; electronics cooling; Thermal management
NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFER FROM CIRCULAR CYL...IAEME Publication
In the present work, the enhancement of natural convection heat transfer utilizing nanofluids as working fluid from horizontal circular cylinder situated in a square enclosure is investigated numerically. The type of the nanofluid is the water-based copper Cu. A model is developed to analyze heat transfer performance of nanofluids inside an enclosure taking into account the solid particle dispersionrs on the flow and heat transfer characteristics. The study uses different Raylieh
numbers (104 , 105 , and 106 ), different enclosure width to cylinder diameter ratios W/D (1.667, 2.5 and 5) and volume fraction of nanoparticles between 0 to 0.2. The work included the solution of the governing equations in the vorticity-stream function formulation which were transformed into body fitted coordinate system
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Effect of Chemical Reaction and Radiation Absorption on Unsteady Convective H...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
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
Effect of Thermo-Diffusion and Chemical Reaction on Mixed Convective Heat And...IJERA Editor
A finite element study of combined heat and mass transfer flow through a porous medium in a circular cylindrical annulus with Soret and Dufour effects in the presence of heat sources has been analyzed. The coupled velocity, energy, and diffusion equations are solved numerically by using Galerkin- finite element technique. Shear stress, Nusslet number and Sherwood number are evaluated numerically for different values of the governing parameters under consideration and are shown in tabular form.
Non-Newtonian Visco-elastic Heat Transfer Flow Past a Stretching Sheet with C...IJERA Editor
In this paper two dimensional flow of a viscoelastic fluid due to stretching surface is considered. Flow analysis is carried out by using closed form solution of fourth order differential equation of motion of viscoelastic fluid. Further (Walters’ liquid B’ model) heat transfer analysis is carried out using convective surface condition. The governing equations of flow and heat transfer are non-linear partial differential equations which are unable to solve analytically hence are solved using Runge-Kutta Numerical Method with efficient shooting technique. The flow and heat transfer characteristics are studied through plots drawn. Numerical values of Wall temperature are calculated and presented in the table and compared with earlier published results which are in good agreement
Double Diffusive Convection and the Improvement of Flow in Square Porous AnnulusIJERA Editor
There has been increased interest shown in recent years to investigate the behavior of heat and mass transfer in a square annulus with a porous medium fixed between the inner and outer walls. This paper aims to evaluate the Soret effect arising in the case of heat and mass transfer in a porous medium bounded by a square annulus and subjected to isothermal heating of the inner surfaces as well as the outer horizontal surfaces. The phenomenon is governed by 3 partial differential equations, the momentum, energy and concentration equations, that are coupled together and result in a situation where change in one variable affects the other equations and vice versa. The partial differential equations are converted into finite element equations with the help of the Galerkin method and then solved to predict solution variables such as temperature, stream function and concentration in the porous medium. It is found that the heat transfer rate at the hot wall decreases with increasing viscous dissipation effect in the porous medium.
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
Fuzzy numbers, Nth - order Fuzzy Initial Value Problems, Runge-Kutta method, ...IOSR Journals
A numerical study is presented of two-dimensional laminar steady-state on megneto-hydrodynamics
(MHD) free convection for heat flow patterns within trapezoidal enclosures. A finite element analysis is
performed to investigate the effects of unifor heating and is also used for solving the Navier-Stokes and
Energybalance equations.In this study, cold bottom walls, uniformly heated left and right (side) walls and
insulated top walls with inclination angles (ф) are considered in a trapezoidal enclosure. The present numerical
procedure adopted in this investigation yields consistent performance over a wide range of parameters, Prandtl
numbers, (Pr = 0.026 - 0.7), and Rayleigh numbers (Ra = 103 – 105), Hartmann number (Ha = 50) with various
tilt angles Ф = 450, 300 and 00(square).Numerical results are presented in terms of streamlines, isotherms, heat
function (total heat flux) and nusselt numbers.for different Ra and Pr. As Ra increases conduction dominant
region changes for different Pr. Complete heat transfer analysis is performed in terms of local and average
nusselt numbers.
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.
Numerical study of natural convection in an enclosed square cavity using cons...eSAT Journals
Abstract In the present study, Constrained Interpolated Profile (CIP) method was used to simulate the natural convection heat transfer and fluid flow in an enclosed square cavity with differentially heated side walls. The fundamental idea of this method is to solve the advection phase equation with CIP method and the non-advection phase equation is calculated with finite difference method. CIPNSE is applied to predict the temperature and velocity profiles in a square cavity for various Rayleigh number: Ra=103, 104 and 105. The streamline and isotherms obtained under these conditions were then compared with those published in literature and found a good agreement. Keywords: Constrained Interpolated Profile (CIP), Finite Difference Method (FDM) and Lattice Boltzmann Method (LBM), Natural Convection, Square Cavity, Stream-Function Vorticity
Transient thermal analysis of phase change material based heat sinkseSAT Journals
Abstract Integrated circuits operate best in a limited range of temperature hence their package must be designed to remove the excessive heat. As an alternative passive cooling technique means, phase change materials or PCMs have been widely investigated for such transient cooling applications considering their advantage such as high latent heat of fusion, high specific heat, controllable temperature stability and small volume change during phase change, etc. This PCM based cooling techniques have great potential application in the devices which are not operated continuously over a long period, but in intermittently using devices like cellular phones, digital cameras, notebook etc. The PCM absorbs heat from the electronic component when it operates and melts, the molten PCM needs to be re-solidified by dissipating heat to the surroundings while the electronic device are idle, such a cooling system is applicable only for intermittent use devices and not those in continuous operation. To achieve effective cooling it is important to ensure that the operating duration of the electronic device does not exceed the time of full melting of PCM. Advanced transient analysis is required for clear understanding of the mechanism behind this method of cooling while practical implementations are considered. Controlled convective cooling techniques can be implemented for continuous operation such kind of systems. The present work is a numerical study consists of thermal analysis of various configurations of finned heat sink with PCM. The configurations considered are finned heat sink without PCM and with PCM, Half-filled PCM towards the fin tip side and cases which includes forced convection for systems with continuous operation. The unsteady analyses were performed to record the transient nature of problem. The characteristic of PCM and the design of operational time of convective cooling are estimated. By analyzing these different configurations a vivid picture of the physics of heat transfer in PCM based heat sink is figured out. Keywords: Phase change materials; Heatsink; electronics cooling; Thermal management
NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFER FROM CIRCULAR CYL...IAEME Publication
In the present work, the enhancement of natural convection heat transfer utilizing nanofluids as working fluid from horizontal circular cylinder situated in a square enclosure is investigated numerically. The type of the nanofluid is the water-based copper Cu. A model is developed to analyze heat transfer performance of nanofluids inside an enclosure taking into account the solid particle dispersionrs on the flow and heat transfer characteristics. The study uses different Raylieh
numbers (104 , 105 , and 106 ), different enclosure width to cylinder diameter ratios W/D (1.667, 2.5 and 5) and volume fraction of nanoparticles between 0 to 0.2. The work included the solution of the governing equations in the vorticity-stream function formulation which were transformed into body fitted coordinate system
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 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.
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.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
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.
Radiation and Mass Transfer Effects on MHD Natural Convection Flow over an In...IJMER
A numerical solution for the unsteady, natural convective flow of heat and mass transfer along an inclined plate is presented. The dimensionless unsteady, coupled, and non-linear partial differential conservation equations for the boundary layer regime are solved by an efficient, accurate and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields have been studied for the effect of Magnetic parameter, buoyancy ratio parameter, Prandtl number, radiation parameter and Schmidt number. The local skin-friction, Nusselt number and Sherwood number are also presented and analyzed graphically.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
HYDROMAGNETIC MIXED CONVECTION MICRO POLAR FLOW DRIVEN BY A POROUS STRETCHING...IAEME Publication
We analyze a finite element solution for the magneto hydrodynamics mixed convection micro polar flow through a porous medium driven by a porous stretching sheet with uniform suction. The governing partial differential equations are solved numerically by using finite element technique. The
effect of Hartmann number, Darcy parameter and surface condition on velocity, micro rotation and temperature functions has been study.
Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow...iosrjce
The unsteady hydromagnetic boundary layer flow of an incompressible and electrically conducting
fluid through a porous medium bounded by a moving surface has been considered. It is assumed that the moving
surface has a velocity profile with respect to time and fluid flow is taken under the influence of a transverse
magnetic field. The similarity solution is used to transform the system of partial differential equations,
describing the problem under consideration, into a boundary value problem of coupled ordinary differential
equations and an efficient numerical technique is implemented to solve the reduced system. The effects of the
parameters such as Magnetic parameter, Prandtl number and Eckert number are discussed graphically on
velocity and temperature distributions
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The 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.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
The Study of Heat Generation and Viscous Dissipation on Mhd Heat And Mass Diffusion Flow Past A Surface
1. IOSR Journal of Applied Physics (IOSR-JAP)
e-ISSN: 2278-4861.Volume 5, Issue 4 (Nov. - Dec. 2013), PP 17-28
www.iosrjournals.org
www.iosrjournals.org 17 | Page
The Study of Heat Generation and Viscous Dissipation on Mhd
Heat And Mass Diffusion Flow Past A Surface
K. Bhagya Lakshmi1
, G.S.S. Raju2
, P.M. Kishore3
and N.V.R.V.Prasada Rao4
1. Department of Mathematics, C.M.R. Technical Campus, Kandlakoya (V), Medchal (M), Hyderabad-501401.
(A.P), India.
2. Department of Mahtematics, J.N.T.U. A. College of Engineering, Pulivendula, Y.S.R. District. (A.P), India.
3. Department of Mathematics, Narayana Engineering College, Nellore - 524001 (A.P), India.
4. Department of Mathematics, S.V.G.S. College, Nellore - 524002 (A.P), India.
Abstract; 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.
Key Words: MHD, heat generation, free convection, viscous dissipation, finite difference method.
I. Introduction
The buoyancy force induced by density differences in a fluid causes natural convection. Natural
convection flows are frequently encountered in physical and engineering problems such as chemical catalytic
reactors, nuclear waste materials etc. Transient free convection is important in many practical applications, such
as furnaces electronic components, solar collectors, thermal regulation process, security of energy systems etc.
when a conductive fluid moves through a magnetic field and an ionized gas is electrically conductive, the fluid
may be influenced by the magnetic field. Magnetohydrodynamic free convection heat transfer flow is of
considerable interest in the technical field due to its frequent occurrence in industrial technology and geothermal
application, liquid metal fluids and MHD power generation systems etc. The change in wall temperature causing
the free convection flow could be a sudden or a periodic one, leading to a variation in the flow. In nuclear
engineering, cooling of medium is more important safety point of view and during this cooling process the plate
temperature starts oscillating about a non-zero constant mean temperature. Further, oscillatory flow has
applications in industrial and aerospace engineering. Viscous mechanical dissipation effects are important in
geophysical flows and also in certain industrial operations and are usually characterized by the Eckert number.
Many practical heat transfer applications involve the conversion of some form of mechanical, electrical, nuclear,
or chemical energy to thermal energy in the medium. Such mediums are said to involve internal heat generation.
For example, a large amount of heat is generated in the fuel elements from atomic reactors as a result of atomic
fission that serves as the heat source for the nuclear power plants. The heat generated in the sun as a result of
fusion of hydrogen into helium makes the sun a large nuclear reactor that supplies heat to the earth. Possible
heat generation effects may modify temperature distribution and, therefore, the particle deposition rate.
In the literature, extensive research work is available to examine the effect of natural convection on
flow past a plate. Examples of this include Vedhanayagam et.al. [1], Kolar et. al. [2] and Li et. al. [3]. Transient
free convection flow past an isothermal vertical plate was first reported by Siegel [4] using an integral method.
The experimental confirmation of these results was discussed by Goldstein et. al. [5]. A review of transient
natural convection presented by Raithby et.al [6] wherein a large number of papers on this topic were reviewed.
In this review, the meaning of transient convection has been explained systematically. They have defined the
conduction regime and the steady state regime and that which lies between these two regimes as the transient
regime. Other studies dealing with transient natural convection are by Das et.al. [7] and Saeid [8]. Simultaneous
heat and mass transfer in laminar free convection boundary layer flows over surface can be found in Lin et .al.
[9] and Mongruel et.al. [10].
Fewer studies have been carried out to investigate the magnetohydrodynamic free convection flow in
the presence of viscous dissipation. In all the investigations mentioned above, viscous mechanical dissipation is
neglected. A number of authors have considered viscous heating effects on Newtonian flows. Isreal-Cookey
et.al. [11] investigated the influence of viscous dissipation and radiation on unsteady MHD free convection flow
2. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 18 | Page
past an infinite heated vertical plate in a porous medium with time dependent suction. Zueco [12] used network
simulation method (NSM) to study the effects of viscous dissipation and radiation on unsteady MHD free
convection flow past a vertical porous plate. Suneetha et.al. [13] have analyzed the thermal radiation effects on
hydromagnetic free convection flow past an impulsively started vertical plate with variable surface temperature
and concentration is analyzed by taking into account of the heat due to viscous dissipation. Recently Hiteesh
[14] studied the boundary layer steady flow and heat transfer of a viscous incompressible fluid due to a
stretching plate with viscous dissipation effect in the presence of a transverse magnetic field.
The heat transfer in a laminar boundary layer flow of a viscous fluid over a linearly stretching
continuous surface with viscous dissipation/frictional heating and internal heat generation was analysed by
Vajravelu and Hadjinicolaou [15]. They considered the volumetric rate of heat generation,
3
'''[ / ]q W m , as
3
0'''[ / ] fq W m Q T T , for fT T and, ''' 0q , for fT T , where 0Q is the heat generation
constant. Madhusudhana Rao et.al. [16] studied the analysis of unsteady free convection heat and mass transfer
flow through a non-homogeneous porous medium with variable permeability bounded by an infinite porous
vertical plate in slip flow regime taking into account the radiation, chemical reaction and temperature gradient
dependent heat source. Mamun et al. [17] investigated MHD-conjugate heat transfer analysis for a vertical flat
plate in presence of viscous dissipation and heat generation. Recently, Azim et al. [18] analyzed viscous Joule
heating MHD-conjugate heat transfer for a vertical flat plate in the presence of heat generation.
Hence, based on the above mentioned investigations and applications, the object of this paper is to
study magnetohydrodynamic transient heat and mass transfer flow by free convection past a vertical plate in the
presence of viscous dissipation and heat generation, when the temperature of the plate oscillates periodically
about a constant mean temperature. The present investigation may be useful for the study of movement of oil or
gas and water through the reservoir of an oil or gas field, underground water in river beds, filtration and water
purification processes. This study of flow past a vertical surface can be utilized as the basis of many scientific
and engineering applications, including earth science, nuclear engineering and metallurgy. In nuclear
engineering, it finds its applications for the design of the blanket of liquid metal around a thermonuclear fusion-
fission hybrid reactor. In metallurgy, it can be applied during the solidification process. The results of the
problem are also of great interest in geophysics, in the study of interaction of geomagnetic field with the fluid in
the geothermal region.
II. Mathematical Analysis
We consider a one – dimensional flow of an incompressible and electrically conducting viscous fluid
along an infinite vertical plate. The 'x -axis is taken along the infinite plate and 'y - axis normal to it. Initially,
the plate and the fluid are at same temperature T
with concentration level C
at all points. At time ' 0t ,
the plate temperature is raised to wT and a periodic temperature variation is assumed to be superimposed on this
mean constant temperature of the plate and the concentration level at the plate is raised to wC . A magnetic field
of uniform strength is applied perpendicular to the plate and the magnetic Reynolds number is assumed to be
small so that the induced magnetic field is neglected [19]. There is no applied electric field. The MHD term is
derived from an order-of-magnitude analysis of the full Navier-Stokes equations. Under these conditions and
assuming variation of density in the body force term (Boussinesq’s approximation), the problem can be
governed by the following set of equations:
22
0
2
'' '
( ' ) ( '
'
c
B uu u
g T T g C C
t y
(1)
22
2
' ' 'p p p
T k T u Q
T T
t c y c y c
(2)
2
2
' '
C C
D
t y
(3)
with the following initial and boundary conditions:
3. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 19 | Page
' '
' ' ' '
' '
' 0, ' , ' ', ' 0
' 0, ' ( )cos ' ', ' ' 0, ' 0
' 0, ' , ' ' , ' 0
w w w
u T T C C for all y t
u T T T T w t C C at y t
u T T C C as y t
(4)
The second term of R.H.S. of the momentum equation (1) denotes buoyancy effects, the third term is
the MHD term. The second term of R.H.S. of the energy equation (2) denotes radiation term, the third term is
viscous dissipation term. The heat generation and heat due to viscous dissipation is taken into an account in
equation (2).
The temperature distribution is independent of the flow and heat transfer is by conduction alone. This is
true for fluids in initial stage due to the absence of convective heat transfer or at small Grashof number flow (Gr
1 ).
Skin – friction is given by
0
'
'
s
y
u
y
(5)
We introduce the non-dimensional variables:
2
' ' '
, , , ' , Pr ,
p R
R
R R R p
C Q Lt y u
t y u w w t Q
t L U k c
2 ' '
' '0
2 ' ' ' '
' '
, , , , ,w
R w w
B T T C C
M Sc C T T T
U D T T C C
(6)
' ' 1/3
1/3
3 2
, , ,
c w
R R
R
g C C g T
Gm U g T L
U
2
2/3 1/3
, R
R
P
U
t g T E
C T
The equations (1) - (3) reduce to following non-dimensional form:
2
2
u u
GmC Mu
t y
(7)
22
2
1
Pr
u
E Q
t y y
(8)
2
2
C C
Sc
t y
(9)
with the following initial and boundary conditions:
0, 0, 0 , 0u C for all y t (10)
0, 1 cos , 1 0, 0u t C at y t
0, 0, 0 , 0u C as y t (11)
where t is phase angle.
Skin – Friction: In non-dimensional form, the skin – friction is given by
4. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 20 | Page
0y
u
y
(12)
Nusselt Number: From temperature field, the rate of heat transfer in non-dimensional form is expressed as
0y
Nu
y
(13)
Sherwood Number: From concentration field, the rate of mass transfer in non-dimensional form is expressed as
0y
C
Sh
y
(14)
All the physical variables are defined in the nomenclature.
III. Numerical Technique
Equations (2 - 7) – (2 - 9) are coupled non-linear partial differential equations and are to be solved
under the initial and boundary conditions of equations (10) – (11). However exact or approximate solutions are
not possible for this set of equations and hence we solve these equations by the implicit finite difference method
of Crank – Nicolson’s type. The finite difference equations corresponding to equations (7) – (9) are as follows:
, 1 , 1, 1 , 1 1, 1 1, , 1,
2
2 2
2
i j i j i j i j i j i j i j i ju u u u u u u u
t y
, 1 , , 1 , , 1 ,
1
2 2 2
i j i j i j i j i j i j
Gm M
C C u u (15)
, 1 , 1, 1 , 1 1, 1 1, , 1,
2
2 21
Pr 2
i j i j i j i j i j i j i j i j
t y
2
1, ,
, 1 ,
2
i j i j
i j i j
u uQ
E
y
(16)
, 1 , 1, 1 , 1 1, 1 1, , 1,
2
2 2
2
i j i j i j i j i j i j i j i jC C C C C C C C
Sc
t y
(17)
Initial and boundary conditions take the following forms
,0 ,0 ,00 0, 0i i iu C for all i
0, 0, 0,0, 1 cos , 1j j ju t C
where L corresponds to , the suffix ' 'i corresponds to y and ' 'j corresponds to t .
Also 1j jt t t and 1i iy y y .
Here we consider a rectangular grid with grid lines parallel to the coordinate axes with spacing ∆y and
∆t in space and time directions respectively. The grid points are given by yi = i.∆y, i = 1,2,3,---,L-1 and tj =j.∆t, j
= 1,2,3,---, P. The spatial nodes on the jth
time grid constitute the jth
layer or level. The maximum value of y was
chosen as 10 after some preliminary investigations, so that the two of the boundary conditions of equation (11)
are satisfied. Here the maximum value of y corresponds to y = . After experimenting with few sets of mesh
sizes, they have been fixed at the level Δy = 0.1 and the time step Δt = 0.02, in this case, special mesh size is
reduced by 50% and the results are compared. It is observed that when mesh size is reduced by 50% in y –
direction, the result differ only in the fifth decimal place.
, , ,0, 0, 0L j L j L ju C
5. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 21 | Page
The values of ‘C, and u’ are known at all grid points at t = 0 from the initial conditions. The values of
‘C, and u’ at time level ‘j+1’ using the known values at previous time level ‘j’ are calculated as follows. The
finite difference equation (17) at every internal nodal point on a particular j- level constitutes a tri-diagonal
system of equations. Such a system of equations is solved by using Thomas algorithm as discussed in Carnahan
et al. [20]. Thus, the values of ‘C’ are known at every nodal point at (j+1)th
time level. Similarly the values of
‘’ are calculated from equation (16). Using the values of ‘C’ and ‘’ at (j+1)th
time level in equation (15), the
values of ‘u’ at (j+1)th
time level are found in similar manner. This process is continued to obtain the solution till
desired time‘t’. Thus the values of ‘C, and u’ are known, at all grid points in the rectangular region at the
desired time level.
The local truncation error is O
2 2
( )t y and it tends to zero when ‘∆t and ∆y’ tend to zero. Hence
the scheme is compatible. The finite difference scheme is unconditionally stable. Compatibility and stability
ensures the convergence of the scheme.
The derivatives involved in equations (12) and (14) are evaluated using five point approximation
formula.
IV. Results And Discussion
The convection flows driven by combinations of diffusion effects are very important in many
applications. The foregoing formulations may be analyzed to indicate the nature of interaction of the various
contributions to buoyancy. In order to gain physical insight into the problem, the value of is chosen 1.0. The
values of Prandtl number are chosen 0.71, 7 which represent air and water respectively at 200
c temperature and
1 atmospheric pressure and the values of Schmidt number are chosen to represent the presence of species by
hydrogen (0.22), water vapour (0.60), ammonia (0.78) and carbon dioxide (0.96) at 250
c temperature and 1
atmospheric pressure.
Extensive computations were performed. Default values of the thermo physical parameters are
specified as follows:
Magnetic parameter M = 2, Modified Grashof number Gc = 10, Phase angle
2
t
, Prandtl number Pr =
0.71(air), Eckert number E = 0.5, Heat generation parameter Q= 0.5, Schmidt number Sc = 0.22 (hydrogen) and
time t = 0.2. All graphs therefore correspond to these values unless otherwise indicated.
Fig (1) represents the velocity profiles due to the variations in t . It is evident from the figure that the
velocity increases and attains its maximum value in the vicinity of the plate and then tends to zero as y .
Moreover, the velocity is marginally affected by the variations in the phase angle. No reverse effect is observed
in the case of heating of the surface (Gm < 0).
Fig (2) reveals the effects of M, on the velocity profiles. It is obvious from the figure that the velocity
near the plate exceeds i.e., the velocity overshoot occurs. It is observed that an increase in the value of ‘M’ leads
to fall in the velocity. It is due to the application of transverse magnetic field will result a resistive type force
(Lorentz force) similar to drag force which tends to resist the flow and thus reducing its velocity. The presence
of a porous medium increases the resistance to flow resulting in decrease in the flow velocity. The opposite
phenomenon is observed for heating of the plate (Gm < 0).
The species concentration is coupled to the velocity via Modified Grashof number as seen in equation
(7). For various values of Modified Grashof number, the velocity profiles are plotted in figure (3). It is obvious
from the figure that the maximum velocity attains in the vicinity of the plate then decreases to zero as y .
It is observed that greater cooling of surface (an increase in Gm) results in an increase in the velocity for air. It is
due to the fact increase in the values of and mass Grashof number has the tendency to increase the mass
buoyancy effect. This gives rise to an increase in the induced flow. The reverse effect is observed in case of Gm
< 0.
Fig. 4 shows the effect of variation of Prandtl number Pr. It is noticed that the velocity decreases as Pr
increases. The velocity for Pr = 0.71 is higher than that of Pr=7. Physically, it is possible because fluids with
high Prandtl number have high viscosity and hence move slowly. It is also observed that no reverse effect for
Gm<0.
Effects of variations of time t on the velocity are presented in fig. 5. It is clear from the fig that the
velocity increases as the time t increases and the reverse effect is noticed if Gm<0.
For different values of the Schmidt number the velocity profiles are plotted in Fig. 6. It is obvious from
the figure that an increase in the Schmidt number results in decrease in the velocity. It is observed that the
reverse effect is obtained for Gm<0.
Figure (7) reveals the transient temperature profiles against y (distance from the plate). The magnitude
of temperature is maximum at the plate and then decays to zero asymptotically. The magnitude of temperature
6. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 22 | Page
for air (Pr=0.71) is greater than that of water (Pr=7). This is due to the fact that thermal conductivity of fluid
decreases with increasing ‘Pr’, resulting a decrease in thermal boundary layer thickness. Also the temperature
falls with an increase in the phase angle t for both air and water.
In figure (8) we depict the effect of Prandtl number (Pr) on the temperature field. It is observed that an
increase in the Prandtl number leads to decrease in the temperature field. Also, temperature field falls more
rapidly for water in comparison to air and the temperature curve is exactly linear for mercury (Pr = 0.025),
which is more sensible towards change in temperature. From this observation it is conclude that mercury is most
effective for maintaining temperature differences and can be used efficiently in the laboratory. Air can replace
mercury, the effectiveness of maintaining temperature changes are much less than mercury. However, air can be
better and cheap replacement for industrial purpose. This is because, either increase of kinematic viscosity or
decrease of thermal conductivity leads to increase in the value of Prandtl number (Pr). Hence temperature
decreases with increasing of Prandtl number (Pr).
The effect of Eckert number ‘E’ on the temperature is shown in fig (9). Eckert number is the ratio of
the kinetic energy of the flow to the boundary layer enthalpy difference. The effect of viscous dissipation on
flow field is to increase the energy, yielding a greater fluid temperature and as a consequence greater buoyancy
force. The increase in the buoyancy force due to an increase in the dissipation parameter enhances the
temperature. It is also observed that the magnitude of temperature for air (Pr=0.71) is greater than that of water
(Pr=7).
Effect of variations in Q on the temperature is presented in Fig. 10. It is clear from the figure that the
temperature increases as heat generation parameter increases. It is also observed that the magnitude of
temperature for air (Pr=0.71) is greater than that of water (Pr=7).
Fig (11) concerns with the effect of ‘t’ on the temperature. We see that temperature ‘θ’ increases as
time ‘t’ increases for both air (Pr = 0.71) and water (Pr = 7).
Fig (12) concerns with the effect of ‘Sc’ on the concentration. It is noted that the concentration at all
points in the flow field decreases exponentially with y and tends to zero as y . A comparison of curves in
the figure shows a decrease in concentration with an increase in Schmidt number. Physically it is true, since the
increase of ‘Sc’ means decrease of molecular diffusivity. That results in decrease of concentration boundary
layer. Hence, the concentration of species is higher for small values of Sc and lower for large values of ‘Sc’.
Fig (13) reveals the skin – friction against time t for varies values of parameters M, Gm, Sc, E, t and
Pr. It is noticed that the skin friction decreases with an increase in Eckert number, modified Grashof number and
heat generator parameter while it increase with an increase in magnetic parameter, phase angle and Schmidt
number for both air and water. The magnitude of the Skin-friction for water is greater than air and this behavior
is reversed after ‘t=0.8’.
Fig (14) depicts the Nusselt number against time‘t’ for various values of parameters ‘ t , Pr, Q, E and
Sc’. It is found that the rate of heat transfer falls with increasing t , Prandtl number, Eckert number, and heat
generation parameter while it increases with an increase in Schmidt number.
It is marked from Fig. (15) that the rate of concentration transfer increases with
increasing values of Schmidt number ‘Sc’.
V. Figures
Fig.(1): Velocity profile for different values of t
7. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 23 | Page
Fig.(2): Velocity profile for different values of M
Fig.(3): Velocity profile for different values of Gm
Fig.(4): Velocity profile for different values of Pr
Fig.(5): Velocity profile for different values of t
8. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 24 | Page
Fig.(6): Velocity profile for different values of Sc
Fig.(7): Temperature profile for different values of t
Fig.(8): Temperature profile for different values of Pr
Fig.(9): Temperature profile for different values of E
9. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 25 | Page
Fig.(10): Temperature profile for different values of Q
Fig.(11): Temperature profile for different values of t
Fig.(12): Concentration profile for different values of Sc
Fig. (13): Skin friction profile
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
Q=0,Pr=0.71
Q=2,Pr=0.71
Q=4,Pr=0.71
Q=0,Pr=07
Q=2,Pr=07
Q=4,Pr=07
10. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 26 | Page
Fig. (14): Nusselt number profile
Fig. (15): Sherwood number profile
VI. Conclusions
This paper presents a numerical study of heat generation effects on the transient hydromagnetic natural
convection flow past a vertical plate with mass diffusion and fluctuating temperature about time at the plate, by
taking into account the heat due to viscous dissipation. The governing equations are solved by an implicit finite
difference method of Crank – Nicolson type. The results are obtained for temperature, velocity, skin – friction,
Nusselt number and Sherwood number. The effects of various parameters are discussed on the flow variables
and presented by graphs.
From the present numerical investigation, following observations have been drawn:
Velocity increases and attains its maximum value in the vicinity of the plate and then tends to ‘0’ as
y→ due to the variations in phase angle (ωt), Schmidt number (Sc), time (t), Modified Grashof number
(Gm) and Prandtl number (Pr).
Velocity decreases with an increase in magnetic parameter (M), Schmidt number (Sc) and the reverse
effect is noticed for Gm<0. Also Velocity decreases with an increase in phase angle (ωt) and Prandtl
number (Pr).
Velocity increases with increasing time‘t’ and modified Grashof number (Gm) and the reverse effect is
noticed for Gm<0.
The magnitude of temperature is maximum at the plate and then decays to zero asymptotically.
Temperature falls with an increase in the phase angle (ωt) and Prandtl number (Pr) for both air and water.
The magnitude of temperature for air (Pr = 0.71) is greater than that of water (Pr = 7).
Temperature enhances due to an increase in the dissipation parameter (E), heat generating parameter (Q)
and time‘t’ for both air and water.
Concentration decreases with an increase in Schmidt number (Sc).
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t
Sh
Sc=0.22
Sc=0.60
Sc=0.78
Sc=0.96
11. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 27 | Page
Skin – friction profile for water is greater than that of air and also it decreases with an increase in Grashof
number (Gm), Eckert number (E) and heat generation parameter (Q)’ while it increases with an increase
in ‘magnetic parameter (M), phase angle (ωt) and Schmidt number (Sc)’.
It is found that the rate of heat transfer falls with increasing t , Prandtl number (Pr), Eckert number (E),
and heat generation parameter (Q) while it increases with an increase in Schmidt number (Sc).
The rate of concentration transfer increases with increasing values of Schmidt number
(Sc).
References
[1]. Vedhanayagam, M., Altenkirch, R.A. and Eichhorn, R. (1980): A transfor-mation of the boundary layer equations for free
convection flow pasta vertical flat plate with rbitrary blowing and wall temperaturevariation, Int. J. Heat Mass Transfer, 23, 1286-
1288.
[2]. Kolar, A.K. and Sastri, V.M. (1988): Free convective transpiration over a vertical plate, a numerical study, Heat and Mass Transfer,
23, 327-336.
[3]. Li Jian, Ingham ,D.B. and Pop, I. (2001): Natural convection from a vertical flat plate with a surface temperature oscillation, Int. J.
Heat Mass Transfer, 44, 2311-2322.
[4]. Siegel, R. (1958): Transient free convection from a vertical flat plate, Trans. ASME, 80, 347-359.
[5]. Goldstein, R.J. and Eckert, E.R.G. (1960): The steady and transient free convection boundary layer on a uniformly heated vertical
plate, Int. J. Heat Mass Transfer, 1, 208-218.
[6]. Raithby, G.D. and Hollands, K.G.T. (1985): Natural Convection In Hand-book of Heat Transfer Fundamentals,(Rohsenow W.M.,
Hartnett, J.D. and Ganic, E.N. (eds.)), McGraw-Hill, New York.
[7]. Das, U.N., Deka, R.K. and Soundalgekar, V.M. (1999): Transient free convection flow past an infinite vertical plate with periodic
temperature variation, Journal of Heat Transfer, Transactions of the American Society Of Mechanical Engineers, 121, 1091-1094.
[8]. Saeid, Nawaf, H. (2003): Transient free convection from vertical wall with oscillating surface temperature, Asean Journal For
Science & Technology Development, 20, 261-269.
[9]. Lin, H.T. and Yu, W.S. (1995): Combined heat and mass transfer by laminar natural convection flow from a vertical plate, Heat and
Mass Transfer, 30, 369-376.
[10]. Mongruel, A., Cloitre, M. and Allain, C. (1996) Scaling of boundary-layer flows driven by double-diffusive convection, Int. J. Heat
Mass Transfer, 39, 3899-3910.
[11]. Israel – cookey, C., Ogulu, A., Omubo – Pepple ,V.M. (2003): The influence of viscous dissipation and radiation on unsteady MHD
free convection flow past an infinite heated vertical plate in a porous medium with time depedent suction. Int. J. Heat Mass
Transfer, 46, 13, 2305 – 2311.
[12] Zueco Jordan, J. (2007): Network Simulation Method Applied to Radiation and Dissipation Effects on MHD Unsteady Free
Convection over Vertical Porous Plate. Applied Mathematical Modelling, 31, 20, 2019 – 2033.
[13]. Suneetha, S. Bhaskar Reddy, N. Ramachandra Prasad, V. (2008): the thermal radiation effects on MHD free convection flow past
an impulsively started vertical plate with variable surface temperature and concentration. Journal of Naval Architecture and Marine
engineering, 2, 57 – 70.
[14]. Hitesh Kumar (2009): Radiative Heat Transfer with Hydro magnetic flow and viscous dissipation over a stretching surface in the
presence of variable heat flux. Thermal Science 13, 2, 163 – 169.
[15]. K. Vajravelu and A. Hadjinicolaou, (1993): Heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and
internal heat generation, International Communications in Heat and Mass Transfer, 20, 3, 417–430.
[16]. B. Madhusudhana Rao, G.Viswanatha Reddy, M.C.Raju, S.V.K.Varma, (2013): MHD transient free convection and chemically
reactive flow past a porous vertical plate with radiation and temperature gradient dependent heat source in slip flow regime. IOSR
Journal of Applied Physics, 3, 6, 22-32.
[17]. A. Mamun, Z. R. Chowdhury, M. A. Azim, and M. M. Molla, (2008): MHD-conjugate heat transfer analysis for a
vertical flat plate in presence of viscous dissipation and heat generation, International Communications in Heat and Mass Transfer,
35, 10, 1275–1280.
[18]. M. A. Azim, A. A.Mamun, andM.M. Rahman,(2010): Viscous Joule heating MHD-conjugate heat transfer for a vertical flat plate in
the presence of heat generation, International Communications in Heat and Mass Transfer, 37, 6, 666–674.
[19]. Cowling ,T.G. (1957): Magnetohydrodynamics Interscience Publishers, New York.
[20]. Carnahan, B., H.A. Luther, J.O. Wilkes, (1969): Applied Numerical Methods, John Wiley & Sons, New York.
NOMENCLATURE
'u velocity component in 'x - axis
't time
0B the magnetic field component along 'y - axis
'C concentration at any point in the flow field
D mass diffusivity
pC specific heat at constant pressure
g gravitational acceleration
'T temperature of the fluid near the plate
Q dimensional heat generating/absorbing parameter
Q dimensionless heat generating/absorbing parameter
12. The Study Of Heat Generation And Viscous Dissipation On Mhd Heat And Mass Diffusion Flow Past
www.iosrjournals.org 28 | Page
k thermal conductivity of fluid
Pr Prandtl number
Gm modified Grashof number
M magnetic parameter
Sc Schmidt number
t time in dimensionless coordinate
E Eckertnumber
RL reference length
Rt reference time
u dimensionless velocity component
RU reference velocity
C dimensionless concentration
GREEK SYMBOLS
coefficient of volume of expansion
c concentration expansion coefficient
density
electrical conductivity
amplitude (constant)
kinematic viscosity
viscosity of fluid
dimensionless temperature
frequency of oscillation
SUBSCRIPTS
w conditions at the wall
conditions in the free stream