This document presents a numerical study of the effects of varying dynamic viscosity coupled with viscous energy dissipation on convective heat transfer in a fluid-filled rectangular cavity. The cavity contains a continuously moving heated horizontal plate and is bounded by fixed walls. The governing equations for momentum, energy and vorticity are solved using the finite difference method for different fluids at varying dynamic viscosities and heat capacities. The results show that increasing dynamic viscosity decreases velocity and increasing specific heat capacity affects energy distribution and heat transfer rates. These findings provide useful baseline data for manufacturing processes involving continuous moving surfaces like wire drawing and rolling.
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.
IRJET-Effect of Thermal-Diffusion, Dissipation and Chemical Reaction on Unste...IRJET Journal
This document discusses heat and mass transfer in an unsteady flow of a viscous electrically conducting fluid through a vertical channel with heat sources and traveling thermal waves on the boundaries. The governing equations for momentum, continuity, energy and diffusion are presented. Boundary conditions are specified where one wall has a traveling thermal wave and the other is maintained at a constant temperature, and both walls are at constant concentrations. Dimensionless variables are introduced and the governing equations are non-dimensionalized. The effects of parameters like the thermal and solutal Rayleigh numbers, Darcy and Prandtl numbers, internal heat generation, and chemical reaction on velocity, temperature and concentration fields are to be analyzed.
This document provides an abstract for a research article on magnetohydrodynamic mixed convection micro polar flow through a porous medium driven by a porous stretching sheet. The following key points are discussed:
- Governing partial differential equations are derived and numerically solved using finite element analysis to study the effects of various parameters on velocity, micro rotation, and temperature profiles.
- Parameters investigated include the Hartmann number, Darcy parameter, surface conditions, and their impact on the fluid flow is explored.
- The study aims to provide insights applicable to processes involving polymer sheet stretching and controlled cooling/heating of strips or filaments during manufacturing.
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
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.
TWO FLUID ELECTROMAGNETO CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN VERTICAL W...IAEME Publication
The mixture of viscous and magneto convective flow and heat transfer between a long vertical wavy wall and a parallel flat wall in the presence of applied electric field parallel to gravity , magnetic field normal to gravity in the presence of source or sink is investigated. The non-linear equations governing the flow are solved using the linearization technique. The effect of Grash of number and width ratio is to promote the flow for both open and short circuits. The effect of Hartmann number is to suppress the flow, the effect of source is to promote and the effect of sink is to suppress the velocity for open and short circuit s. Conducting ratio decreases the temperature where as width ratio increases the temperature.
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
MHD Free Convection from an Isothermal Truncated Cone with Variable Viscosity...IJERA Editor
This paper presents a study of MHD free convection flow of an electrically conducting incompressible fluid with
variable viscosity about an isothermal truncated cone in the presence of heat generation or absorption. The fluid
viscosity is assumed to vary as a inverse linear function of temperature. The non-linear coupled partial
differential equations governing the flow and heat transfer have been solved numerically by using an implicit
finite - difference scheme along with quasilinearization technique. The non-similar solutions have been obtained
for the problem, overcoming numerical difficulties near the leading edge and in the downstream regime. Results
indicate that skin friction and heat transfer are strongly affected by, both, viscosity-variation parameter and
magnetic field. In fact, the transverse magnetic field influences the momentum and thermal fields, considerably.
Further, skin friction is found to decrease and heat transfer increases near the leading edge. Also, it is found that
the direction of heat transfer gets reversed during heat generation.
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.
IRJET-Effect of Thermal-Diffusion, Dissipation and Chemical Reaction on Unste...IRJET Journal
This document discusses heat and mass transfer in an unsteady flow of a viscous electrically conducting fluid through a vertical channel with heat sources and traveling thermal waves on the boundaries. The governing equations for momentum, continuity, energy and diffusion are presented. Boundary conditions are specified where one wall has a traveling thermal wave and the other is maintained at a constant temperature, and both walls are at constant concentrations. Dimensionless variables are introduced and the governing equations are non-dimensionalized. The effects of parameters like the thermal and solutal Rayleigh numbers, Darcy and Prandtl numbers, internal heat generation, and chemical reaction on velocity, temperature and concentration fields are to be analyzed.
This document provides an abstract for a research article on magnetohydrodynamic mixed convection micro polar flow through a porous medium driven by a porous stretching sheet. The following key points are discussed:
- Governing partial differential equations are derived and numerically solved using finite element analysis to study the effects of various parameters on velocity, micro rotation, and temperature profiles.
- Parameters investigated include the Hartmann number, Darcy parameter, surface conditions, and their impact on the fluid flow is explored.
- The study aims to provide insights applicable to processes involving polymer sheet stretching and controlled cooling/heating of strips or filaments during manufacturing.
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
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.
TWO FLUID ELECTROMAGNETO CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN VERTICAL W...IAEME Publication
The mixture of viscous and magneto convective flow and heat transfer between a long vertical wavy wall and a parallel flat wall in the presence of applied electric field parallel to gravity , magnetic field normal to gravity in the presence of source or sink is investigated. The non-linear equations governing the flow are solved using the linearization technique. The effect of Grash of number and width ratio is to promote the flow for both open and short circuits. The effect of Hartmann number is to suppress the flow, the effect of source is to promote and the effect of sink is to suppress the velocity for open and short circuit s. Conducting ratio decreases the temperature where as width ratio increases the temperature.
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
MHD Free Convection from an Isothermal Truncated Cone with Variable Viscosity...IJERA Editor
This paper presents a study of MHD free convection flow of an electrically conducting incompressible fluid with
variable viscosity about an isothermal truncated cone in the presence of heat generation or absorption. The fluid
viscosity is assumed to vary as a inverse linear function of temperature. The non-linear coupled partial
differential equations governing the flow and heat transfer have been solved numerically by using an implicit
finite - difference scheme along with quasilinearization technique. The non-similar solutions have been obtained
for the problem, overcoming numerical difficulties near the leading edge and in the downstream regime. Results
indicate that skin friction and heat transfer are strongly affected by, both, viscosity-variation parameter and
magnetic field. In fact, the transverse magnetic field influences the momentum and thermal fields, considerably.
Further, skin friction is found to decrease and heat transfer increases near the leading edge. Also, it is found that
the direction of heat transfer gets reversed during heat generation.
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
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Journals
Abstract This paper examines the problem of two dimensional unsteady free convection oscillatory flow of an Elastico-viscous fluid past an impulsively started infinite vertical plate with constant heat flux and heat generation. The flow is subjected to a constant suction. Using perturbation techniquethe expressions are obtained for velocity, temperature distribution and skin friction. The results obtained are discussed for various numerical values of the parameters entered into the equations governing the flow, with help of graphs.
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.
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.
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.
Thermal radiation effects on mhd free convection flow of a micropolar fluid p...Alexander Decker
This document summarizes a study on thermal radiation effects on magnetohydrodynamic (MHD) free convection flow of a micropolar fluid past a stretching surface embedded in a non-Darcian porous medium. The governing equations for momentum, angular momentum, and energy are solved numerically. Key findings include:
1) The micropolar fluid model helps reduce drag forces and acts as a cooling agent compared to classical fluids.
2) Parameters like the Darcy number, radiation, magnetic field, and porous medium properties influence the velocity, microrotation, temperature, skin friction, and heat transfer.
3) Increased microrotation constant and coupling constant reduce skin friction but increase heat transfer.
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 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.
Effects of Variable Viscosity and Thermal Conductivity on MHD free Convection...theijes
This document summarizes a study that numerically investigates the effects of variable viscosity and thermal conductivity on magnetohydrodynamic (MHD) free convection and mass transfer flow over an inclined vertical surface in a porous medium with heat generation. The governing equations are reduced to ordinary differential equations using similarity transformations and then solved numerically using a shooting method. The results show that increasing the viscosity variation parameter, thermal conductivity parameter, magnetic parameter, permeability parameter, or Schmidt number decreases the fluid velocity, while increasing the heat generation parameter, local Grashof number, or mass Grashof number increases the fluid velocity. Skin friction, Nusselt number, and Sherwood number are also computed and presented in tabular form.
International Journal of 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.
Analysis of mhd non darcian boundary layer flow and heat transfer over an exp...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Effects of conduction on magneto hydrodynamics mixed convection flow in trian...Alexander Decker
This document summarizes research on magnetohydrodynamic (MHD) mixed convection flow in triangular enclosures. Key points:
1) The study investigates the effects of conduction on MHD mixed convection flow in triangular enclosures using a finite element method.
2) Parameters like the Hartmann number, Prandtl number, Reynolds number, and Rayleigh number are found to strongly influence the flow and thermal fields.
3) Validation of the numerical code is done by comparing average Nusselt numbers to previous research on natural convection in triangular enclosures.
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
The Effect of Hall Current on an Unsteady MHD Free Convective Flow along a Ve...iosrjce
In this paper we investigate the effect of hall current on an unsteady magnetohydrodynamic free
convective flow along a vertical porous plate with thermal radiation. The dimensionless governing coupled,
non-linear boundary layer partial differential equations are solved by an efficient finite element method for
velocity, temperature, concentration field and also the expression for shearing stress, Nusselt number and
Sherwood number are obtained. The effect of thermal Grashof Number, solutal Grashof Number, Transpiration
cooling parameter, Prandtl Number, Schmidt Number, Hartmann number, Thermal radiation parameter, Eckert
number and Hall parameter on the flow field are shown through graphs and tables.
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.
Magnetic field effect on mixed convection flow in a nanofluid under convectiv...IAEME Publication
An analysis is carried out to investigate the influence of the prominent magnetic effect on mixed convection heat and mass transfer in the boundary layer region of a semi-infinite vertical flat plate in a nanofluid under the convective boundary conditions. The transformed boundary layer,ordinary differential equations are solved numerically using Runge-Kutta Fourth order method.
This document presents a numerical solution for unsteady heat and mass transfer flow past an infinite vertical plate with variable thermal conductivity, taking into account Dufour number and heat source effects. The governing equations are non-linear and coupled, and were solved numerically using an implicit finite difference scheme. Various parameters, including Dufour number and heat source, were found to influence the velocity, temperature, and concentration profiles. Skin friction, Nusselt number, and Sherwood number were also calculated.
This document summarizes a study on mixed convective heat and mass transfer flow of a visco-elastic fluid over a stretching sheet. The study considers the effects of chemical reaction, temperature gradient dependent heat sink, and magnetic field. The governing equations for momentum, energy and diffusion are presented and non-linear similarity transformations are applied to reduce the equations to an ordinary differential equation system. The system is solved numerically using a Runge-Kutta method with shooting techniques to determine skin friction coefficient, Nusselt number, and Sherwood number.
This document presents an analysis of free convective flow and heat transfer of a viscous incompressible fluid over a linearly moving vertical porous plate with suction and viscous dissipation. The fluid viscosity is assumed to vary linearly with temperature. Governing boundary layer equations are non-dimensionalized using similarity transformations and reduced to a boundary value problem. The problem is solved numerically using a shooting method along with the 4th order Runge-Kutta method. Results for dimensionless velocity and temperature distributions are presented for various parameter values, and skin friction and Nusselt numbers are reported in tables.
The 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 results show that, with proper selection of physical parameters, significant heat transfer
enhancements and pressure drop reductions can be achieved simultaneously with porous pin fins and
the overall heat transfer performances in porous pin fin channels are much better than those in
traditional solid pin fin channels. The effects of pore density are significant. As PPI increases, the
pressure drops and heat fluxes in porous pin fin channels increase while the overall heat transfer
efficiencies decrease and the maximal overall heat transfer efficiencies are obtained at PPI 20.
Furthermore, the effects of pin fin form are also remarkable. With the same physical parameters, the
overall heat transfer efficiencies in the long elliptic porous pin fin channels are the highest while they
are the lowest in the short elliptic porous pin fin channels
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
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Journals
Abstract This paper examines the problem of two dimensional unsteady free convection oscillatory flow of an Elastico-viscous fluid past an impulsively started infinite vertical plate with constant heat flux and heat generation. The flow is subjected to a constant suction. Using perturbation techniquethe expressions are obtained for velocity, temperature distribution and skin friction. The results obtained are discussed for various numerical values of the parameters entered into the equations governing the flow, with help of graphs.
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.
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.
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.
Thermal radiation effects on mhd free convection flow of a micropolar fluid p...Alexander Decker
This document summarizes a study on thermal radiation effects on magnetohydrodynamic (MHD) free convection flow of a micropolar fluid past a stretching surface embedded in a non-Darcian porous medium. The governing equations for momentum, angular momentum, and energy are solved numerically. Key findings include:
1) The micropolar fluid model helps reduce drag forces and acts as a cooling agent compared to classical fluids.
2) Parameters like the Darcy number, radiation, magnetic field, and porous medium properties influence the velocity, microrotation, temperature, skin friction, and heat transfer.
3) Increased microrotation constant and coupling constant reduce skin friction but increase heat transfer.
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 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.
Effects of Variable Viscosity and Thermal Conductivity on MHD free Convection...theijes
This document summarizes a study that numerically investigates the effects of variable viscosity and thermal conductivity on magnetohydrodynamic (MHD) free convection and mass transfer flow over an inclined vertical surface in a porous medium with heat generation. The governing equations are reduced to ordinary differential equations using similarity transformations and then solved numerically using a shooting method. The results show that increasing the viscosity variation parameter, thermal conductivity parameter, magnetic parameter, permeability parameter, or Schmidt number decreases the fluid velocity, while increasing the heat generation parameter, local Grashof number, or mass Grashof number increases the fluid velocity. Skin friction, Nusselt number, and Sherwood number are also computed and presented in tabular form.
International Journal of 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.
Analysis of mhd non darcian boundary layer flow and heat transfer over an exp...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Effects of conduction on magneto hydrodynamics mixed convection flow in trian...Alexander Decker
This document summarizes research on magnetohydrodynamic (MHD) mixed convection flow in triangular enclosures. Key points:
1) The study investigates the effects of conduction on MHD mixed convection flow in triangular enclosures using a finite element method.
2) Parameters like the Hartmann number, Prandtl number, Reynolds number, and Rayleigh number are found to strongly influence the flow and thermal fields.
3) Validation of the numerical code is done by comparing average Nusselt numbers to previous research on natural convection in triangular enclosures.
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
The Effect of Hall Current on an Unsteady MHD Free Convective Flow along a Ve...iosrjce
In this paper we investigate the effect of hall current on an unsteady magnetohydrodynamic free
convective flow along a vertical porous plate with thermal radiation. The dimensionless governing coupled,
non-linear boundary layer partial differential equations are solved by an efficient finite element method for
velocity, temperature, concentration field and also the expression for shearing stress, Nusselt number and
Sherwood number are obtained. The effect of thermal Grashof Number, solutal Grashof Number, Transpiration
cooling parameter, Prandtl Number, Schmidt Number, Hartmann number, Thermal radiation parameter, Eckert
number and Hall parameter on the flow field are shown through graphs and tables.
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.
Magnetic field effect on mixed convection flow in a nanofluid under convectiv...IAEME Publication
An analysis is carried out to investigate the influence of the prominent magnetic effect on mixed convection heat and mass transfer in the boundary layer region of a semi-infinite vertical flat plate in a nanofluid under the convective boundary conditions. The transformed boundary layer,ordinary differential equations are solved numerically using Runge-Kutta Fourth order method.
This document presents a numerical solution for unsteady heat and mass transfer flow past an infinite vertical plate with variable thermal conductivity, taking into account Dufour number and heat source effects. The governing equations are non-linear and coupled, and were solved numerically using an implicit finite difference scheme. Various parameters, including Dufour number and heat source, were found to influence the velocity, temperature, and concentration profiles. Skin friction, Nusselt number, and Sherwood number were also calculated.
This document summarizes a study on mixed convective heat and mass transfer flow of a visco-elastic fluid over a stretching sheet. The study considers the effects of chemical reaction, temperature gradient dependent heat sink, and magnetic field. The governing equations for momentum, energy and diffusion are presented and non-linear similarity transformations are applied to reduce the equations to an ordinary differential equation system. The system is solved numerically using a Runge-Kutta method with shooting techniques to determine skin friction coefficient, Nusselt number, and Sherwood number.
This document presents an analysis of free convective flow and heat transfer of a viscous incompressible fluid over a linearly moving vertical porous plate with suction and viscous dissipation. The fluid viscosity is assumed to vary linearly with temperature. Governing boundary layer equations are non-dimensionalized using similarity transformations and reduced to a boundary value problem. The problem is solved numerically using a shooting method along with the 4th order Runge-Kutta method. Results for dimensionless velocity and temperature distributions are presented for various parameter values, and skin friction and Nusselt numbers are reported in tables.
The 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 results show that, with proper selection of physical parameters, significant heat transfer
enhancements and pressure drop reductions can be achieved simultaneously with porous pin fins and
the overall heat transfer performances in porous pin fin channels are much better than those in
traditional solid pin fin channels. The effects of pore density are significant. As PPI increases, the
pressure drops and heat fluxes in porous pin fin channels increase while the overall heat transfer
efficiencies decrease and the maximal overall heat transfer efficiencies are obtained at PPI 20.
Furthermore, the effects of pin fin form are also remarkable. With the same physical parameters, the
overall heat transfer efficiencies in the long elliptic porous pin fin channels are the highest while they
are the lowest in the short elliptic porous pin fin channels
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vibrational displacement was studied experimentally. The study was carried for a
Raghiely number between (3.77 - 10.8)*107 with applied heat flux between (20 - 45)
Watt. The vibrational experimental measurements were carried out for different
frequency ratio (0.87-1.6) and vibrational Rayleigh number ranged between (0.12 -
2.7)*107. The results of the heat transfer inside the enclosure without vibration show
a very close agreement with the published one. The vibrational heat transfer results
show that the behavior of different heat transfer convection parameters can be
affected by applying a forced vibration condition. It is shown that the high heat
transfer can be achieved at frequencies near to the system natural frequency at
constant heat flux. Also, it is concluded that a careful attention should be given to the
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This document summarizes a study on cooling power converters on trains through natural convection. It discusses a numerical analysis of the effect of discrete heat flux distributions on a vertical aluminum plate representing the train wall. The plate contains 12 resistors in 3 rows and 4 columns representing electronic components. Simulations were run to determine the influence of heat flux distribution and channel dimensions on maximum temperature. Experimental work was also conducted to validate the numerical model. In conclusion, the heat flux distribution significantly impacts heat transfer, with distributed fluxes allowing better cooling than uniform fluxes.
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ordinary differential equations by similarity transformation, before being solved numerically using Runge-Kutta-Fehlberg
method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are
presented for different values of the governing parameters. Effects of the Magnetic parameter, melting parameter, stretching
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Numerical analysis of dynamic viscosity effect associated with a continuously moving heated horizontal plate
1. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.9, 2013-Special Issue - 2nd International Conference on Engineering and Technology Research
52
Co-published with the Faculty of Engineering and Technology of Ladoke, Akintola University of Technology, Ogbomoso-Nigeria
Numerical Analysis of Dynamic Viscosity Effect Associated with a
Continuously Moving Heated Horizontal plate
Waheed Mufutau Adekojo1
Sangotayo Emmanuel Olayimika2*
1. Department of Mechanical Engineering, College of Engineering, Federal University of Agriculture, P.
M. B. 2240, Abeokuta, Nigeria.
2. Department of Mechanical Engineering, Ladoke Akintola University of Technology, P. M. B. 4000,
Ogbomoso, Nigeria.
* E-mail of the corresponding author: olemsangotayo@gmail.com
Abstract
This work presents numerical studies of the effects of varying dynamic viscosity coupled with viscous-energy
dissipation function on the convective heat transfer in a fluid-filled rectangular cavity. The cavity was filled with
fluids as quenching media.
The flow governing equations including the momentum and energy equations were solved using the finite
difference method. The study was carried out for different fluids such as oil with Prandtl number, Pr = 10, air
with Prandtl number, Pr = 0.7 and liquid metal with Prandtl number, Pr = 0.01, for various dynamic viscosity
parameters in the range 5*10-1
≤ µ ≤ 9*10-1
, and heat capacity in the range 1≤Cp≤10 in order to characterize the
nature of the flow patterns and energy distribution.
The results revealed that the dynamic viscosity has significant influence on the velocity and temperature profiles
for a particular specific heat capacity and Prandtl number higher than unity at fixed viscous dissipation. Further
results show that an increase in the dynamic viscosity for a Prandtl number greater than unity leads to a
significant decrease in the maximum velocity attainable in the cavity. It was concluded that the dynamic
viscosity and specific heat capacity have significance influence on energy distribution and the rate of heat
transfer in the enclosure. The results would be useful as baseline design data for manufacturing and material
processing industries involved with wire drawing, continuous rolling.
Keywords: Mixed Convection, Heat transfer, Dynamic Viscosity, Isotherms, Finite difference scheme
1. Introduction
Heat transfer in the boundary layer adjacent to continuous moving surfaces has many important applications in
many manufacturing processes including the cooling and drying of paper and textiles, wire drawing, continuous
casting, metal extrusion, glass fiber production and hot rolling (Sami et al., 2003; Chen, 2000; Ali and Al-
Yousef, 1998). The flow over a material moving continuously through a fluid is induced by the movement of the
solid materials and by thermal buoyancy. Hence surface motion and buoyancy effect will determine the
momentum and thermal transport processes. Thermal buoyancy effect due to the heating or cooling of a
continuously moving surface, under some circumstances may alter significantly the flow pattern, thermal field
and heat transfer behaviour in the manufacturing process (Chen, 2000).
Many researchers have investigated the effects of buoyancy force caused by continuously moving surfaces on
quiescent fluid for different orientations. The numerical simulation of thermal transport associated with a
continuously moving flat sheet in materials processing was carried out by Karwe and Jaluria (1991; 1998). Ali
and Al-Yousef (1998) examined this effect on vertical surfaces and Chen (2000) examined its effect on vertical
and inclined surfaces. They concluded that the buoyancy force has significance on the velocity and temperature
distribution and hence on the heat transfer rate from the surface. Wong (2007) investigated the effect of the
combined buoyancy- and lid-driven convection in a square cavity in which the influence of pressure on the flow
was studied. Al-Sanea and Ali (2000) investigated the effect of buoyancy parameter on moving plate in rolling
and extrusion processes. The laminar mixed convection adjacent to a vertical, continuously stretching sheet was
studied numerically by Chen (1998). Fan et al., (1997) carried out the numerical investigation of the mixed
convective heat and mass transfer over a horizontal plate. A numerical study of the flow and heat transfer
characteristics associated with a heated continuously stretching surface being cooled by a mixed convection flow
was carried out by Chen (2000). They concluded that the buoyancy force has pronounced effects on the flow
field, the local Nusselt number and friction coefficient.
In all the above cited works, the effect of viscous-energy dissipation function on heat transfer at high
temperature was not considered. However, high speed and temperature have appreciable effects on the energy
distributions and flow field (Oosthuizen and Naylor, 1999). From the findings, it is clear that the effect of
varying dynamic viscosity coupled with viscous-energy dissipation function on temperature profile, heat transfer
rate and flow field in a rectangular enclosure bounded by a continuously moving heated horizontal plate have not
been considered. This fact motivates the present study.
2. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.9, 2013-Special Issue - 2nd International Conference on Engineering and Technology Research
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The aim of this work is thus to present the effect of varying dynamic viscosity coupled with viscous-energy
dissipation function on the flow patterns, energy distribution and heat transfer rate within rectangular cavity.
2. The Physical and the Mathematical Models
Figure 1 shows a continuously moving horizontal plate emerging from a slot at a velocity Uw and temperature Tw
into an otherwise quiescent fluid. The plate forms the upper wall of the rectangular enclosure under
consideration. The enclosure is also bounded by a fixed horizontal isothermal wall on the lower part, a fixed
isothermal vertical wall bordering the extrusion die surface on the left and an adiabatic vertical wall on the right.
The temperature Tw of the upper horizontal wall is higher than that of the lower horizontal wall (i.e. Tw > T∞) as a
result of which free convective motion ensued in the enclosure.
The flow is assumed steady, incompressible, laminar, and two-dimensional and the fluid is Newtonian. The heat
transfer by radiation and the internal heat generation are assumed negligible, while viscous-energy dissipation
function effect is considered. The fluid properties are assumed independent of temperature except for the
buoyancy term in the momentum equation for which the Boussinesq approximation is used. The extrusion die
wall is stationary and impermeable for which the non-slip boundary conditions applied.
The flow governing equations at every point of the continuum comprise the expressions for the conservation of
mass, momentum and energy with viscous dissipation term included. These equations for a two-dimensional
rectangular domain are (Ozisik, 1985) :
2.1 Continuity equation:
0=
∂
∂
+
∂
∂
y
v
x
u
(1)
2.2 the Navier-Stokes equations in the x- and y-directions:
x- direction:
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
2
2
2
2
1
-
y
u
x
u
x
p
y
u
v
x
u
u ν
ρ
(2)
Y - directions:
( )∞+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
TTβg
y
v
x
v
y
p
y
v
v
x
v
u -
1
- 2
2
2
2
ν
ρ
(3)
where ( )∞T-Tgβ is the body force per unit volume in y direction.
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2.3 The thermal energy transport equation
µ2
2
2
2
φρ +
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
y
T
x
T
k
y
T
v
x
T
ucp (4)
where φ is the viscous-energy-dissipation function, defined as:
222
2
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
y
u
x
v
y
v
x
u
φ (5)
The consideration of this function becomes important if either the fluid viscosity or the flow velocities are high
[10].
The prescribed boundary conditions for the velocities and the temperature are:
;Lx0H,yatTT0,v,Uu ww ≤≤====
;Lx00,yatTT0,v0,u ≤≤==== ∞
;Hy00,xat0T0,v0,u ≤≤====
.Hy0L,xat0
x
T
x
v
x
u
≤≤==
∂
∂
=
∂
∂
=
∂
∂
(6)
3. Method of Analysis and the Solution Techniques
The Navier-Stokes equations are class of partial differential equations that could be classified as elliptic,
parabolic or hyperbolic depending on the problem under consideration. These equations in their incompressible
form can be solved by using either the vorticity-stream function approach or in their primitive-variable form. In
this work, the former approach is adopted and so equations (2) and (3) are reduced to vorticity transport equation
by eliminating the pressure gradient terms between them using the continuity equation (1), and the expression for
scalar value of the vorticity, ω, in the two-dimensional Cartesian coordinate system defined as
y
u
x
v
∂
∂
−
∂
∂
=ω . (7)
The resulting expression is the dimensional vorticity transport equation:
∂
∂
+
∂
∂
+
∂
∂
− 2
2
2
2
yxx
T
g
ωω
υβ (8)
The velocity components in a two-dimensional Cartesian coordinates are defined as the derivatives of the
stream-function, ψ , as follows:
x
v
y
u
∂
∂
−=
∂
∂
=
ψψ
, (9)
which on substitution in equation (7) gives the Poisson equation for the stream function
- 2
2
2
2
∂
∂
+
∂
∂
=
yx
ψψ
ω (10)
The derived and energy equations and the prescribed boundary conditions were cast in non-dimensional form so
that the results obtained could be generalized for a wide range of physical situations using L, (Tw - T∞), WU ,
LUW and LUW respectively for length, temperature, velocity, stream function and vorticity following (Sami
et al.,2003):
,,
,,
ww U
v
V
U
u
U
L
y
Y
L
x
X
==
==
( )
( )
,
,,
LU
LUTT
TT
w
ww
ω
ψ
θ
=Ω
=Ψ
−
−
=
∞
∞
(11)
The normalized form of the X- and Y-velocity components, stream function, vorticity and energy transport
equations are:
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Co-published with the Faculty of Engineering and Technology of Ladoke, Akintola University of Technology, Ogbomoso-Nigeria
X
V
Y
U
∂
Ψ∂
−=
∂
Ψ∂
= , (12)
2
2
2
2
YX ∂
Ψ∂
+
∂
Ψ∂
=Ω− (13)
XRe
Gr
YXReY
V
X
U
∂
∂
−
∂
Ω∂
+
∂
Ω∂
=
∂
Ω∂
+
∂
Ω∂ θ
22
2
2
2
1
(14)
2
Y
θ
X
θ
Pr
1
Y
θ
X
θ
222
2
2
2
2
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
Y
U
X
V
Y
V
X
U
Re
Ec
Re
VU
(15)
In the above equations, Ec stands for Eckert number, Re Reynolds number, Gr Grashof number, and Pr, Prandtl
number. The Eckert number relates the flow viscous-dissipation term to the energy distributions. The number is
a criterion for deciding whether the viscous-energy dissipation effect should be considered in the heat transfer
analysis or not. The Prandtl number relates the rates of diffusion of heat and momentum and it is function
specific heat capacity, Cp, dynamic viscosity, µ, and thermal conductivity, k. The Grashof number is a
dimensionless parameter representing the ratio of the buoyancy force to the viscous force in the free-convection
flow problem. It indicates whether the flow is laminar or turbulent and the dynamic process that is dominant.
The boundary conditions in non-dimensional form are:
;1X01;Yat1U
0;V0;0;
≤≤===
=≠Ψ≠Ω
θ
;X;at Y
θVU; ΨΩ
100
00
≤≤=
====≠
100
00
;Y;at X
θVU; ΨΩ
≤≤=
====≠
1Y01;Xat
0
XX
V
X
U
;0
≤≤=
=
∂
∂
=
∂
∂
=
∂
∂
=Ψ≠Ω
θ
. (16)
The vorticity and energy transport equations (14) and (15) are non-linear in character. There are no known
general analytical solutions for these coupled equations. One of the most fruitful approaches to the solution of
equations (12) – (15) is the finite difference method, which involves the approximation of each term of the
differential equations by their corresponding differential quotient. The resulting linear equations were solved
simultaneously by adopting the relaxation technique.
The convective heat transfer into the enclosure is computed in terms of the Nusselt number, which is a
dimensionless number that describes the ratio of the heat transfer by convection and conduction across the fluid
layer. The temperature gradient that would result from the heat exchange process between the fluid and the wall
can be related to the local Nusselt number, Nux, through the following expression:
1=
∂
∂
−==
Y
x
x
Yk
xh
Nu
θ
(17)
The average Nusselt number is obtained by the integration of the local Nusselt number over the entire length of
the heated wall:
∫
∂
∂
−==
=
1
0
10
dX
YQ
Q
uN
orYcond
conv θ
&
&
(18)
The flow steady state was determined by monitoring the convergence of the temperature and vortex field, using
the following criterion:
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-
2
1
2
2
n
ij
1n
ij
2
δ
φ
φφ
<
∑∑
∑∑
=
+
=
=
+
=
M
j
n
ij
N
i
M
j
N
i (19)
The parameter φ stands for Ψ , θ or Ω and n denotes the number of iterations before convergence of the
results. The value of δ used in different literatures varies between 10-3
and 10-8
(Chung, 2002)
4. Discussion of Numerically Generated Results
The effect of the convergence criterion on the numerical results was studied by computing the average Nusselt
number at different value of convergence parameter, δ, between 10-1
and 10-8
. The results which are presented
show that a value of δ = 10-4
was adequate for convergence. The results of the grid independence tests show that
a 41 by 41 grid system is sufficient for good numerical stability, field resolution and accurate results as reported
in a similar work carried out by Waheed (2009).
In order to ascertain the validity of the code used in this work and the accuracy of the present simulation, the
Nusselt number for a convective flow of a classical problem was computed for a Prandtl number, Pr = 0.7 and
Rayleigh number, Ra = 1000. The Nusselt number computed with the help of the program used in this work is
Nu = 1.1210 which compared very well with Nu = 1.132 computed by Waheed (2006) for the same Rayleigh and
Prandtl numbers with about 2% discrepancy. Further validation of the results was done as follow: for Ra = 105
and Pr = 0.7, the computed Nusselt number by Hong (1992) is Nu = 4.5885 and the one computed by Waheed
(2006) is 4.6201. The computed Nusselt number from the present simulation is 4.7438 which is in good
agreement with the above two results.
Figures 2, 3 and 4 show the temperature profiles respectively for Pr = 0.01, Pr = 0.7 and Pr = 10.0 at Eckert
number, Ec = 1.0 and Richardson number, Gr/Re2
= 1.0 for various values of dynamic viscosity, µ. It is evident
from figures 3 and 4 that an increase in the dynamic viscosity results in negligible effect on the wall temperature
gradient close to the moving plate for liquid metal and gases, and hence produces an insignificant effect in the
surface heat transfer rate. Also Figure 5 reveals that the effects of dynamic viscosity are found to be more
pronounced for a fluid with a high Prandtl number, Pr = 10.
Figure 2: Plot of Temperature, θ versus vertical coordinate, Y for viscosities; µ=0.5, µ=0.9 and Pr = 0.01
-12.0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Y coordinate
LogofTemperatureDistribution
µ = 0.5
µ = 0.9
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Figure 3: Plot of Temperature, θ versus vertical coordinate, Y for viscosities; µ=0.5, µ=0.9 and Pr = 1.0
Figure 4: Plot of Temperature, θ versus vertical coordinate, Y for various viscosity, µ, Pr = 10.0
Figure 5 shows the variations of Nusselt number with the dynamic viscosity. It is observed that for a particular
value of Pr, Pr >>1 the Nusselt number increases with the dynamic viscosity. Also an increase in the Prandtl
number results in an increase in the Nusselt number. Figure 7 shows that an increase in dynamic viscosity results
in significant enhancement in the Nusselt number at high value of specific heat capacity, Cp but it has negligible
effect at low values.
-12.0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Y coordinate
LogofTemperatureDistribution
µ = 0.5
µ = 0.9
-12.0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Y coordinate
LogofTemperaturedistribution
µ = 0.5
µ = 0.7
µ = 0.9
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Figure 5: Plot of Average Nusselt number, Nu versus viscosities, µ, for different fluids
It is clear from Figure 6 that an increase in dynamic viscosity at high Prandtl number produces a decrease in the
velocity gradient that deteriorates the momentum transport, which in turn decreases the flow rate. For liquid
metal, i.e. fluid with Pr << 1, the effect of convection on the fields is very weak. Small Pr could result from very
small momentum diffusivity (i.e. very weak convection) or very high thermal diffusivity. These diffusion rates
are precisely the quantities that determine the thickness of the boundary layers for a given external flow field.
Large momentum or thermal diffusivity means that the viscous or temperature influence is felt farther out in the
flow field. It implies that positive increase in the dynamic viscosity decelerates the flow at a particular Prandtl
number.
Figure 6: Plot of Average Nusselt number, Nu versus viscosities, µ, for heat capacities; Cp=1.0 and Cp=10.0
5. Conclusion
The flow and thermal fields along a continuously moving horizontal sheet of extruded material were studied for
different quenching media, i.e. oil, air and liquid metal with Pr = 10, 0.7 and 0.01 respectively for varying
dynamic viscosity, in order to characterize the flow patterns and energy distribution with viscous-energy
dissipation function. The numerical model based on finite difference procedure was validated with published
results. It can be concluded that the dynamic viscosity and specific heat capacity have significance influence on
energy distribution and the rate heat transfer in the enclosure. The dynamic viscosity has significance effect on
energy distribution and heat transfer rate from the surface for particular Prandtl number and specific heat
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Viscosity
AverageNusseltnumber,Nu
Liquid metals
Gasses
Oils
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Viscosity
AverageNusseltnumber,Nu
Cp = 1
Cp = 10
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capacity.
References
Ali, M.E. and Al-Yousef. F., (1998). “Laminar mixed convection from a continuously moving vertical surface
with suction or injection”. Heat and Mass Transfer, 33, pp 79-86.
Al-Sanea, S.A. and Ali, M.E., (2000). “The effect of extrusion slit on the flow and heat-transfer characteristics
from a continuously moving material with suction or injection”. International Journal of Heat and Fluid Flow,
21, pp 84-91.
Chen, C.H., (1998). “Laminar mixed convection adjacent to vertical, continuously sheet sheets”. Heat and Mass
Transfer, 33, pp 471-476.
Chen, C.H., (2000). “Mixed convection cooling of a heated, continuously stretching surface”. Heat and Mass
Transfer, 36, pp 79-86
Chung, T.J., (2002). Computational Fluid Dynamics. Cambrdge University Press, Cambridge.
Fan, J.R., Shi, J.M. and Xu, X.Z., (1997). “Similarity solution of mixed convection over a horizontal moving
plate”. Heat and Mass Transfer, 32, pp 199-206.
Karwe, M.V. and Jaluria, Y., (1991). “Numerical simulation of thermal transport associated with a continuously
moving flat sheet in material processing”. ASME Journal of Heat Transfer 113, Pp 612-619.
Karwe, M.V. and Jaluria, Y., (1998). “Fluid flow and mixed convention transport from a moving plate in rolling
and extrusion process” ASME Journal of Heat Transfer, 110, pp 625-661.
Oosthuizen, P. H. and Naylor, D. (1999). An Introduction to Convective Heat Transfer Analysis, McGraw-Hill,
New York.
Ozisik, M.N., (1985). Heat Transfer, A Basic Approach. International Textbook Company, Seraton, Pa.
Sami, A., Ali, M.E. and Al-Sanea, (2003). “Convection regimes and heat-transfer characteristics along a
continuously moving heated vertical plate”. International Journal of Heat and Fluid Flow, 24, pp 888-901.
Waheed, M.A. (2009). “Mixed convective heat transfer in rectangular enclosures driven by a continuously
moving horizontal plate”. International Journal of Heat and Mass Transfer, 52, pp. 5055–5063.
Waheed, M.A., (2006). “Temperature dependent fluid properties effect on the heat function formulation of
natural convective flow and heat transfer”. International Journal of Numerical Methods for Heat and Fluid Flow,
16(2), pp 240-257.
Wong, J.C.F., (2007). “Numerical simulation of two-dimensional laminar mixed-convection in a lid-driven
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Flow, 17(1), pp.46-93.
Nomenclature
Bf - Buoyancy force parameter ( )( )uTgReGr 3
WW
3
xx νβ ∞−== T
cp - Specific heat capacity of fluid, J/kgK
Ec - Eckert number ( )pw cTU ∆= 2
g - Acceleration due to gravity, m/s2
Gr - Grashof number, ( = ( ) 23
g νβ /LTT cw− )
Pr – Prandtl number, kCpµ
H - Height of the enclosure
h - Heat transfer coefficient
k - Fluid thermal conductivity, W/mK
L - Length of the enclosure
M - Number of horizontal grid lines
n - Number of iterations
N - Number of vertical grid lines
Nu - Average Nusselt number,( khLTkQ =∆= & )
Nux- Local Nusselt number ( )khx=
p - Pressure, N/m2
Pr- Prandtl number ( = ν / α)
Ra - Rayleigh number ( = ( ) ανβ /g 3
LTTGrPr w ∞−= )
Re - Reynolds number ( = Uw L/ν)
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Ri - Richardson number, ( ) 22
/Re ww ULTTgGr ∞−== β
T - Temperature, K
u - Horizontal velocity component, m/s
U - Dimensionless horizontal velocity component ( )wUu=
v - Vertical velocity component, m/s
V - Dimensionless vertical velocity component ( )wUv=
x - Horizontal coordinate, m
X - Dimensionless coordinate ( )Lx=
y - Vertical coordinate, m
Y - Dimensionless coordinate ( )Ly=
Greek Alphabet
α - Thermal diffusivity ( )pck ρ=
β - Volumetric thermal expansion coefficient, 1/K
δ - Convergence parameter
θ - Dimensionless temperature = ( ) ( )∞∞ −− TTTT w
µ - Dynamic viscosity, Pas
ν - Kinematic viscosity, m2
/s
ρ - Density, kg/m3
φ - Viscous-energy-dissipation function
ψ - Stream function, m2
/s
Ψ - Dimensionless Stream-function ( )( )LUWψ=
ω - Vorticity, 1/s
Ω - Dimensionless vorticity ( )WULω=
Sub- and Superscripts
w - Condition at surface
∞ - Condition at ambient medium
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