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
Non-NewtonianFluid Flow and Heat Transfer over a Non- Linearly Stretching Sur...IJERA Editor
This paper investigates the MHD flow and heat transfer of an electrically conducting non-newtonian power-law
fluid over a non-linearly stretching surface along with porous plate in porous medium. The governing equations
are reduced to non-linear ordinary differential equations by means of similarity transformations. These
equations are then solved numerically with the help ofRunge – Kutta shooting method. The effect of various
flow parameters in the form of dimensionless quantities on the flow field are discussed and presented
graphically.
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.
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.
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.
Non-Darcy Convective Heat and Mass Transfer Flow in a Vertical Channel with C...IJERA Editor
In this paper, We made an attempt to study thermo-diffusion and dissipation effect on non-Darcy convective
heat and Mass transfer flow of a viscous fluid through a porous medium in a vertical channel with Radiation and
heat sources. The governing equations of flow, heat and mass transfer are solved by using regular perturbation
method with δ, the porosity parameter as a perturbation parameter. The velocity, temperature, concentration,
shear stress and rate of Heat and Mass transfer are evaluated numerically for different variations of parameter.
Non-NewtonianFluid Flow and Heat Transfer over a Non- Linearly Stretching Sur...IJERA Editor
This paper investigates the MHD flow and heat transfer of an electrically conducting non-newtonian power-law
fluid over a non-linearly stretching surface along with porous plate in porous medium. The governing equations
are reduced to non-linear ordinary differential equations by means of similarity transformations. These
equations are then solved numerically with the help ofRunge – Kutta shooting method. The effect of various
flow parameters in the form of dimensionless quantities on the flow field are discussed and presented
graphically.
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.
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.
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.
Non-Darcy Convective Heat and Mass Transfer Flow in a Vertical Channel with C...IJERA Editor
In this paper, We made an attempt to study thermo-diffusion and dissipation effect on non-Darcy convective
heat and Mass transfer flow of a viscous fluid through a porous medium in a vertical channel with Radiation and
heat sources. The governing equations of flow, heat and mass transfer are solved by using regular perturbation
method with δ, the porosity parameter as a perturbation parameter. The velocity, temperature, concentration,
shear stress and rate of Heat and Mass transfer are evaluated numerically for different variations of parameter.
Radiation Effects on MHD Free Convective Rotating Flow with Hall EffectsIJERA Editor
In this paper, we have studied the unsteady an incompressible MHD rotating free convection flow of Viscoelastic fluid through a porous medium with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter. The expressions for the velocity, temperature, concentration have been derived analytically and also its behaviour is computationally discussed with reference to different flow parameters with the help of graphs. The skin friction, the Nusselt number and the Sherwood number are also obtained and their behaviour discussed.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
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.
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.
MHD Natural Convection Flow of an incompressible electrically conducting visc...IJERA Editor
We consider a two-dimensional MHD natural convection flow of an incompressible viscous and electrically
conducting fluid through porous medium past a vertical impermeable flat plate is considered in presence of a
uniform transverse magnetic field. The governing equations of velocity and temperature fields with appropriate
boundary conditions are solved by the ordinary differential equations by introducing appropriate coordinate
transformations. We solve that ordinary differential equations and find the velocity profiles, temperature profile,
the skin friction and nusselt number. The effects of Grashof number (Gr), Hartmann number (M) and Prandtl
number (Pr), Darcy parameter (D-1) on velocity profiles and temperature profiles are shown graphically.
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.
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
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.
Effect of Mass Transfer and Hall Current on Unsteady MHD Flow with Thermal Di...IJERA Editor
The paper investigated the effect of mass transfer and Hall current on unsteady MHD flow with Thermal Diffusivity of a viscoelastic fluid in a porous medium. The resultant equations have been solved analytically. The velocity, temperature and concentration distributions are derived, and their profiles for various physical parameters are shown through graphs. The coefficient of Skin friction, Nusselt number and Sherwood number at the plate are derived and their numerical values for various physical parameters are presented through tables. The influence of various parameters such as the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter, Hartmann number, Hall parameter, and the frequency of oscillation on the flow field are discussed. It is seen that, the velocity increases with the increase in Gc, Gr, m and K, and it decreases with increase in Sc,M, n and Pr, temperature decreases with increase in Pr and n, Also, the concentration decreases with the increase in Sc and n.
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.
MHD CASSON FLUID FLOW AND HEAT TRANSFER WITH PST AND PHF HEATING CONDITIONS D...IAEME Publication
In the present paper two-dimensional flow of non-Newtonian MHD flow of Casson fluid heat transfer with PST & PHF is considered. Using Nevier Stoke’s Equations of Motion the momentum and energy equations of Casson fluid are derived. These governing equations of motion and temperature are non-linear partial differential equations which are tedious to solve as they are, hence these partial differential equations are converted into Ordinary differential equations using suitable similarity transformations. These ODE’s are then solved numerically by the efficient Runge Kutta Fehlberg method. Effects of various governing parameters on the flow and heat transfer profiles are analyzed. Further the numerical Values of Wall temperature and Wall temperature Profiles are calculated and discussed in detail through graphs in both the cases of PST and PHF.
STEADY FLOW OF A VISCOUS FLUID THROUGH A SATURATED POROUS MEDIUM AT A CONSTAN...Journal For Research
In this paper the Steady flow of a viscous fluid through a porous medium over a fixed horizontal, impermeable and thermally insulated bottom. The flow through the porous medium satisfies the general momentum and energy equations are obtained when the temperature on the fixed bottom and on free surface prescibed. By using Galerkin Method, the expression for Velocity and Drag force are obtained. The Galerkin Method endowed with distinct features that account for its superiority over competing methods. The effect of different parameters on Velocity and Drag force are discussed with the help of graphs.
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.
NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFE...IAEME Publication
In the present paper an analysis of mixed convection of an unsteady magneto hydrodynamic (MHD) flow of an incompressible viscous fluid through porous media due to a vertical porous stretching sheet in the presence of viscous dissipation, thermal radiation and heat source /sink has been carried out. The fluid considered is viscous and incompressible. The governing partial differential equations of the flow, mass and heat transfer are highly non linear hence are converted into a system of ordinary differential equations using suitable similarity transformations. These ordinary differential equations are further converted into 7 first order ordinary differential equations and are solved numerically by Matlab ode-45 solver via shooting method.
This Slides talk about the transient state and how to solve such problems
SECTION 1
Theory
-Transient State Review
-Accumulation Concept
-General Methodology for Transient State
-Differential and Integral MB
SECTION 2
Exercises
-Mass Balances (Analytical, “by hand”)
-Mass Balances (Computer solved)
Problems and Exercises in my webpage
www.ChemicalEngineeringGuy.com
Suscribe to my channel:
www.youtube.com/ChemEngineeringGuy
Visit my Facebook Page:
www.facebook.com/Chemical.Engineering.Guy
e-Mail me:
chemical.engineering.guy@gmail.com
Radiation Effects on MHD Free Convective Rotating Flow with Hall EffectsIJERA Editor
In this paper, we have studied the unsteady an incompressible MHD rotating free convection flow of Viscoelastic fluid through a porous medium with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter. The expressions for the velocity, temperature, concentration have been derived analytically and also its behaviour is computationally discussed with reference to different flow parameters with the help of graphs. The skin friction, the Nusselt number and the Sherwood number are also obtained and their behaviour discussed.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
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.
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.
MHD Natural Convection Flow of an incompressible electrically conducting visc...IJERA Editor
We consider a two-dimensional MHD natural convection flow of an incompressible viscous and electrically
conducting fluid through porous medium past a vertical impermeable flat plate is considered in presence of a
uniform transverse magnetic field. The governing equations of velocity and temperature fields with appropriate
boundary conditions are solved by the ordinary differential equations by introducing appropriate coordinate
transformations. We solve that ordinary differential equations and find the velocity profiles, temperature profile,
the skin friction and nusselt number. The effects of Grashof number (Gr), Hartmann number (M) and Prandtl
number (Pr), Darcy parameter (D-1) on velocity profiles and temperature profiles are shown graphically.
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.
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
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.
Effect of Mass Transfer and Hall Current on Unsteady MHD Flow with Thermal Di...IJERA Editor
The paper investigated the effect of mass transfer and Hall current on unsteady MHD flow with Thermal Diffusivity of a viscoelastic fluid in a porous medium. The resultant equations have been solved analytically. The velocity, temperature and concentration distributions are derived, and their profiles for various physical parameters are shown through graphs. The coefficient of Skin friction, Nusselt number and Sherwood number at the plate are derived and their numerical values for various physical parameters are presented through tables. The influence of various parameters such as the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter, Hartmann number, Hall parameter, and the frequency of oscillation on the flow field are discussed. It is seen that, the velocity increases with the increase in Gc, Gr, m and K, and it decreases with increase in Sc,M, n and Pr, temperature decreases with increase in Pr and n, Also, the concentration decreases with the increase in Sc and n.
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.
MHD CASSON FLUID FLOW AND HEAT TRANSFER WITH PST AND PHF HEATING CONDITIONS D...IAEME Publication
In the present paper two-dimensional flow of non-Newtonian MHD flow of Casson fluid heat transfer with PST & PHF is considered. Using Nevier Stoke’s Equations of Motion the momentum and energy equations of Casson fluid are derived. These governing equations of motion and temperature are non-linear partial differential equations which are tedious to solve as they are, hence these partial differential equations are converted into Ordinary differential equations using suitable similarity transformations. These ODE’s are then solved numerically by the efficient Runge Kutta Fehlberg method. Effects of various governing parameters on the flow and heat transfer profiles are analyzed. Further the numerical Values of Wall temperature and Wall temperature Profiles are calculated and discussed in detail through graphs in both the cases of PST and PHF.
STEADY FLOW OF A VISCOUS FLUID THROUGH A SATURATED POROUS MEDIUM AT A CONSTAN...Journal For Research
In this paper the Steady flow of a viscous fluid through a porous medium over a fixed horizontal, impermeable and thermally insulated bottom. The flow through the porous medium satisfies the general momentum and energy equations are obtained when the temperature on the fixed bottom and on free surface prescibed. By using Galerkin Method, the expression for Velocity and Drag force are obtained. The Galerkin Method endowed with distinct features that account for its superiority over competing methods. The effect of different parameters on Velocity and Drag force are discussed with the help of graphs.
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.
NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFE...IAEME Publication
In the present paper an analysis of mixed convection of an unsteady magneto hydrodynamic (MHD) flow of an incompressible viscous fluid through porous media due to a vertical porous stretching sheet in the presence of viscous dissipation, thermal radiation and heat source /sink has been carried out. The fluid considered is viscous and incompressible. The governing partial differential equations of the flow, mass and heat transfer are highly non linear hence are converted into a system of ordinary differential equations using suitable similarity transformations. These ordinary differential equations are further converted into 7 first order ordinary differential equations and are solved numerically by Matlab ode-45 solver via shooting method.
This Slides talk about the transient state and how to solve such problems
SECTION 1
Theory
-Transient State Review
-Accumulation Concept
-General Methodology for Transient State
-Differential and Integral MB
SECTION 2
Exercises
-Mass Balances (Analytical, “by hand”)
-Mass Balances (Computer solved)
Problems and Exercises in my webpage
www.ChemicalEngineeringGuy.com
Suscribe to my channel:
www.youtube.com/ChemEngineeringGuy
Visit my Facebook Page:
www.facebook.com/Chemical.Engineering.Guy
e-Mail me:
chemical.engineering.guy@gmail.com
Similar to Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow over a continuous Surface in a Porous Medium with Hydromagnetic Field
Magnetohydrodynamic mixed convection flow and boundary layer control of a nan...IAEME Publication
This research work is focused on the numerical solution of steady MHD mixed convection boundary layer flow of a nanofluid over a semi-infinite flat plate with heat generation/absorption and viscous dissipation effects in the presence of suction and injection. Gyarmati’s variational principle developed on the thermodynamic theory of irreversible processes is employed to solve the problem
numerically. The governing boundary layer equations are approximated as simple polynomial functions, and the functional of the variational principle is constructed.
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.
Melting Heat Transfer in MHD Boundary Layer Stagnation-Point Flow towards a S...iosrjce
An analysis is carried out to study the MHD steady two-dimensional stagnation-point flow and heat transfer from
a warm, laminar liquid flow to a melting stretching sheet. The governing partial differential equations are transformed into
ordinary differential equations by similarity transformation, before being solved numerically using Runge-Kutta-Fehlberg
method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are
presented for different values of the governing parameters. Effects of the Magnetic parameter, melting parameter, stretching
parameter and prandtl number on the flow and heat transfer characteristics are thoroughly examined.
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.
Homotopy Analysis to Soret and Dufour Effects on Heat and Mass Transfer of a ...iosrjce
The objective of this paper is to study the Soret and Dufour effects on the free convection boundary
layer flow of an incompressible, viscous and chemically reacting fluid over a vertical plate in the presence of
viscous dissipation. The governing partial differential equations are converted to a set of ordinary differential
equations using suitable similarity transformations. The resulting equations are solved analytically using
homotopy analysis method (HAM). The convergence of obtained analytical solutions is explicitly discussed. The
effects of various parameters on dimensionless velocity, temperature and concentration profiles are discussed
with the help of graphs. The numerical values of skin friction, Nusselt number and Sherwood number for
different parameters are presented in tabular form. Our results are compared with the previously published
results and are found to be in good agreement.
International Journal of 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.
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.
MHD convection flow of viscous incompressible fluid over a stretched vertical...IJERA Editor
The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of 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 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.
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.
Boundary layer flow and heat transfer of a dusty fluid over a vertical permea...eSAT Journals
Abstract
The steady boundary layer free convective flow of a dusty fluid past a vertical permeable stretching surface is studied .The governing equations are converted into first order ordinary differential equations using similarity transformations. These equations are solved numerically by using Runge kutta forth order method. The effects of physical parameters like fluid-particle interaction parameter, local Grashof number, suction parameters, Prandtl number, radiation parameter and Eckert number on the flow and heat transfer characteristics are computed and presented graphically. Also the rate of heat transfer at the surface is discussed. The present results are compared with the previous study and there is a good agreement.
AMS classification 76T10, 76T15
Keywords: Volume fraction, Interaction parameter, Dusty fluid, Thermal radiation, suction parameter, steady flow and heat transfer, Boundary layer flow, Numerical solution.
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.
Similar to Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow over a continuous Surface in a Porous Medium with Hydromagnetic Field (18)
An Examination of Effectuation Dimension as Financing Practice of Small and M...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Does Goods and Services Tax (GST) Leads to Indian Economic Development?iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Childhood Factors that influence success in later lifeiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Emotional Intelligence and Work Performance Relationship: A Study on Sales Pe...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Customer’s Acceptance of Internet Banking in Dubaiiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study of Employee Satisfaction relating to Job Security & Working Hours amo...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Consumer Perspectives on Brand Preference: A Choice Based Model Approachiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Student`S Approach towards Social Network Sitesiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Broadcast Management in Nigeria: The systems approach as an imperativeiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study on Retailer’s Perception on Soya Products with Special Reference to T...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study Factors Influence on Organisation Citizenship Behaviour in Corporate ...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Consumers’ Behaviour on Sony Xperia: A Case Study on Bangladeshiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Design of a Balanced Scorecard on Nonprofit Organizations (Study on Yayasan P...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Public Sector Reforms and Outsourcing Services in Nigeria: An Empirical Evalu...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Media Innovations and its Impact on Brand awareness & Considerationiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Customer experience in supermarkets and hypermarkets – A comparative studyiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Social Media and Small Businesses: A Combinational Strategic Approach under t...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Secretarial Performance and the Gender Question (A Study of Selected Tertiary...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Implementation of Quality Management principles at Zimbabwe Open University (...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Organizational Conflicts Management In Selected Organizaions In Lagos State, ...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
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.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow over a continuous Surface in a Porous Medium with Hydromagnetic Field
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. II (Nov. - Dec. 2015), PP 01-06
www.iosrjournals.org
DOI: 10.9790/5728-11620106 www.iosrjournals.org 1 | Page
Similarity Solution of an Unsteady Heat and Mass Transfer
Boundary Layer Flow over a continuous Surface in a Porous
Medium with Hydromagnetic Field
Md.Jashim Uddin
Department of Electrical & Electronic Engineering,
International Islamic University Chittagong,Bangladesh
Abstract: 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.
Keywords: Boundary layer, Heat and Mass transfer, Hydromagnetic Field, Similarity solution, Unsteady
I. Introduction
The study of boundary layer flow of an electrically conducting fluid has many applications in
manufacturing and natural process which include cooling of electronic devices by fans, cooling of nuclear
reactors during emergency shutdown, cooling of an infinite metallic plate in a cooling bath, textile and paper
industries, glass-fiber production, manufacture of plastic and rubber sheets, the utilization of geothermal energy,
the boundary layer control in the field of aerodynamics, food processing, plasma studies and in the flow of
biological fluids.
The porous media heat transfer problems have numerous thermal engineering applications such as
geothermal energy recovery, crude oil extraction, thermal insulation, ground water pollution, oil
extraction, thermal energy storage, thermal insulations and flow through filtering devices. Excellent reviews on
this topic are provided in the literature by Nield and Bejan [1], Vafai [2], Ingham and Pop [3] and Vadasz [4].
Recently, Cheng and Lin [5] examined the melting effect on mixed convective heat transfer from a permeable
over a continuous Surface embedded in a liquid saturated porous medium with aiding and opposing external
flows. The unsteady boundary layer flow over a stretching sheet has been studied by Devi et al. [6], Elbashbeshy
and Bazid [7], Tsai et al. [8] and Ishak [9]. Fluid heating and cooling are important in many industries such as
power, manufacturing, transportation, and electronics. Johnson and Cheng [10] examined the necessary and
sufficient condition under which similarity solution exist for free convection boundary layer flow over a
continuous surface in porous media. Williams et al., [11] studied the unsteady free convection flow over a
vertical flat plate under the assumption of variations of the wall temperature with time and distance. They found
possible semi-similar solutions for a verity of classes of wall temperature distributions. Kumari et al., [12]
observed that the unsteadiness in the flow field was caused by the time dependent velocity of the moving sheet.
The constant temperature and the constant heat flux conditions were consideration in their investigation. An
excellent summary of applications is given by Huges and Young [13] Raptis [14] studied mathematically the
case of time varying two dimensional natural convective flow of an incompressible, electrically conducting fluid
along an infinite vertical porous plate embedded in a porous medium. Helmy[15] analyzed MHD unsteady free
convection flow past a vertical plate embedded in a porous medium. Elabashbeshy [16] studied heat and mass
transfer along a vertical plate in the presence of magnetic field. Chamkha and Khaled[17] investigated the
problem of coupled heat and mass transfer by magnetohydrodynamic free convection from an inclined plate in
the presence of internal heat generation or absorption. Transport processes through porous media play important
roles in diverse applications, such as in geothermal operations, petroleum industries, thermal insulation, design
of solid-matrix heat exchangers, chemical catalytic reactors, and many others. Bejan and Khair[18] reported on
the natural convection boundary layer flow in a saturated porous medium with combined heat and mass transfer.
Raptis A. et al [19] have discussed the importance of inertia effects for flows in porous media. Elbashbeshy and
Bazid [20] presented an exact similarity solution for momentum and heat transfer in an unsteady flow whose
motion is caused solely by the linear stretching of a horizontal continuous surface. Ali and Magyari [21]
presented the problem of unsteady fluid and heat flow induced by a submerged continuous surface while its
2. Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow over a…
DOI: 10.9790/5728-11620106 www.iosrjournals.org 2 | Page
steady motion is slowed down gradually. It will be demonstrated that the system of time-dependent governing
equations can be reduced by introducing a suitable transformation variables. Accurate numerical solutions are
generated by employing shooting method. A comprehensive parametric study is conducted and a representative
set of graphical results for the velocity and temperature profiles are reported and discussed. Computed
numerical results are plotted and the characteristics of the flow and heat transfer are analyzed in detail. Pertinent
results are displayed graphically and discussed quantitatively.
II. Mathematical Analysis
Let us consider the unsteady heat and mass transfer boundary layer flow of a viscous and
incompressible fluid through a porous medium bounded by a continuously moving surface with uniform
velocity U in the presence of uniform transverse magnetic field of strength B0. The magnetic Reynolds number
is assumed to be small so that the induced magnetic field can be neglected. Using boundary layer
approximation, the governing equations of continuity, momentum and energy under unsteady condition for the
flow and the temperature are written in usual notation as:
∂u
∂x
+
∂v
∂y
= 0 (1)
∂u
∂t
+ u
∂u
∂x
+ v
∂v
∂y
=
1
ρ
∂
∂y
(μ
∂u
∂y
) −
σe B0
2
ρ
u (2)
∂T
∂t
+ u
∂T
∂x
+ v
∂T
∂y
=
k
ρcp
∂2T
∂y2 +
μ
ρcp
(
∂u
∂y
)2
+
σe B0
2
u2
ρcp
(3)
Where ρis the density, μ be the coefficient of viscosity, σe the electrical conductivity, k thermal conductivity of
the fluid and cp the specific heat at constant pressure. The other symbols have their usual meanings.
The corresponding boundary conditions are:
u = U, v = 0 ; T = Tw at y = 0 (4)
and u = 0, T = T∞ at y = ∞; (5)
The continuity equation (1) is satisfied by introducing the stream function (x, y),
such that u =
∂ψ
∂y
, v = −
∂ψ
∂x
The momentum and energy equations (2)and (3) can be transformed to the corresponding ordinary differential
equations by introducing the following similarity transformations:
ψ = Uxν∞t f(η)
η = y
U
ν∞ xt
, θ η =
T−T∞
Tw −T∞
(6)
where is a reference kinematic viscosity.
It will be assumed that the temperature difference between the moving surface and the free stream varies as
Cxτ
.that is, Tw − T∞ = Cxτ
(7)
Where C is a constant, τ is exponent, and x is measured from the leading edge of the surface.
If U = u = ax, x = ξ, t = δ then the momentum and energy equations (2) – (3) after some simplifications,
reduce to the following forms:
f′′′
+ A1Prff′′
+ AηPrf′′
− AA1Mpf′
= 0 (8)
where Pr =
ρν∞
2μ
(Prandtl number) and Mp =
ν∞σe B0
2
μ
(Magnetic Parameter),
A1 = δ Unsteadiness parameter and A =
1
a
θ′′
+ A1Pr ηθ′
+ Afθ′
Ec + Pr AA1Mpf′
+ Af′′2
= 0 (9)
where Ec =
2μcp ξ
k
(Eckert number)
3. Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow over a…
DOI: 10.9790/5728-11620106 www.iosrjournals.org 3 | Page
The corresponding boundary conditions are:
f = 0; f′
= 1; θ = 1 as η = 0
f′
= 0; θ = 0 as η = ∞ (10)
Where the prime (′) denotes differentiation with respect to η
It is important to note that for liquids (Pr > 1.0) and for gases (Pr < 1.0).
III. Methodology And Solution Of The Problem:
We have applied free parameter method to solve governing partial differential equations to find
similarity solution. The free parameter method is the way of finding “Similarity Solution” of PDE by assuming
that the dependent variables in the equations are to express in terms “similarity parameters” known as similarity
variables, finally a single variable. Similarity variables must be constructed in such way that the number of
independent variables that occur in the equations reduced by one (at least) from the total number of the
independent variables. The governing boundary layer equations (2) - (3) subject to boundary conditions (4) and
(5) are solved numerically by using shooting method. First of all higher order non-linear differential equations
(2) -(3) are converted into simultaneous linear differential equations of first order and they are further
transformed into initial value problem by applying the shooting technique with the Runge-kutta Gill method.
The corresponding velocity and temperature profiles are shown in figures.
IV. Result And Discussion
In order to get a physical insight into the problem, a representative set of numerical results is shown
graphically in Figs.1-10, to illustrate the influence of physical parameters viz., magnetic parameterMp , Prandtl
number Pr Eckert number Ec ,Unsteadiness parameter and variation (exponent) τ on the velocity f′
and
temperature θ. The profiles for velocity and temperature are shown in fig.1 to fig. 10.
Effect for velocity profile:
4.1 Effect of the Prandtl number:
From the fig.1 it is observed that the velocity f′
(0) decreases as the power-law index of the surface temperature
variation (exponent) τ and the magnetic parameter(Mp) increases with Prandtl number Pr = 0.05
4.2 Effect of the viscosity Parameter:.
It is seen from fig.1 that the velocity decreases as the power-law index of the surface temperature variation τ and
the magnetic parameter (Mp) increases with the variable viscosity parameter θ = 3.0
4.3 Effect of the Eckert number:
From the fig.2-3, it is observed that the velocity profiles are almost identical for different values of temperature
variation τ and the magnetic parameter (Mp) with Eckert number Ec = 0.0 andEc = 0.75
4.4 Effect of the Magnetic Parameter:
In fig.4 it is observed that the velocity decreases as magnetic parameter (Mp) increases.
Effect for temperature profile:
4.5 Effect of variation (exponent):
From the fig.5 it is observed that as variation (exponent) τ increases the temperature decreases for fixed value of
magnetic parameter (Mp).
4.6 Effect of the Magnetic Parameter:
It is seen from fig.6-7 that the temperature decreases as magnetic parameter Mp increases.
4.7 Effect of the Prandtl and Eckert number:
From the fig.8 it is observed that the temperature decreases as the magnetic parameter increases with Pr=0.05,
τ = 0.3 and Ec = 0.75
4.8 Effect of Heat transfer:
It is seen from fig.9 that as Mp increases the heat transfer rate - θ′
(0), decreases but as τ increases the heat
transfer rate increases.That is why, the parameters τ and Mp have considerable influence on the heat transfer rate
- θ′
(0).
4.9 Effect of the Unsteadiness Parameter:
In the fig.10(a), it is observed that the velocity profiles are approximately symmetrical for different values of
unsteadiness parameter𝐴1, temperature variation τ and the magnetic parameter (Mp) with Pr = 0.05, Eckert
number Ec = 0.75,𝐴1 = 0.5.
It is seen from fig.10(b) that as variation (exponent) τ increases the temperature decreases for fixed value of
magnetic parameter (Mp) and unsteadiness parameter𝐴1with Pr = 0.05, 𝐴1 = 0.5 and θ = 3.0
ESULTS AND DISC
4. Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow over a…
DOI: 10.9790/5728-11620106 www.iosrjournals.org 4 | Page
5. Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow over a…
DOI: 10.9790/5728-11620106 www.iosrjournals.org 5 | Page
.
6. Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow over a…
DOI: 10.9790/5728-11620106 www.iosrjournals.org 6 | Page
V. Conclusion
An analysis is here made of unsteady laminar boundary Layer flow for heat and mass transfer over a
continuous surface for magnetic field under which similarity solution are possible. The effects of power law
index of the surface temperature variation (exponent), magnetic parameter and variable parameter on a
hydromagnetic flow and heat transfer on a continuously moving surface have been studied numerically using the
the shooting technique. The magnetic field stabilizes the flow which in turn delays the boundary layer separation
from the continuous surface in a porous medium. The effect of the Eckert number Ec , variation (exponent) τ,
unsteadiness parameter A1 and Prandtl number Pr on the heat transfer were studied. From the present numerical
investigations the following majors conclusions may be drawn:
i. Velocity and temperature in the unsteady case is observed to be lesser than those of the steady case.
ii. It is observed that an increment in unsteadiness parameter increases the Prandtl number and decreases the
Eckert number
iii. The velocity decreases with the increase of power law index of the surface temperature variation
(exponent) and the magnetic parameter.
iv. The temperature decreases with the increase of the power law index of the surface temperature variation
(exponent) and the magnetic parameter .
v. The heat transfer rate increases rapidly with the increase of power law index of the surface temperature
variation (exponent) whereas when the magnetic parameter increases the heat transfer rate decreases.
vi. Temperature decreases with an increasing in the value of unsteadiness parameter A1.
vii. Increasing the Prandtl number leads to a decrease in the surface temperature.
References
[1]. Nield, D.A. and A. Bejan, 2006. Convection in Porous Media (3rd Edn.). Springer, New York.
[2]. Vafai, K. 2005. Handbook of Porous Media (2nd
Edn.). Taylor & Francis, New York.
[3]. Ingham, D.B. and I. Pop, 2005. Transport Phenomena in Porous Media III. Elsevier, Oxford.
[4]. Vadasz, P., 2008. Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York.
[5]. Cheng, W.T. and C.H. Lin, 2007. Melting effect on mixed convective heat transfer with aiding and opposing external flows from
the vertical plate in a liquidsaturated porous medium. Int. J. Heat MassTransfer 50: 3026-3034.
[6]. C. D. S. Devi, H. S. Takhar, G. Nath, “Unsteady mixed convection flow in stagnation region adjacent to a vertical surface,” Heat
Mass Transfer, 26(1991) 71-79.
[7]. E. M. A. Elbashbeshy, M. A. A. Bazid, “Heat transfer over an unsteady stretching surface,” Heat Mass Transfer, 41(2004) 1-4.
[8]. R. Tsai, K. H. Huang, J. S. Huang, “Flow and heat transfer over an unsteady stretching surface with non-uniform heat source,” Int.
Commun. Heat Mass Transfer, 35(2008) 1340-1343.
[9]. A. Ishak, Unsteady, “MHD flow and heat transfer over a stretching plate,” J. Appied Sci. 10(18)(2010) 2127-2131.
[10]. Johnson C. H. and Cheng P. 1978. Possible similarity solutions for free convection boundary Layers adjacent to flat plates in porous
media. Int. J. Heat Mass Transfer. 21: 709-718.
[11]. Williams J. C., Mulligan J. C. and Rhyne T. B. 1987. Semi-similar solutions for unsteady free convection boundary Layer flow on a
vertical flat plate. 175: 309-332.
[12]. Kumari M., Slaouti A., Nakamura S., Takhar H. S. and Nath G. 1986. Unsteady free convection flow over a continuous moving
vertical surface. Acta Mechanica. 116: 75-82.
[13]. Huges W.F. and Young F.J. The Electro-Magneto Dynamics of fluids, John Wiley and Sons, New York, (1996).
[14]. Raptis A. Flow through a porous medium in the presence of magnetic field, Int. J. Energy Res., Vol.10, (1986)
[15]. Helmy K.A.. MHD unsteady free convection flow past a vertical porous plate, ZAMM, Vol.78, (1998).
[16]. Elabashbeshy E.M.A.. Heat and mass transfer along a vertical plate with variable temperature and concentration in the presence of
magnetic filed, Int. J. Eng. Sci., Vol. 34, (1997).
[17]. Chamkha A.J. and Khaled A.R.A.. Similarity solutions for hydromagnetic simultaneous heat and mass transfer by natural
convection from an inclined plate with internal heat generation or absorption, Heat Mass Transfer, Vol. 37, (2001).
[18]. Bejan A. and Khair K.R.. Heat and mass transfer by natural convection in a porous medium, Int.Commun. Heat Mass Transfer, Vol.
28, (1985).
[19]. Raptis A., Tzivanidis G. and Kafousias N. Free convection and mass transfer flow through a porous medium bounded by an infinite
vertical limiting surface with constant suction, Letter Heat Mass Transfer, Vol. 8, (1981).
[20]. E.M.A. Elbashbeshy, M.A.A. Bazid, Heat transfer over an unsteady continuous surface, Heat Mass Transfer 41 (2004) 1–4.
[21]. M.E. Ali, E. Magyari, Unsteady fluid and heat flow induced by a submerged continuous surface while its steady motion is slow
down gradually, Int. J. Heat Mass Transfer 50 (2007) 188–195.