1) The document analyzes heat transfer in a viscous, free convective MHD flow through a porous medium past a vertical porous plate with variable temperature.
2) Governing equations are derived for the flow considering various parameters like porosity, magnetic field, temperature variation in space and time.
3) Solutions are obtained using perturbation methods by decomposing the velocity and temperature into power series of the Eckert number. Expressions for velocity, temperature and skin friction are derived.
Thermal instability of incompressible non newtonian viscoelastic fluid with...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.
Mathematical Modeling of Quasi Static Thermoelastic Transient behavior of thi...IJERA Editor
The present paper deals with the determination of displacement and thermal transient stresses in a thick circular plate with internal heat generation. External arbitrary heat supply is applied at the upper surface of a thick circular plate, whereas the lower surface of a thick circular plate is insulated and heat is dissipated due to convection in surrounding through lateral surface. Here we compute the effects of internal heat generation of a thick circular plate in terms of stresses along radial direction. The governing heat conduction equation has been solved by using integral transform method. The results are obtained in series form in terms of Bessel’s functions and the results for temperature change and stresses have been computed numerically and illustrated graphically.
Effects Of Heat Source And Thermal Diffusion On An Unsteady Free Convection F...IOSR Journals
The present analysis is made to investigate the effects of heat source and thermal diffusion on an unsteady free convection flow along a porous vertical plate in a rotating system. The plate is subjected to constant heat and mass flux also. The problem is solved analytically and expressions for velocity. Energy and temperature profiles, skin friction and Nusselt number are obtained. The effects of different parameter entered in the problem are discussed on the primary and secondary velocities, temperature and concentration distributions, primary and secondary skin frictions and Nusselt number with the help of tables and graphs.
Thermal instability of incompressible non newtonian viscoelastic fluid with...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.
Mathematical Modeling of Quasi Static Thermoelastic Transient behavior of thi...IJERA Editor
The present paper deals with the determination of displacement and thermal transient stresses in a thick circular plate with internal heat generation. External arbitrary heat supply is applied at the upper surface of a thick circular plate, whereas the lower surface of a thick circular plate is insulated and heat is dissipated due to convection in surrounding through lateral surface. Here we compute the effects of internal heat generation of a thick circular plate in terms of stresses along radial direction. The governing heat conduction equation has been solved by using integral transform method. The results are obtained in series form in terms of Bessel’s functions and the results for temperature change and stresses have been computed numerically and illustrated graphically.
Effects Of Heat Source And Thermal Diffusion On An Unsteady Free Convection F...IOSR Journals
The present analysis is made to investigate the effects of heat source and thermal diffusion on an unsteady free convection flow along a porous vertical plate in a rotating system. The plate is subjected to constant heat and mass flux also. The problem is solved analytically and expressions for velocity. Energy and temperature profiles, skin friction and Nusselt number are obtained. The effects of different parameter entered in the problem are discussed on the primary and secondary velocities, temperature and concentration distributions, primary and secondary skin frictions and Nusselt number with the help of tables and graphs.
The International Institute for Science, Technology and Education (IISTE) , International JoThe International Institute for Science, Technology and Education (IISTE) , International Journals Call for papaers: http://www.iiste.org/Journalsurnals Call for papaers: http://www.iiste.org/Journals
EFFECT OF SLIP PARAMETER OF A BOUNDARY-LAYER FLOW FOR NANOFLUID OVER A VERTIC...IAEME Publication
In this paper we analyze the effect of momentum slip, thermal slip and solutal slip on stagnation point flow of MHD nanofluid towards stretching sheet .The governing partial differential equation of flow, heat and mass transfer on considered flow are converted into the ordinary differential equations by means of similarity trans formations .The resulting equations are solved by the Runge-Kutta fourth order method with efficient shooting technique. Effects of various governing parameters on flow, heat and mass transfer are studied through the plots. The various numerical tables which are calculated and tabulated. A comparison of our present results with a previous study has been done and we found that an excellent agreement is there with the earlier results and of ours.
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
In this paper, first we evaluate a finite
integral involving general class of polynomials and the
product of two
H
-functions and then we make its
application to solve boundary value problem on heat
conduction in a rod under the certain conditions and
further we establish an expansion formula involving
about product of
H
-function. In view of generality of
the polynomials and products of
H
-function occurring
here in, on specializing the coefficients of polynomials
and parameters of the
H
-function, our results would
readily reduce to a large number of results involving
known class of polynomials and simpler functions.
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.
Analysis and Design of One Dimensional Periodic Foundations for Seismic Base ...IJERA Editor
Periodic foundationis a new type of seismic base isolation system. It is inspired by the periodic material crystal
lattice in the solid state physics. This kind of material has a unique property, which is termed as frequency band
gap that is capable of blocking incoming waves having frequencies falling within the band gap. Consequently,
seismic waves having frequencies falling within the frequency band gap are blocked by the periodic foundation.
The ability to block the seismic waveshas put this kind of foundation as a prosperous next generation of seismic
base isolators. This paper provides analytical study on the one dimensional (1D) type periodic foundations to
investigate their seismic performance. The general idea of basic theory of one dimensional (1D) periodic
foundations is first presented.Then, the parametric studies considering infinite and finite boundary conditions are
discussed. The effect of superstructure on the frequency band gap is investigated as well. Based on the analytical
study, a set of equations is proposed for the design guidelines of 1D periodic foundations for seismic base
isolation of structures.
Three dimensional static analysis of two dimensional functionally graded platesijmech
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated. The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it shows very good agreement.
Three dimensional static analysis of two dimensional functionally graded platesrtme
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional
theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The
effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated.
The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it
shows very good agreement
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
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Study of Instability of Streaming Rivlin-Ericksen Fluid in Porous MediumIOSR Journals
There are many elastoviscous fluids that can be characterised neither by Maxwell’s constitutive relations nor by Oldroyd’s constitutive relations. One such class of viscoelastic fluids is the Rivlin– Ericksen’s fluid. RIVLIN and ERICKSEN] have proposed a theoretical model for such viscoelastic fluid. The behaviour of surface waves propagating between two Rivlin–Ericksen elastico-viscous fluids is examined. The investigation is made in the presence of a vertical electric field and a relative horizontal constant velocity. The influence of both surface tension and gravity force is taken into account. Due to the inclusion of streaming flow a mathematical simplification is considered. The viscoelastic contribution is demonstrated in the boundary conditions. From this point of view the approximation equations of motion are solved in the absence of viscoelastic effects. The solutions of the linearized equations of motion under nonlinear boundary conditions lead to derivation of a nonlinear equation governing the interfacial displacement and having damping terms with complex coefficients.
Effects of Variable Fluid Properties and MHD on Mixed Convection Heat Transfe...IOSR Journals
The effects of variable Fluid Properties like variation of permeability, porosity, thermal conductivity and magnetic field on Mixed Convection Heat transfer from Vertical Heated Plate Embedded in a Sparsely Packed Porous Medium have been approached numerically. The boundary layer flow in the porous medium is governed by Lapwood – Forchheimer – Brinkman extended Darcy model and the Lorentz force. The natures of these equations are highly non-linear and coupled each other. The non-linear differential equations are non-dimensionalised using the non-dimensional parameter involving Grashoff number Gr, Prandtl number Pr, Hartmann number M, Eckert number E and so on. Similarity transformations are employed and the resulting ordinary differential equations are solved numerically by using shooting algorithm with Runge – Kutta and Newton – Raphson method to obtain velocity and temperature distributions. Besides, skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence in decreasing the flow field, whereas its influence is reversed on the rate of heat transfer for all values of permeability considered. The effect of Magnetic field is diminution with velocity of the fluid flow. Further, the obtained results under the two limiting conditions were found to be in good agreement with the existing results
The International Institute for Science, Technology and Education (IISTE) , International JoThe International Institute for Science, Technology and Education (IISTE) , International Journals Call for papaers: http://www.iiste.org/Journalsurnals Call for papaers: http://www.iiste.org/Journals
EFFECT OF SLIP PARAMETER OF A BOUNDARY-LAYER FLOW FOR NANOFLUID OVER A VERTIC...IAEME Publication
In this paper we analyze the effect of momentum slip, thermal slip and solutal slip on stagnation point flow of MHD nanofluid towards stretching sheet .The governing partial differential equation of flow, heat and mass transfer on considered flow are converted into the ordinary differential equations by means of similarity trans formations .The resulting equations are solved by the Runge-Kutta fourth order method with efficient shooting technique. Effects of various governing parameters on flow, heat and mass transfer are studied through the plots. The various numerical tables which are calculated and tabulated. A comparison of our present results with a previous study has been done and we found that an excellent agreement is there with the earlier results and of ours.
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
In this paper, first we evaluate a finite
integral involving general class of polynomials and the
product of two
H
-functions and then we make its
application to solve boundary value problem on heat
conduction in a rod under the certain conditions and
further we establish an expansion formula involving
about product of
H
-function. In view of generality of
the polynomials and products of
H
-function occurring
here in, on specializing the coefficients of polynomials
and parameters of the
H
-function, our results would
readily reduce to a large number of results involving
known class of polynomials and simpler functions.
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.
Analysis and Design of One Dimensional Periodic Foundations for Seismic Base ...IJERA Editor
Periodic foundationis a new type of seismic base isolation system. It is inspired by the periodic material crystal
lattice in the solid state physics. This kind of material has a unique property, which is termed as frequency band
gap that is capable of blocking incoming waves having frequencies falling within the band gap. Consequently,
seismic waves having frequencies falling within the frequency band gap are blocked by the periodic foundation.
The ability to block the seismic waveshas put this kind of foundation as a prosperous next generation of seismic
base isolators. This paper provides analytical study on the one dimensional (1D) type periodic foundations to
investigate their seismic performance. The general idea of basic theory of one dimensional (1D) periodic
foundations is first presented.Then, the parametric studies considering infinite and finite boundary conditions are
discussed. The effect of superstructure on the frequency band gap is investigated as well. Based on the analytical
study, a set of equations is proposed for the design guidelines of 1D periodic foundations for seismic base
isolation of structures.
Three dimensional static analysis of two dimensional functionally graded platesijmech
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated. The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it shows very good agreement.
Three dimensional static analysis of two dimensional functionally graded platesrtme
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional
theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The
effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated.
The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it
shows very good agreement
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
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Study of Instability of Streaming Rivlin-Ericksen Fluid in Porous MediumIOSR Journals
There are many elastoviscous fluids that can be characterised neither by Maxwell’s constitutive relations nor by Oldroyd’s constitutive relations. One such class of viscoelastic fluids is the Rivlin– Ericksen’s fluid. RIVLIN and ERICKSEN] have proposed a theoretical model for such viscoelastic fluid. The behaviour of surface waves propagating between two Rivlin–Ericksen elastico-viscous fluids is examined. The investigation is made in the presence of a vertical electric field and a relative horizontal constant velocity. The influence of both surface tension and gravity force is taken into account. Due to the inclusion of streaming flow a mathematical simplification is considered. The viscoelastic contribution is demonstrated in the boundary conditions. From this point of view the approximation equations of motion are solved in the absence of viscoelastic effects. The solutions of the linearized equations of motion under nonlinear boundary conditions lead to derivation of a nonlinear equation governing the interfacial displacement and having damping terms with complex coefficients.
Effects of Variable Fluid Properties and MHD on Mixed Convection Heat Transfe...IOSR Journals
The effects of variable Fluid Properties like variation of permeability, porosity, thermal conductivity and magnetic field on Mixed Convection Heat transfer from Vertical Heated Plate Embedded in a Sparsely Packed Porous Medium have been approached numerically. The boundary layer flow in the porous medium is governed by Lapwood – Forchheimer – Brinkman extended Darcy model and the Lorentz force. The natures of these equations are highly non-linear and coupled each other. The non-linear differential equations are non-dimensionalised using the non-dimensional parameter involving Grashoff number Gr, Prandtl number Pr, Hartmann number M, Eckert number E and so on. Similarity transformations are employed and the resulting ordinary differential equations are solved numerically by using shooting algorithm with Runge – Kutta and Newton – Raphson method to obtain velocity and temperature distributions. Besides, skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence in decreasing the flow field, whereas its influence is reversed on the rate of heat transfer for all values of permeability considered. The effect of Magnetic field is diminution with velocity of the fluid flow. Further, the obtained results under the two limiting conditions were found to be in good agreement with the existing results
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.
Chemical Reaction Effects on Free Convective Flow of a Polar Fluid from a Ver...IOSR Journals
This article deals with a study of two dimensional free convective flow of a polar fluid through a porous medium due to combined effects of thermal and mass diffusion in presence of a chemical reaction of first order. The objective of the present investigation is to analyze the free convective flow in the presence of prescribed wall heat flux and mass flux condition. The governing partial differential equations are non-dimensionalized and transformed into a system of non-similar equations. The resulting coupled nonlinear partial differential equations are solved numerically under appropriate transformed boundary conditions using an implicit finite difference scheme in combination with quasilinearisation technique. Computations are performed for a wide range of values of the various governing flow parameters of the velocity, angular velocity, temperature and species concentration profiles and results are presented graphically. The numerical results for local skin friction coefficient, couple stress coefficient, local Nusselt number and local Sherwood number are also presented. The obtained results are compared with previously published work and were to be in excellent agreement. The study reveals that the flow characteristics are profoundly influenced by the polar effects
Amplitude Effects on Natural Convection in a Porous Enclosure having a Vertic...IOSR Journals
Numerical investigation of unsteady natural convection flow through a fluid-saturated porous
medium in a cubic enclosure which is induced by time-periodic variations in the surface temperature of a
vertical wall was considered. The governing equations were written under the assumption of Darcy-law and
then solved numerically using finite difference method. The problem is analyzed for different values of
amplitude a in the range 0.2 ≤ a ≤ 0.8, the Rayleigh number, Ra=200, Period, τ = 0.01, time, 0 ≤ t ≤ 0.024. It
was found that heat transfer increases with increasing the amplitude. The location of the maximum fluid
temperature moves with time according to the periodically changing heated wall temperature. Two main cells
rotating in opposite direction to each other were observed in the cavity for all values of the parameters
considered. The amplitude of Nusselt number increases with the increase in the oscillating amplitude. All the
results of the problem were presented in graphical form and discussed.
Asphaltic Material in the Context of Generalized Porothermoelasticity ijsc
In this work, a mathematical model of generalized porothermoelasticity with one relaxation time for poroelastic half-space saturated with fluid will be constructed in the context of Youssef model (2007). We will obtain the general solution in the Laplace transform domain and apply it in a certain asphalt material which is thermally shocked on its bounding plane. The inversion of the Laplace transform will be obtained numerically and the numerical values of the temperature, stresses, strains and displacements will be illustrated graphically for the solid and the liquid.
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.
ASPHALTIC MATERIAL IN THE CONTEXT OF GENERALIZED POROTHERMOELASTICITYijsc
In this work, a mathematical model of generalized porothermoelasticity with one relaxation time for
poroelastic half-space saturated with fluid will be constructed in the context of Youssef model (2007). We
will obtain the general solution in the Laplace transform domain and apply it in a certain asphalt material
which is thermally shocked on its bounding plane. The inversion of the Laplace transform will be obtained
numerically and the numerical values of the temperature, stresses, strains and displacements will be
illustrated graphically for the solid and the liquid.
Thermal instability of incompressible non newtonian viscoelastic fluid with...eSAT Journals
Abstract The thermal convection of incompressible Walters’B′ Rotating viscoelastic fluid is considered in the presence of uniform vertical magnetic field. By applying normal mode analysis method, the dispersion relation has been derived and solved analytically. For stationary convection, Walters’B′ viscoelastic fluid behaves like an ordinary (Newtonian) fluid. It is studied that rotation has a stabilizing effect, whereas the magnetic field has both stabilizing and destabilizing effects. The rotation and magnetic field are found to introduce oscillatory mode in the system which were non – existent in their absence. The results are presented through graphs. Key words: Walters’B′ Fluid, thermal convection, viscoelasticity, Rotation, Magnetic field.
Effects of Hall and thermal on MHD Stokes’ second problem for unsteady second...IJERA Editor
In this paper, we investigated the combined effects of Hall and thermal on MHD Stokes’ second problem for
unsteady second grade fluid flow through porous medium. The expressions for the velocity field and the
temperature field are obtained analytically. The effects of various pertinent parameters on the velocity field and
temperature field are studied in detail with the aid of graphs.
Magneto convective flowand heat transfer of two immiscible fluids between ver...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.
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.
nternational Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
Magnetohydrodynamic Rayleigh Problem with Hall Effect in a porous PlateIJERA Editor
This paper gives very significant analytical and numerical results to the magnetohydrodynamic flow version of
the classical Rayleigh problem including Hall Effect in a porous plate. An exact solution of the MHD flow of
incompressible, electrically conducting, viscous fluid past an uniformly accelerated and insulated infinite porous
plate has been presented. Numerical values for the effects of the Hall parameter N, the Hartmann number M and
the Porosity parameter P0 on the velocity components u and v are tabulated and their profiles are shown
graphically. The numerical results show that the velocity component u and v increases with the increase of N,
decreases with the increase of P0 and u decreases and v increases with the increase of M.
Thermal Effects in Stokes’ Second Problem for Unsteady Second Grade Fluid Flo...IOSR Journals
In this paper, we investigated the effects of magnetic field and thermal in Stokes’ second problem for unsteady second grade fluid flow through a porous medium. The expressions for the velocity field and the temperature field are obtained analytically. The effects of various pertinent parameters on the velocity field and temperature field are studied through graphs in detail.
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.
Similar to Heat transfer in viscous free convective fluctuating mhd flow through porous media past a vertical porous plate with variable temperature (20)
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
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Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
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The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
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Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Neuro-symbolic is not enough, we need neuro-*semantic*
Heat transfer in viscous free convective fluctuating mhd flow through porous media past a vertical porous plate with variable temperature
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.6, 2012
Heat Transfer in Viscous Free Convective Fluctuating MHD Flow
Through Porous Media Past a Vertical Porous Plate with Variable
Temperature
S.R.Mishra, G.C.Dash and M.Acharya
Department of Mathematics,
Institute of Technical Education and Research,
Siksha ‘O’ Anusandhan University, Khandagiri,
Bhubaneswar-751030, Orissa, INDIA
E-mail: satyaranjan_mshr@yahoo.co.in
Abstract
In this paper, free convective Magnetohydrodynamics (MHD) flow of a viscous incompressible and
electrically conducting fluid past a hot vertical porous plate embedded in a porous medium has been studied. The
temperature of the plate varies both in space and time. The main objective of this paper is to study the effect of
porosity of the medium coupled with the variation of plate temperature with regards to space and time. The effect of
pertinent parameters characterizing the flow has been presented through the graph. The most interesting finding is
that presence of porous media has no significant contribution to the flow characteristics. Further, heating and cooling
of the plate due to convective current is compensated by the viscous dissipation.
Keywords: MHD flow/Span wise Co-sinusoidal/Free Convection /Heat transfer, porous medium.
1. Introduction
Many Industrial applications use Magnetohyrodynamics (MHD) effects to resolve the complex problems
very often occurred in industries. The available hydrodynamics solutions include the effects of magnetic field which
is possible as because the most of the industrial fluids are electrically conducting. For example, liquid metal MHD
takes its root in hydrodynamics of incompressible media which gains importance in the metallurgical industry,
Nuclear reactor, sodium cooling system ,every storage and electrical power generation[1974-76].Free convective
flows are of great interest in a number of industrial applications such as fiber and granular insulation, geothermal
system etc. Buoyancy is also of importance in an environment where difference between land and air temperature
can give rise to complicated flow patterns. The unsteady free convection flow past an infinite porous plate and
semi-infinite plate were studied by Nanda and Sharma [1962] .In their first paper they assumed the suction velocity
−1/2
at the plate varying in time as t ,where as in the second paper the plate temperature was assumed to oscillate in
time about a constant non-zero mean. Free convective flow past a vertical plate has been studied extensively by
Ostrach [1953] and many others. The free convective heat transfer on vertical semi-infinite plate was investigated by
Berezovsky [1977]. Martynenko et al. [1984] investigated the laminar free convection from a vertical plate.
The basic equations of incompressible MHD flow are non-linear. But there are many interesting cases where the
equations became linear in terms of the unknown quantities and may be solved easily. Linear MHD problems are
accessible to exact solutions and adopt the approximations that the density and transport properties be constant. No
fluid is incompressible but all may be treated as such whenever the pressure changes are small in comparison with
the bulk modulus. Mention may be made to the works in [1975, 72]. Ferdows et al. [2004] analysed free convection
flow with variable suction in presence of thermal radiation. Alam et al. [2006] studied Dufour and Soret effect with
variable suction on unsteady MHD free convection flow along a porous plate. Majumdar et al. [2007] gave an exact
solution for MHD flow past an impulsively started infinite vertical plate in the presence of thermal radiation.
Muthucumaraswamy et al. [2009] studied unsteady flow past an accelerated infinite vertical plate with variable
temperature and uniform mass diffusion. Recently, Dash et al[2009] have studied free convective MHD flow of a
visco-elastic fluid past an infinite vertical porous plate in a rotating frame of reference in the presence of
chemical reaction. In the present study we have set the flow through porous media with uniform porous matrix with
suction and blowing at the plate surface besides the free convective MHD effects and fluctuating surface temperature.
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From the established result it is clear that the suction prevents the imposed non-torsional oscillations spreading away
from the oscillating surface (disk) by viscous diffusion for all values of frequency of oscillations. On the contrary the
blowing promotes the spreading of the oscillations far away from the disk and hence the boundary layer tends to be
infinitely thick when the disk is forced to oscillate with resonant frequency. In other words, in case of blowing and
resonance the oscillatory boundary layer flows are no longer possible.
Therefore, in the present study it aims at finding a meaningful solution for a non-linear coupled equation to bring
out the effects of suction/blowing with varying span- wise co-sinusoidal time dependent temperature in the presence of
uniform porous matrix in a free convective Magnetohydrodynamic flow past a vertical porous plate.
2. Formulation of the problem
An unsteady flow of a viscous incompressible electrically conducting fluid through a porous medium past an
insulated, infinite, hot, porous plate lying vertically on the x − z plane is considered. The x -axis is oriented in the
* * *
*
direction of the buoyancy force and y -axis is taken perpendicular to the plane of the plate. A uniform magnetic field
* * *
of strength B0 is applied along the y*-axis. Let (u , v , w ) be the component of velocity in the direction
( x , y , z ) respectively. The plate being considered infinite in x* direction, hence all the physical quantities are
* * *
independent of x . Thus, following Acharya and Padhi [1983], ω is independent of z and the equation of continuity
* * *
gives v = −V (constant) throughout.
*
We assume the span wise co-sinusoidal temperature of the form
T = T * + ε (T0* − T∞ ) cos(π z * / l − ω *t * ),
*
(2.1)
*
The mean temperature T of the plate is supplemented by the secondary temperature
ε (T0 − T∞ ) cos(π z / l − ω *t * ) varying with space and time. Under the usual Boussineques approximation the free
* * *
convective flow through porous media is governed by the following equations:
ut* + v*u * = υ (u * + u * ) + g β (T * − T∞ ) − σ B02 u * / ρ − υ u * / K P* ,
y yy zz
*
(2.2)
K µ
Tt * + v*Ty * = (Tyy * + Tzz * ) + ((u * )2 + (u z * ) 2 ) (2.3)
ρ CP ρ CP y
The boundary conditions are given by
y* = 0 : u * = 0, v* = V (Const.), T * = T0* + ε (T0* − T∞* ) cos(π z * / l − ω *t * ) (2.4)
y → ∞ : u → 0, T → T∞ .
* * * *
Introducing the non dimensional quantities defined in the nomenclature, we get,
ω 1 1 1
ut − u y = (u yy + u zz ) + Grθ − (M 2 + )u , (2.5)
Re Re Re KP
ω 1
θt − θ y = (θ yy + θ zz ) + Re Ec(u y 2 + u z 2 ), (2.6)
Re Pr Re
with corresponding boundary conditions:
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y = 0 : u = 0,θ = 1 + ε cos(π z − t ),
(2.7)
y → ∞ : u → 0,θ → 0.
Since the amplitude, ε (<< 1), of the plate temperature is very small, we represent the velocity and temperature in the
neighborhood of the plate as
u ( y, z , t ) = u0 ( y ) + ε u1 ( y, z , t ) + o(ε 2 ),
(2.8)
θ ( y, z , t ) = θ 0 ( y ) + εθ1 ( y, z, t ) + o(ε 2 ),
Comparing the coefficient of like powers of ε after substituting (2.8) in (2.5) and (2.6), we get the following zeroth
order equations:
1
u0 yy + Re u0 y − ( M 2 + )u0 = − Re Grθ 0 , (2.9)
KP
θ0 yy + Pr Re θ 0 y = − Pr Re2 Ecu0 y 2 (2.10)
For solving the above coupled equations we use the following perturbation equations with perturbation parameter Ec,
the Eckert number,
u0 = u01 + Ecu02 + o( Ec 2 )
2
(2.11)
θ0 = θ01 + Ecθ02 + o( Ec )
Substituting (2.11) into (2.9) and (2.10) we get the following zeroth and first order equations of Ec .
1
′′ ′
u01 + Re u0 − ( M 2 + )u01 = − Re Grθ 01 , (2.12)
KP
θ 01 + Pr Re θ 01 = 0,
′′ ′ (2.13)
1
′′ ′
u02 + Re u02 − ( M 2 + )u02 = − Re Grθ 02 , (2.14)
KP
′′ ′ ′
θ 02 + Pr Re θ 02 = − Pr Re 2 u012 , (2.15)
The corresponding boundary conditions are:
y = 0; u01 = 0, θ 01 = 1, u02 = 0,θ 02 = 0,
(2.16)
y → ∞; u01 = 0,θ 01 = 0, u02 = 0,θ 02 = 0.
The solution of the equations (2.12) to (2.15) under the boundary conditions (2.16) are
u01 ( y ) = C3 (e− Pr Re y − e − m1 y ), (2.17)
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θ 01 ( y ) = e − Pr Re y , (2.18)
u02 ( y ) = C4e− m1 y + A7e− Pr Re y + A8e−2Pr Re y + A9e −2 m1 y + A10e− (Pr Re + m1 ) y , (2.19)
θ 02 ( y ) = A3e − Pr Re y + A4e−2Pr Re y + A5e−2 m y + A6e − (Pr Re +m ) y ,
1 1
(2.20)
The terms of the coefficient of ε give the following first order equations:
1
ωu1t − Re u1 y = (u1 yy + u1zz ) + Gr Re θ1 − ( M 2 + )u1 , (2.21)
KP
1
ωθ1t − Re θ1 y = (θ1 yy + θ1zz ) + Re2 u0 y u1 y . (2.22)
Pr
In order to solve (2.21) and (2.22),it is convenient to adopt complex notations for velocity and temperature profile as,
u1 ( y, z , t ) = φ ( y )ei (π z −t ) . (2.23)
θ1 ( y, z, t ) = φ ( y )ei (π z −t ) .
The solutions obtained in terms of complex notations, the real part of which have physical significance.
Now, substituting (2.23) into (2.21) and (2.22) we get the following coupled equations:
1
φ ′′( y ) + Re φ ′( y ) + [ωi − π 2 − ( M 2 + )]φ = − Re Grψ ( y ), (2.24)
KP
ψ ′′( y ) + Pr Reψ ′( y ) + (ω Pr i − π 2 )ψ = −2 Re Pr uoyφ ′, (2.25)
Again, to uncouple above equations we assume the following perturbed forms:
φ = φ0 + Ecφ1 + o( Ec 2 ),
(2.26)
ψ = ψ 0 + Ecψ 1 + o( Ec 2 ),
Substituting (2.26) into (2.24) and (2.25) and equating the coefficient of like powers of Ec we get the subsequent
equations:
1
φ0′′ + Re φ0′ + [ωi − π 2 − ( M 2 + )]φ0 = − Re Grψ 0 , (2.27)
KP
′′ ′
ψ 0 + Pr Reψ 0 + (ω Pr i − π 2 )ψ 0 = 0, (2.28)
1
φ1′′+ Re φ1′ + [ωi − π 2 − ( M 2 + )]φ1 = − Re Grψ 1 , (2.29)
K
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ψ 1′′ + Pr Reψ 1′ + (ω Pr i − π 2 )ψ 1 = −2 Re2 Pr u01φ0′ ,
′ / (2.30)
Where the prime denotes the differentiation with respect to y.
The corresponding boundary conditions are:
y = 0; φ0 = 0, φ1 = 0,ψ 0 = 1,ψ 1 = 0,
(2.31)
y → ∞;φ0 = 0, φ1 = 0,ψ 0 = 0,ψ 1 = 0,
The solutions of equations (2.27) to (2.30) under the boundary conditions (2.31) are:
φ0 ( y ) = A11 (e− m y − e− m y ),
2 3
(2.32)
ψ 0 ( y) = e−m y ,
2
(2.33)
ψ 1 ( y ) = A12e− (Pr Re +m ) y + A13e− (Pr Re + m ) y + A14e − ( m + m ) y + A15e − ( m + m ) y + A16e − m y ,
2 3 1 2 1 2 4
(2.34)
φ1 ( y ) = A17 e− (Pr Re + m ) y + A18e − (Pr Re + m ) y + A19e − ( m + m ) y + A20e − ( m + m ) y + A21e − m y + A22e − m y . (2.35)
2 3 1 2 1 3 4 3
The important flow characteristics of the problem are the plate shear stress and the rate of heat transfer at the plate.
The expressions for shear stress ( τ ) and Nusselt number ( Nu ) are given by
du
τ = τ *l / µV = Re al at y = 0
dy
= C3 (− Pr Re+ m1 ) − Ec[C4 m1 + A7 Pr Re+ 2 A8 Pr Re+ 2m1 A9 + A10 (Pr Re+ m1 )]
(2.36)
+ε F Cos (π z − t + α )
Where
F = Fr + iFi
= A11 (m3 − m2 ) − Ec[(m3 A22 + m4 A21 + A17 (Pr Re+ m2 ) + A18 (Pr Re+ m3 ) + A19 (m1 + m2 ) + A20 (m3 + m1 )]
The amplitude and the phase angle are given by
Fi
F = Fr2 + Fi 2 , α = tan −1
Fr
q*l dθ
Nu = − =Real at y = 0
k (T0 − T∞ )
* *
dy
= − Pr Re− Ec[Pr Re( A3 + 2 A4 ) + 2m1 A5 + A6 (Pr Re+ m1 + β )]
(2.37)
−ε G Cos (π z − t + β )
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Where G = Gr + iGi
= m2 + Ec[(m14 A16 + A12 (Pr Re+ m2 ) + A13 (Pr Re+ m3 ) + A14 (m1 + m2 ) + A14 (m1 + m3 )]
The amplitude and the phase angle are given by
Gi
G = Gr2 + Gi2 , β = tan −1 .
Gr
3. Result and Discussions
This section analyses the velocity, temperature, and amplitude, phase angle of shear stress and rate of heat
transfer. The present discussion brings the following cases as particular case.
• M = 0, represents the case of non-conducting fluid without magnetic field.
• K P → ∞, represents without porous medium.
The most important part of the discussion is due to the presence of the sinusoidal variation of surface
temperature with space and time and the forcing forces such as Lorentz force, resistive force, due to porosity of the
medium, thermal buoyancy and cross flow due to permeable surface.
From equation (2.5) the following results follows.
Vl
In the absence of cross flow, V = 0 ⇒ Re = = 0, the u component of velocity remains unaffected by
υ
convective acceleration and thermal buoyancy force and the equation (2.5) reduces to
1 2 1
ut = ( u yy + u zz ) − M + u
ω KP
More over, the viscosity contributes significantly with a combined retarding effect caused by magnetic force and
resistance due to porous medium with an inverse multiplicity of the frequency of the temperature function. Further,
M = 0, K P → ∞, reduces the problem to a simple unsteady motion given by ut =
ω
1
(u yy + u zz ) .
Thus, in the absence of cross flow, that is, in case of a non-permeable surface, frequency of the fluctuating
ω *l 2
temperature parameter ω = acts as a scaling factor.In case of impermeable surface, that is, in absence of
υ
cross flow, equation (6) reduces to θt =
1
(θ + θ ) .
ω Pr yy zz
This shows that the unsteady temperature gradient is reduced by high prandtle number fluid as well as
increasing frequency of the fluctuating temperature and viscous dissipation fails to effect the temperature
distribution.
From fig.1 it is observed that maximum velocity occurs in case of cooling of the plate ( Gr > 0 ) without
magnetic field (curve IX) and reverse effect is observed exclusively due to heating of the plate ( Gr < 0 ) with a
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back flow (curve X). Thermal buoyancy effect has a significant contribution over the flow field.
From curves (VIII & IX), it is seen that the Lorentz force has a retarding effect which is in conformity with the
earlier reported result [14].
In the present study, velocity decrease is 7 times and there by steady state is reached within a few layers of the
flow domain. Further, it is seen that flow reversal occurs only in case of heating of the plate (curve X& XX) both in
porous and non porous medium.
Moreover, it is interesting to note that increase in Pr leads to significant decrease of the velocity (curve I (air)
and IV (water)) throughout the flow field but an increase in cross flow Reynolds number, increases the velocity in
the vicinity of the plate. Afterwards, it decreases rapidly. From curves, I & VIII it is observed that an increase in
viscous dissipation increases the velocity at all points. Thus, the energy loss sets a cooling current vis-à-vis
accelerating the velocity to reach the steady state.
These observations are clearly indicated by the additive and subtractive terms in the equation discussed earlier.
Fig.2 exhibits asymptotically decreasing behavior of the temperature distribution. For higher Pr fluid (Pr =7.0),
the temperature reduces drastically (curves I&IV).
It is interesting to remark that the cooling and heating of the plate (curves I & X) makes no difference in temperature
distribution which is compensated due to increase in dissipative energy loss.
Fig.3 shows the variation of amplitude of shear stress in both the cases i.e. presence and absence of porous
medium. Shear stress is almost linear for all values of frequency of fluctuation. For high Reynolds number as well as
greater buoyancy force shearing stress increases (curves II, XII, VII & XVII).
From fig.4 a steady increasing behavior is marked with an increasing frequency parameter but when ω
exceed 15.0 (approx.), slight decrease is marked. There is always a phase lead for all the parameters and for all
values of ω . A decrease in phase angle is marked due to increase in Reynolds number and magnetic parameter but
the reverse effect is observed in case of Prandtl number .Phase angle remains unaffected due to buoyancy effect,
dissipative loss and presence of porous medium.
Fig.5 and fig.6 exhibits the variation of amplitude G and phase angle tan β in case of heat transfer.
Further, G exhibits three layer characters due to high, medium and low value of Pr.
4. Conclusion
• Heating of the plate leads to back flow.
• Lorentz force has retarding effects.
• Presence of porous medium has no significant contribution.
• Viscous dissipation generates a cooling current which accelerates the velocity.
• There are three layer variation of amplitude of heat transfer with a phase lead.
• A phase lead is marked in case of shear stress and a phase lag for heat transfer which remains
unaffected by magnetic parameter, Grashoff number and Eckert number
Reference:
Stangeby,P.C(1974), A review of the status of MHD power generation technology including generation for a
Canadian MHD research programme, UTIAS rev, vol.39.
Lielausis,O.A,(1975), Liquid metal magnetohydrodynamics, Atomic Energy rev.,vol.13, pp.527
Ham, J.C.R and Moreu, R, (1976), Liquid metal magnetohydrodynamics with strong magnetic field”, a report on
Euromech, J.Fluid mech. vol.78, pp.261.
Nanda,R.S. and Sharma,V.P,(1962), Possibility similarity solutions of unsteady free convection flow past a vertical
plate with suction, J. phys. Soc. Japan ; vol.17(10),pp.1651-1656.
Ostrach,S.(1953), New aspects of natural convection heat transfer, Trans.Am.Soc.Mec.Engrs.; vol.75,pp.1287-1290.
Berezovsky, A. A., Martynenko, O. G. and Sokovishin,Yu. A, (1977) Free convective heat transfer on a vertical semi
infinite plate, J. Engng. Phys., vol.33, pp.32-39.
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Vol.2, No.6, 2012
Martynenko,O. G., Berezovsky, A. A. and Sokovishin ,Yu. A,(1984), Laminar free convection from a vertical plate”,
Int. J. Heat Mass Transfer, vol.27, pp.869-881.
Murty ,S.N,(1975),MHD Ekman layer on a porous plate, Nuow cimento, vol. 29 ,pp.296.
Debnath ,L,(1972) “On unsteady MHD boundary layers in a rotating flow” ZAAM ,vol.52,pp.623.
Acharya,B.P. and Padhy,S,(1983), Free convective viscous flow along a hot vertical porous plate with periodic
temperature. Indian J. Pure appl.Math., vol.14(7),pp.838-849.
Ferdows, M., Sattar,M.A and Siddiki,M.N.A,(2004), Numerical approach on parameters of the thermal radiation
interaction with convection in boundary layer flow at a vertical plate with variable suction.
Thammasat International Journal of Science and Technology, vol.9, pp. 19-28.
Alam,M.S., Rahman ,M.M. and Samad ,M.A.,(2006), Dufour and Soret effects on unsteady MHD free convection
and mass transfer flow past a vertical plate in a porous medium. Nonlinear Analysis: Modeling and Control ,vol.11,
pp.271-226.
Majumder,M.K. and Deka,R.K,(2007), MHD flow past an impulsively started infinite vertical plate in the presence
of thermal radiation,Rom. Jour. Phys., vol.52 (5-7), pp.565-573.
Dash,G.C., Rath,P.K., Mohapatra,N. and Dash,P.K,(2009), Free convective MHD flow through porous media of a
rotating visco-elastic fluid past an infinite vertical porous plate with heat and mass transfer in the presence of a
chemical reaction, AMSE Modeling B,vol.78,No.4,pp.21-37.
Muthucumaraswamy R, Sundarraj M and Subhramanian V.S.A,(2009) Unsteady flow past an accelerated infinite
vertical plate with variable temperature and uniform mass diffusion. Int. J. of Appl. Math. and Mech. 5(6): 51-56.
Nomenclature:
( x* , y * , z * ) Cartesian Co-ordinate system V Suction velocity
l Wave length ε Amplitude of the span
wise co-sinusoidal temperature
υ Kinematics coefficient of viscosity µ Coefficient of viscosity
y = y* / l
Dimensionless space variable θ Dimensionless temperature
z = z /l
*
(T * − T∞ ) (T0* − T∞ )
* *
β Coefficient of volumetric expansion B0 Uniform magnetic field strength
g Acceleration due to gravity β Coefficient of volume approximation
T0* Temperature of the plate *
T∞ ambient temperature
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9. Mathematical Theory and Modeling www.iiste.org
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Vol.2, No.6, 2012
ρ Density CP Specific heat at constant pressure
K Thermal conductivity ω Frequency of temperature fluctuation-
ω *l 2 / υ
Re Reynolds number- Vl / υ Pr Prandtl number- µCP / K
Gr Grashoff number- υ g β (T0* − T∞ ) / V 3
*
Ec Eckert number- V 2 CP (T0* − T∞ )
*
M Hartmann number- B0l σ
µ τ Dimensionless skin friction
Nu Nusselt number K P Permeability of the medium
Appendix
1
Re+ Re 2 + 4( M 2 + )
KP Pr Re+ Pr 2 Re2 − 4(Pr iω − π 2 )
m1 = , m2 =
2 2
1
Re+ Re2 − 4(iω − π 2 − ( M 2 + ))
KP Pr Re+ Pr 2 Re2 − 4(Pr iω − π 2 )
m3 = , m4 =
2 2
− Re Gr 6
Pr Re2 C32
C3 = , A3 = −∑ Ai , A4 = − ,
1 2
Pr Re − Pr Re − ( M +
2 2 2 2
) i =4
KP
Pr Re 2 C32 m12 2 Pr 2 Re3 C32 m1 − Re GrA3
A5 = − , A6 = , A7 =
4m12 − 2m12 Pe Re (Pr Re+ m1 ) 2 − 2m12 Pr Re Pr 2 Re 2 − Pr Re2 − ( M 2 +
1
)
KP
− Re GrA4 − Re GrA5
A8 = , A9 =
1 1
4 Pr 2 Re2 − 2 Pr Re2 − ( M 2 + ) 4m12 Re2 − 2m1 Re− ( M 2 + )
KP KP
− Re GrA6 10
A10 = , C4 = −∑ Ai ,
1
(Pr Re+ m1 ) 2 − Re(Pr Re+ m1 ) − ( M 2 + ) i =7
KP
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Vol.2, No.6, 2012
Re Pr M Gr Ec
3.5 I 2.0 0.71 2.0 5.0 0.01
IX
II 4.0 0.71 2.0 5.0 0.01
XIX III 2.0 2.00 2.0 5.0 0.01
3.0 VII IV 2.0 7.00 2.0 5.0 0.01
VIII,XVIII V 2.0 0.71 4.0 5.0 0.01
VI 2.0 7.00 4.0 5.0 0.01
IV,XIV VII 2.0 0.71 2.0 10.0 0.01
2.5 I,XI VIII 2.0 0.71 2.0 5.0 0.02
XVII IX 2.0 0.71 0.0 5.0 0.02
X 2.0 0.71 2.0 -5.0 0.02
V,XV
2.0 II,XII XI 2.0 0.71 2.0 5.0 0.01
XII 4.0 0.71 2.0 5.0 0.01
XIII 2.0 2.00 2.0 5.0 0.01
XIV 2.0 7.00 2.0 5.0 0.01
1.5 III,XIII XV 2.0 0.71 4.0 5.0 0.01
XVI 2.0 7.00 4.0 5.0 0.01
X,XX
u
XVII 2.0 0.71 2.0 10.0 0.01
XVIII 2.0 0.71 2.0 5.0 0.02
1.0 XIX 2.0 0.71 0.0 5.0 0.02
XX 2.0 0.71 2.0 -5.0 0.02
0.5
0.0
0.0 0.5 y 1.0 1.5 2.0
-0.5
POROUS MEDIUM
-1.0
WITOUT POROUS
MEDIUM
Fig.1 Velocity disrtibution.Kp=1(I-X),Kp=1000(XI-XX)
1.0 Re Pr M Gr Ec
I. 2.0 0.71 2.0 5.0 0.01
II. 4.0 0.71 2.0 5.0 0.01
IV,VI,XIV,XVI III. 2.0 2.00 2.0 5.0 0.01
0.8 IV. 2.0 7.00 2.0 5.0 0.01
III,XIII V. 2.0 0.71 4.0 5.0 0.01
VI. 2.0 7.00 4.0 5.0 0.01
II,XII VII. 2.0 0.71 2.0 10.0 0.01
0.6 VIII. 2.0 0.71 2.0 5.0 0.02
I,V,VIII,X,XI, IX. 2.0 0.71 0.0 5.0 0.02
XV,XVIII,XX X. 2.0 0.71 2.0 -5.0 0.02
θ
IX,XIX Re Pr M Gr Ec
0.4 XI. 2.0 0.71 2.0 5.0 0.01
VII,XVII XII. 4.0 0.71 2.0 5.0 0.01
XIII. 2.0 2.00 2.0 5.0 0.01
XIV. 2.0 7.00 2.0 5.0 0.01
XV. 2.0 0.71 4.0 5.0 0.01
0.2
XVI. 2.0 7.00 4.0 5.0 0.01
XVII. 2.0 0.71 2.0 10.0 0.01
XVIII. 2.0 0.71 2.0 5.0 0.02
XIX. 2.0 0.71 0.0 5.0 0.02
0.0
XX. 2.0 0.71 2.0 -5.0 0.02
0.0 0.5 1.0 1.5 2.0
y porous medium
Non-porous
Fig.2.Temperature distribution.Kp=1(I-X),Kp=1000(XI-XX)
medium
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Vol.2, No.6, 2012
II,XII
3.0 VII,XVII Re Pr M Gr Ec
I. 2.0 0.71 2.0 5.0 0.01
II. 4.0 0.71 2.0 5.0 0.01
III. 2.0 2.00 2.0 5.0 0.01
IV. 2.0 7.00 2.0 5.0 0.01
2.5 V. 2.0 0.71 4.0 5.0 0.01
I,VIII,X,XI, VI. 2.0 7.00 4.0 5.0 0.01
XVIII,XX VII. 2.0 0.71 2.0 10.0 0.01
VIII. 2.0 0.71 2.0 5.0 0.02
2.0 V,XV IX. 2.0 0.71 0.0 5.0 0.02
VI,XVI X. 2.0 0.71 2.0 -5.0 0.02
IX,XIX
IV,XIV III,XIII
|F|
1.5
Re Pr M Gr Ec
XI. 2.0 0.71 2.0 5.0 0.01
XII. 4.0 0.71 2.0 5.0 0.01
XIII. 2.0 2.00 2.0 5.0 0.01
1.0 XIV. 2.0 7.00 2.0 5.0 0.01
XV. 2.0 0.71 4.0 5.0 0.01
XVI. 2.0 7.00 4.0 5.0 0.01
XVII. 2.0 0.71 2.0 10.0 0.01
XVIII. 2.0 0.71 2.0 5.0 0.02
0.5
XIX. 2.0 0.71 0.0 5.0 0.02
XX. 2.0 0.71 2.0 -5.0 0.02
0.0
5 10 15 20 25 30
ω
Fig. 3. The amplitude |F| of the shear stress.Kp=1(I-X),Kp=1000(XI-XX)
IX XIX
0.7 Re Pr M Gr Ec
I,VII,VIII,X,XVII, I. 2.0 0.71 2.0 5.0 0.01
XVIII,XI,XX III,XIII II. 4.0 0.71 2.0 5.0 0.01
III. 2.0 2.00 2.0 5.0 0.01
IV. 2.0 7.00 2.0 5.0 0.01
0.6
IV,XIV V. 2.0 0.71 4.0 5.0 0.01
VI. 2.0 7.00 4.0 5.0 0.01
VII. 2.0 0.71 2.0 10.0 0.01
VI,XVI II,XII VIII. 2.0 0.71 2.0 5.0 0.02
0.5 IX. 2.0 0.71 0.0 5.0 0.02
X. 2.0 0.71 2.0 -5.0 0.02
V,XV
Tan α
0.4 Re Pr M Gr Ec
XI. 2.0 0.71 2.0 5.0 0.01
XII. 4.0 0.71 2.0 5.0 0.01
XIII. 2.0 2.00 2.0 5.0 0.01
XIV. 2.0 7.00 2.0 5.0 0.01
0.3
XV. 2.0 0.71 4.0 5.0 0.01
XVI. 2.0 7.00 4.0 5.0 0.01
XVII. 2.0 0.71 2.0 10.0 0.01
XVIII. 2.0 0.71 2.0 5.0 0.02
0.2 XIX. 2.0 0.71 0.0 5.0 0.02
XX. 2.0 0.71 2.0 -5.0 0.02
0.1
5 10 15 20 25 30
ω
Fig. 4. The phase angle Tanα of shear stress,Kp=1(I-X),Kp=1000(XI-XX)
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21.0 Re Pr M Gr Ec
I. 2.0 0.71 2.0 5.0 0.01
II. 4.0 0.71 2.0 5.0 0.01
19.0 III. 2.0 2.00 2.0 5.0 0.01
IV. 2.0 7.00 2.0 5.0 0.01
17.0 V. 2.0 0.71 4.0 5.0 0.01
VI. 2.0 7.00 4.0 5.0 0.01
VII. 2.0 0.71 2.0 10.0 0.01
15.0 I,V,VII,VIII,IX,X,XI,XV,XVII, VIII. 2.0 0.71 2.0 5.0 0.02
IX. 2.0 0.71 0.0 5.0 0.02
XVIII,XIX,XX X. 2.0 0.71 2.0 -5.0 0.02
13.0
|G|
III,XIII II,XII
Re Pr M Gr Ec
11.0 IV,VI,XIV,X XI. 2.0 0.71 2.0 5.0 0.01
VI XII. 4.0 0.71 2.0 5.0 0.01
9.0 XIII. 2.0 2.00 2.0 5.0 0.01
XIV. 2.0 7.00 2.0 5.0 0.01
XV. 2.0 0.71 4.0 5.0 0.01
7.0 XVI. 2.0 7.00 4.0 5.0 0.01
XVII. 2.0 0.71 2.0 10.0 0.01
XVIII. 2.0 0.71 2.0 5.0 0.02
5.0
XIX. 2.0 0.71 0.0 5.0 0.02
XX. 2.0 0.71 2.0 -5.0 0.02
3.0
5 10 15 20 25 30
ω
Fig.5. The amplitude |G| of the ate of heat transfer.
Kp=1(I-X),Kp=1000(XI-XX)
Re Pr M Gr Ec
-0.1
I. 2.0 0.71 2.0 5.0 0.01
5 10 15 20 25 30 II. 4.0 0.71 2.0 5.0 0.01
III. 2.0 2.00 2.0 5.0 0.01
IV. 2.0 7.00 2.0 5.0 0.01
-0.2 V. 2.0 0.71 4.0 5.0 0.01
VI. 2.0 7.00 4.0 5.0 0.01
VII. 2.0 0.71 2.0 10.0 0.01
VIII. 2.0 0.71 2.0 5.0 0.02
-0.3 IX. 2.0 0.71 0.0 5.0 0.02
X. 2.0 0.71 2.0 -5.0 0.02
Tan β
Re Pr M Gr Ec
-0.4 XI. 2.0 0.71 2.0 5.0 0.01
XII. 4.0 0.71 2.0 5.0 0.01
XIII. 2.0 2.00 2.0 5.0 0.01
XIV. 2.0 7.00 2.0 5.0 0.01
-0.5 III,XIII XV. 2.0 0.71 4.0 5.0 0.01
XVI. 2.0 7.00 4.0 5.0 0.01
XVII. 2.0 0.71 2.0 10.0 0.01
XVIII. 2.0 0.71 2.0 5.0 0.02
I,V,VII,VIII,IX,X,XI, XIX. 2.0 0.71 0.0 5.0 0.02
-0.6 XV,XVII,XVIII,XIX,XX IV,VI,XIV,XVI XX. 2.0 0.71 2.0 -5.0 0.02
II,XII
-0.7
ω
Fig.6.The phase angle Tanβ of the rate of heat transfer.
Kp=1(I-X),Kp=1000(XI-XX)
78
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