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.
International journal of engineering and mathematical modelling vol2 no3_2015_1IJEMM
A weak nonlinear stability analysis has been performed for an oscillatory mode of convection, heat and mass transports in terms of
Nusselt, Sherwood numbers are derived and evaluated by a non$-$autonomous complex Ginzburg-Landau equation. The porous layer boundaries are heated sinusoidally with time. The basic state temperature is defined in terms of study and oscillatory parts, where study part show nonlinear throughflow effect on the system and time dependant part show modulation effect. The generalized Darcy model is employed for the momentum equation. The disturbances of the flow are expanded in power series of amplitude of modulation, which is assumed to be small and employed using normal mode technics. The effect of vertical throughflow is found to stabilize or destabilize the system depending on its direction. The time relaxation parameter $\lambda_1$ has destabilizing effect, while time retardation parameter $\lambda_2$ has stabilizing effect on the system. Three types of modulations have been analyzed, and found that, OPM, LBMO cases are effective on heat and mass transfer than IPM case. The effects of amplitude and frequency of modulation on heat and mass transports have been analyzed and depicted graphically. The study establishes that the heat and mass transports can be controlled effectively by a mechanism that is external to the system.
Chemical reaction and radiation effect on mhd flow past an exponentially acce...Alexander Decker
This document describes a mathematical analysis of MHD fluid flow past an exponentially accelerated vertical plate embedded in a porous medium. The analysis considers the effects of variable temperature, mass diffusion, radiation, and a heat source on the flow characteristics. The governing equations for this problem are derived and non-dimensionalized. The non-dimensional equations are then solved using the Laplace transform technique. The effects of various physical parameters like the magnetic field, radiation, heat generation, and chemical reaction on the velocity, temperature, and concentration profiles are determined.
Thermal radiation effects on mhd free convection flow of a micropolar fluid p...Alexander Decker
This document summarizes a study on thermal radiation effects on magnetohydrodynamic (MHD) free convection flow of a micropolar fluid past a stretching surface embedded in a non-Darcian porous medium. The governing equations for momentum, angular momentum, and energy are solved numerically. Key findings include:
1) The micropolar fluid model helps reduce drag forces and acts as a cooling agent compared to classical fluids.
2) Parameters like the Darcy number, radiation, magnetic field, and porous medium properties influence the velocity, microrotation, temperature, skin friction, and heat transfer.
3) Increased microrotation constant and coupling constant reduce skin friction but increase heat transfer.
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.
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.
This document summarizes a research article that investigates the steady, two-dimensional Falkner-Skan boundary layer flow over a stationary wedge with momentum and thermal slip boundary conditions. The flow considers a temperature-dependent thermal conductivity in the presence of a porous medium and viscous dissipation. Governing partial differential equations are non-dimensionalized and transformed into ordinary differential equations using similarity transformations. The equations are highly nonlinear and cannot be solved analytically, so a numerical solver is used. Numerical results are presented for the skin friction coefficient, local Nusselt number, velocity and temperature profiles for varying parameters like the Falkner-Skan parameter and Eckert number.
Non-Newtonian Visco-elastic Heat Transfer Flow Past a Stretching Sheet with C...IJERA Editor
In this paper two dimensional flow of a viscoelastic fluid due to stretching surface is considered. Flow analysis is carried out by using closed form solution of fourth order differential equation of motion of viscoelastic fluid. Further (Walters’ liquid B’ model) heat transfer analysis is carried out using convective surface condition. The governing equations of flow and heat transfer are non-linear partial differential equations which are unable to solve analytically hence are solved using Runge-Kutta Numerical Method with efficient shooting technique. The flow and heat transfer characteristics are studied through plots drawn. Numerical values of Wall temperature are calculated and presented in the table and compared with earlier published results which are in good agreement
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.
International journal of engineering and mathematical modelling vol2 no3_2015_1IJEMM
A weak nonlinear stability analysis has been performed for an oscillatory mode of convection, heat and mass transports in terms of
Nusselt, Sherwood numbers are derived and evaluated by a non$-$autonomous complex Ginzburg-Landau equation. The porous layer boundaries are heated sinusoidally with time. The basic state temperature is defined in terms of study and oscillatory parts, where study part show nonlinear throughflow effect on the system and time dependant part show modulation effect. The generalized Darcy model is employed for the momentum equation. The disturbances of the flow are expanded in power series of amplitude of modulation, which is assumed to be small and employed using normal mode technics. The effect of vertical throughflow is found to stabilize or destabilize the system depending on its direction. The time relaxation parameter $\lambda_1$ has destabilizing effect, while time retardation parameter $\lambda_2$ has stabilizing effect on the system. Three types of modulations have been analyzed, and found that, OPM, LBMO cases are effective on heat and mass transfer than IPM case. The effects of amplitude and frequency of modulation on heat and mass transports have been analyzed and depicted graphically. The study establishes that the heat and mass transports can be controlled effectively by a mechanism that is external to the system.
Chemical reaction and radiation effect on mhd flow past an exponentially acce...Alexander Decker
This document describes a mathematical analysis of MHD fluid flow past an exponentially accelerated vertical plate embedded in a porous medium. The analysis considers the effects of variable temperature, mass diffusion, radiation, and a heat source on the flow characteristics. The governing equations for this problem are derived and non-dimensionalized. The non-dimensional equations are then solved using the Laplace transform technique. The effects of various physical parameters like the magnetic field, radiation, heat generation, and chemical reaction on the velocity, temperature, and concentration profiles are determined.
Thermal radiation effects on mhd free convection flow of a micropolar fluid p...Alexander Decker
This document summarizes a study on thermal radiation effects on magnetohydrodynamic (MHD) free convection flow of a micropolar fluid past a stretching surface embedded in a non-Darcian porous medium. The governing equations for momentum, angular momentum, and energy are solved numerically. Key findings include:
1) The micropolar fluid model helps reduce drag forces and acts as a cooling agent compared to classical fluids.
2) Parameters like the Darcy number, radiation, magnetic field, and porous medium properties influence the velocity, microrotation, temperature, skin friction, and heat transfer.
3) Increased microrotation constant and coupling constant reduce skin friction but increase heat transfer.
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.
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.
This document summarizes a research article that investigates the steady, two-dimensional Falkner-Skan boundary layer flow over a stationary wedge with momentum and thermal slip boundary conditions. The flow considers a temperature-dependent thermal conductivity in the presence of a porous medium and viscous dissipation. Governing partial differential equations are non-dimensionalized and transformed into ordinary differential equations using similarity transformations. The equations are highly nonlinear and cannot be solved analytically, so a numerical solver is used. Numerical results are presented for the skin friction coefficient, local Nusselt number, velocity and temperature profiles for varying parameters like the Falkner-Skan parameter and Eckert number.
Non-Newtonian Visco-elastic Heat Transfer Flow Past a Stretching Sheet with C...IJERA Editor
In this paper two dimensional flow of a viscoelastic fluid due to stretching surface is considered. Flow analysis is carried out by using closed form solution of fourth order differential equation of motion of viscoelastic fluid. Further (Walters’ liquid B’ model) heat transfer analysis is carried out using convective surface condition. The governing equations of flow and heat transfer are non-linear partial differential equations which are unable to solve analytically hence are solved using Runge-Kutta Numerical Method with efficient shooting technique. The flow and heat transfer characteristics are studied through plots drawn. Numerical values of Wall temperature are calculated and presented in the table and compared with earlier published results which are in good agreement
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.
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.
Validity Of Principle Of Exchange Of Stabilities Of Rivilin- Ericksen Fluid...IRJET Journal
This document presents research on establishing the validity of the Principle of Exchange of Stabilities (PES) for thermal convection of a Rivlin-Ericksen fluid layer in porous medium heated from below with variable gravity. The fluid layer contains suspended particles and is subjected to rotation. The linearized stability equations are formulated using an operator method. It is established that PES is valid for this problem under sufficient conditions, when the gravity field g(z) is nonnegative throughout the fluid layer. This is done by analyzing the resolvent of the linearized stability operator as a composition of integral operators and applying the method of positive operators to show the system has a single greatest eigenvalue.
Effects of conduction on magneto hydrodynamics mixed convection flow in trian...Alexander Decker
This document summarizes research on magnetohydrodynamic (MHD) mixed convection flow in triangular enclosures. Key points:
1) The study investigates the effects of conduction on MHD mixed convection flow in triangular enclosures using a finite element method.
2) Parameters like the Hartmann number, Prandtl number, Reynolds number, and Rayleigh number are found to strongly influence the flow and thermal fields.
3) Validation of the numerical code is done by comparing average Nusselt numbers to previous research on natural convection in triangular enclosures.
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.
Mhd effects on non newtonian micro polar fluid with uniform suctioneSAT Journals
Abstract In this present work, the problem of heat and mass transfer on the flow of Non-newtonian micropolar fluid with uniform suction/blowing with heat generation, radiation, thermophoresis and chemical reaction effects are studied. The non-linear partial differential equation arising from the flow modeling were transformed into coupled non linear ordinary differential equations and subsequently solved by using the implicit finite difference method. The effect of relevant thermo physical parameters such as material parameter K, suction parameter A, heat generation/absorption parameter B, Prandtl number Pr, radiation parameter R, thermophoretic parameter τ, chemical reaction parameter δ , magnetic parameter Ha and Schmidt number Sc are also being numerically investigated and analyzed. Keywords: MHD, micro polar fluid, heat generation, thermophoresis, radiation and finite difference scheme.
Moving Lids Direction Effects on MHD Mixed Convection in a Two-Sided Lid-Driv...A Behzadmehr
Magnetohydrodynamic (MHD) mixed convection flow of Cu–water nanofluid inside a two-sided lid-driven square enclosure with adiabatic horizontal walls and differentially heated sidewalls has been investigated numerically. The effects of moving lids direction, variations of Richardson number, Hartmann number, and volume fraction of nanoparticles on flow and temperature fields have been studied. The obtained results show that for a constant Grashof number (), the rate of heat transfer increases with a decrease in the Richardson and Hartmann numbers. Furthermore, an increase of the volume fraction of nanoparticles may result in enhancement or deterioration of the heat transfer performance depending on the value of the Hartmann and Richardson numbers and the configuration of the moving lids. Also, it is found that in the presence of magnetic field, the nanoparticles have their maximum positive effect when the top lid moves rightward and the bottom one moves leftward.
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.
Heat transefer of a dusty fluid over a stretching sheet with internal heat ge...eSAT Journals
Abstract This paper pertains to investigate the heat transfer characteristics of two dimensional two phase flow in a porous medium over a stretching sheet with internal heat generation. The novelty of the present study is to consider the permeability parameter, space and temperature dependent internal heat generation along with various parameters like Froud number ,heat source/sink parameter, Grashof number, Prandtl number, Eckert number, Volume fraction, fluid interaction parameter etc. The method of solution involves similarity transformation which reduces the partial differential equations into non linear ordinary differential equations. These non linear ordinary differential equations have been solved by applying Runge-Kutta 4-th order method with help of shooting technique. The temperature profiles for different values of flow parameters are presented in figures. It is observed from all the figures that the boundary conditions are satisfied asymptotically in all the cases which supporting the accuracy of the numerical results. All the figures shows that increasing values of any parameter increase the thermal boundary layer except the prandtl number and permeability parameter. AMS classification 76T10, 76T15 Keywords: Heat source/sink, Volume fraction, Fluid – particle interaction parameter, Dusty fluid, Boundary layer flow, Stretching sheet, Eckert number, Grashof number, Prandtle number, Shooting techniques, Froud number, permeability parameter.
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.
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
Boundary layer flow and heat transfer of a dusty fluid over a vertical permea...eSAT Journals
This document discusses boundary layer flow and heat transfer of a dusty fluid over a vertical permeable stretching surface. The governing equations for steady, two-dimensional flow are presented and non-dimensionalized. The equations are then solved numerically using the Runge-Kutta method. Results are presented graphically showing the effects of various parameters like fluid-particle interaction, local Grashof number, radiation parameter, and Eckert number on flow and heat transfer characteristics. Comparisons are made to previous studies to validate the numerical method.
Numerical Analysis of Inlet Gas-Mixture Flow Rate Effects on Carbon Nanotube ...A Behzadmehr
The growth rate and uniformity of Carbon Nano Tubes (CNTs) based on Chemical Vapor Deposition (CVD)
technique is investigated by using a numerical model. In this reactor, inlet gas mixture, including xylene as
carbon source and mixture of argon and hydrogen as carrier gas enters into a horizontal CVD reactor at
atmospheric pressure. Based on the gas phase and surface reactions, released carbon atoms are grown as CNTs on the iron catalysts at the reactor hot walls. The effect of inlet gas-mixture flow rate, on CNTs growth rate and its uniformity is discussed. In addition the velocity and temperature profile and also species concentrations throughout the reactor are presented.
Effects on Study MHD Free Convection Flow Past a Vertical Porous Plate with H...IJMTST Journal
This paper deals with the combined soret effect of thermal radiation and heat generation on the MHD free
convection heat and mass transfer flow of a viscous incompressible fluid past a continuously moving infinite
plate. Closed form of solution for the velocity, temperature and concentration field are obtained and
discussed graphically for various values of the physical parameters present. In addition, expressions for the
skin friction and Sherwood number is also derived and finally discussed with the graphs.
This document presents an analysis of free convective flow and heat transfer of a viscous incompressible fluid over a linearly moving vertical porous plate with suction and viscous dissipation. The fluid viscosity is assumed to vary linearly with temperature. Governing boundary layer equations are non-dimensionalized using similarity transformations and reduced to a boundary value problem. The problem is solved numerically using a shooting method along with the 4th order Runge-Kutta method. Results for dimensionless velocity and temperature distributions are presented for various parameter values, and skin friction and Nusselt numbers are reported in tables.
This document summarizes a study that investigated the effects of Hall current, chemical reaction, and viscous dissipation on unsteady magnetohydrodynamic (MHD) flow, heat, and mass transfer along a porous flat plate. The governing equations were solved numerically using the Galerkin finite element method. Parameters like the magnetic field, Prandtl number, Schmidt number, and modified Grashof numbers for heat and mass transfer were varied to examine their impacts on velocity, temperature, and other variables. The results provide insight into complex coupled transport processes with potential applications in areas like material processing, energy systems, and industrial equipment.
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
This document provides an abstract for a research article on magnetohydrodynamic mixed convection micro polar flow through a porous medium driven by a porous stretching sheet. The following key points are discussed:
- Governing partial differential equations are derived and numerically solved using finite element analysis to study the effects of various parameters on velocity, micro rotation, and temperature profiles.
- Parameters investigated include the Hartmann number, Darcy parameter, surface conditions, and their impact on the fluid flow is explored.
- The study aims to provide insights applicable to processes involving polymer sheet stretching and controlled cooling/heating of strips or filaments during manufacturing.
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Journals
Abstract This paper examines the problem of two dimensional unsteady free convection oscillatory flow of an Elastico-viscous fluid past an impulsively started infinite vertical plate with constant heat flux and heat generation. The flow is subjected to a constant suction. Using perturbation techniquethe expressions are obtained for velocity, temperature distribution and skin friction. The results obtained are discussed for various numerical values of the parameters entered into the equations governing the flow, with help of graphs.
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.
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.
Efficient distributed detection of node replication attacks in mobile sensor ...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
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.
Validity Of Principle Of Exchange Of Stabilities Of Rivilin- Ericksen Fluid...IRJET Journal
This document presents research on establishing the validity of the Principle of Exchange of Stabilities (PES) for thermal convection of a Rivlin-Ericksen fluid layer in porous medium heated from below with variable gravity. The fluid layer contains suspended particles and is subjected to rotation. The linearized stability equations are formulated using an operator method. It is established that PES is valid for this problem under sufficient conditions, when the gravity field g(z) is nonnegative throughout the fluid layer. This is done by analyzing the resolvent of the linearized stability operator as a composition of integral operators and applying the method of positive operators to show the system has a single greatest eigenvalue.
Effects of conduction on magneto hydrodynamics mixed convection flow in trian...Alexander Decker
This document summarizes research on magnetohydrodynamic (MHD) mixed convection flow in triangular enclosures. Key points:
1) The study investigates the effects of conduction on MHD mixed convection flow in triangular enclosures using a finite element method.
2) Parameters like the Hartmann number, Prandtl number, Reynolds number, and Rayleigh number are found to strongly influence the flow and thermal fields.
3) Validation of the numerical code is done by comparing average Nusselt numbers to previous research on natural convection in triangular enclosures.
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.
Mhd effects on non newtonian micro polar fluid with uniform suctioneSAT Journals
Abstract In this present work, the problem of heat and mass transfer on the flow of Non-newtonian micropolar fluid with uniform suction/blowing with heat generation, radiation, thermophoresis and chemical reaction effects are studied. The non-linear partial differential equation arising from the flow modeling were transformed into coupled non linear ordinary differential equations and subsequently solved by using the implicit finite difference method. The effect of relevant thermo physical parameters such as material parameter K, suction parameter A, heat generation/absorption parameter B, Prandtl number Pr, radiation parameter R, thermophoretic parameter τ, chemical reaction parameter δ , magnetic parameter Ha and Schmidt number Sc are also being numerically investigated and analyzed. Keywords: MHD, micro polar fluid, heat generation, thermophoresis, radiation and finite difference scheme.
Moving Lids Direction Effects on MHD Mixed Convection in a Two-Sided Lid-Driv...A Behzadmehr
Magnetohydrodynamic (MHD) mixed convection flow of Cu–water nanofluid inside a two-sided lid-driven square enclosure with adiabatic horizontal walls and differentially heated sidewalls has been investigated numerically. The effects of moving lids direction, variations of Richardson number, Hartmann number, and volume fraction of nanoparticles on flow and temperature fields have been studied. The obtained results show that for a constant Grashof number (), the rate of heat transfer increases with a decrease in the Richardson and Hartmann numbers. Furthermore, an increase of the volume fraction of nanoparticles may result in enhancement or deterioration of the heat transfer performance depending on the value of the Hartmann and Richardson numbers and the configuration of the moving lids. Also, it is found that in the presence of magnetic field, the nanoparticles have their maximum positive effect when the top lid moves rightward and the bottom one moves leftward.
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.
Heat transefer of a dusty fluid over a stretching sheet with internal heat ge...eSAT Journals
Abstract This paper pertains to investigate the heat transfer characteristics of two dimensional two phase flow in a porous medium over a stretching sheet with internal heat generation. The novelty of the present study is to consider the permeability parameter, space and temperature dependent internal heat generation along with various parameters like Froud number ,heat source/sink parameter, Grashof number, Prandtl number, Eckert number, Volume fraction, fluid interaction parameter etc. The method of solution involves similarity transformation which reduces the partial differential equations into non linear ordinary differential equations. These non linear ordinary differential equations have been solved by applying Runge-Kutta 4-th order method with help of shooting technique. The temperature profiles for different values of flow parameters are presented in figures. It is observed from all the figures that the boundary conditions are satisfied asymptotically in all the cases which supporting the accuracy of the numerical results. All the figures shows that increasing values of any parameter increase the thermal boundary layer except the prandtl number and permeability parameter. AMS classification 76T10, 76T15 Keywords: Heat source/sink, Volume fraction, Fluid – particle interaction parameter, Dusty fluid, Boundary layer flow, Stretching sheet, Eckert number, Grashof number, Prandtle number, Shooting techniques, Froud number, permeability parameter.
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.
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
Boundary layer flow and heat transfer of a dusty fluid over a vertical permea...eSAT Journals
This document discusses boundary layer flow and heat transfer of a dusty fluid over a vertical permeable stretching surface. The governing equations for steady, two-dimensional flow are presented and non-dimensionalized. The equations are then solved numerically using the Runge-Kutta method. Results are presented graphically showing the effects of various parameters like fluid-particle interaction, local Grashof number, radiation parameter, and Eckert number on flow and heat transfer characteristics. Comparisons are made to previous studies to validate the numerical method.
Numerical Analysis of Inlet Gas-Mixture Flow Rate Effects on Carbon Nanotube ...A Behzadmehr
The growth rate and uniformity of Carbon Nano Tubes (CNTs) based on Chemical Vapor Deposition (CVD)
technique is investigated by using a numerical model. In this reactor, inlet gas mixture, including xylene as
carbon source and mixture of argon and hydrogen as carrier gas enters into a horizontal CVD reactor at
atmospheric pressure. Based on the gas phase and surface reactions, released carbon atoms are grown as CNTs on the iron catalysts at the reactor hot walls. The effect of inlet gas-mixture flow rate, on CNTs growth rate and its uniformity is discussed. In addition the velocity and temperature profile and also species concentrations throughout the reactor are presented.
Effects on Study MHD Free Convection Flow Past a Vertical Porous Plate with H...IJMTST Journal
This paper deals with the combined soret effect of thermal radiation and heat generation on the MHD free
convection heat and mass transfer flow of a viscous incompressible fluid past a continuously moving infinite
plate. Closed form of solution for the velocity, temperature and concentration field are obtained and
discussed graphically for various values of the physical parameters present. In addition, expressions for the
skin friction and Sherwood number is also derived and finally discussed with the graphs.
This document presents an analysis of free convective flow and heat transfer of a viscous incompressible fluid over a linearly moving vertical porous plate with suction and viscous dissipation. The fluid viscosity is assumed to vary linearly with temperature. Governing boundary layer equations are non-dimensionalized using similarity transformations and reduced to a boundary value problem. The problem is solved numerically using a shooting method along with the 4th order Runge-Kutta method. Results for dimensionless velocity and temperature distributions are presented for various parameter values, and skin friction and Nusselt numbers are reported in tables.
This document summarizes a study that investigated the effects of Hall current, chemical reaction, and viscous dissipation on unsteady magnetohydrodynamic (MHD) flow, heat, and mass transfer along a porous flat plate. The governing equations were solved numerically using the Galerkin finite element method. Parameters like the magnetic field, Prandtl number, Schmidt number, and modified Grashof numbers for heat and mass transfer were varied to examine their impacts on velocity, temperature, and other variables. The results provide insight into complex coupled transport processes with potential applications in areas like material processing, energy systems, and industrial equipment.
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
This document provides an abstract for a research article on magnetohydrodynamic mixed convection micro polar flow through a porous medium driven by a porous stretching sheet. The following key points are discussed:
- Governing partial differential equations are derived and numerically solved using finite element analysis to study the effects of various parameters on velocity, micro rotation, and temperature profiles.
- Parameters investigated include the Hartmann number, Darcy parameter, surface conditions, and their impact on the fluid flow is explored.
- The study aims to provide insights applicable to processes involving polymer sheet stretching and controlled cooling/heating of strips or filaments during manufacturing.
Natural convection heat transfer oscillatory flow of an elastico viscous flui...eSAT Journals
Abstract This paper examines the problem of two dimensional unsteady free convection oscillatory flow of an Elastico-viscous fluid past an impulsively started infinite vertical plate with constant heat flux and heat generation. The flow is subjected to a constant suction. Using perturbation techniquethe expressions are obtained for velocity, temperature distribution and skin friction. The results obtained are discussed for various numerical values of the parameters entered into the equations governing the flow, with help of graphs.
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.
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.
Efficient distributed detection of node replication attacks in mobile sensor ...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
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
Available transfer capability computations in the indian southern e.h.v power...eSAT Publishing House
This document summarizes a research paper that compares three methods for computing available transfer capability (ATC) in the Indian southern extra high voltage power system: 1) the conventional continuation repeated power flow method, 2) a radial basis function neural network technique, and 3) an adaptive neuro fuzzy inference system technique. It provides background on ATC calculations and describes the radial basis function neural network and adaptive neuro fuzzy inference system techniques in detail. The paper aims to show that intelligent techniques can more accurately compute ATC online compared to conventional methods. It tests the three ATC computation methods on the 72-bus Indian southern power system.
Study on shear strength characteristics of coir mat reinforced sandeSAT 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.
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
Nutrients retention in functional beef burgers with especial emphasis on lipi...eSAT Publishing House
This document summarizes a study that investigated the retention of nutrients, particularly fatty acids, in healthier beef burgers with different amounts of added dry soybean sprouts (DSS). Several burger formulations were prepared with partial replacement of pork fat with soybean oil and 0g/kg, 5g/kg, 10g/kg or 20g/kg of added DSS. Cooking yield increased with higher DSS amounts, indicating it improved water and fat retention. Higher DSS concentrations also led to higher protein and fat retention. Burgers with 5g/kg or 10g/kg DSS had improved ratios of polyunsaturated to saturated fatty acids after cooking due to reduced saturated fat losses. Multivariate analysis identified
This document summarizes approaches to numerically solving fuzzy differential equations. It begins by introducing fuzzy differential equations and discussing previous work in the area. It then defines key concepts related to fuzzy sets, fuzzy derivatives, and quasi-differential equations. The document proposes using finite difference methods to build fuzzy analogs for solving initial-boundary value problems involving fuzzy differential equations. It describes setting up spatial and temporal grids and defining discrete fuzzy number solutions that approximate the exact solutions to the fuzzy differential equations.
Production of lactic acid from sweet meat industry waste by lactobacillus del...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.
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.
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.
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.
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.
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.
Lake sediment thickness estimation using ground penetrating radareSAT Publishing House
This document summarizes a study that used ground penetrating radar (GPR) to estimate the thickness and volume of sediments in Punem Lake, India. GPR profiles identified two sediment layers in the lake. The thickness of the first layer ranged from 0.02m to 1.16m with an average of 0.5m, while the second layer ranged from 0.08m to 0.99m with an average of 0.37m. Sediment volume was estimated at 260,303 cubic meters for the first layer and 188,171 cubic meters for the second layer. Thicker sediments were found toward the north, northeast, and east of the lake for the first layer and toward the north, northeast, and west for
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.
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
This document discusses cell search and synchronization procedures in Long Term Evolution (LTE) cellular networks. It describes how a mobile unit detects and locks onto base stations when powering on or moving between cells. The mobile unit first detects the Primary Synchronization Signal (PSS) to synchronize on the subframe boundary and determine the physical layer cell identity. It then detects the Secondary Synchronization Signal (SSS) to obtain the cell identity group number and complete cell ID detection. Simulations are presented showing the probability of detecting neighboring cells under different channel conditions and signal-to-noise ratios, with better detection at cell centers compared to edges.
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
Thermal Instability of Chemically Reacting Maxwell Fluid in a Horizontal Poro...IJERA Editor
The effect of chemical reaction on the linear stability of a viscoelastic fluid saturated horizontal densely-packed
porous layer is investigated. The viscoelastic properties are given by Maxwell constitutive relations. The porous
layer is cooled from the upper boundary while an adiabatic thermal boundary condition is imposed at the lower
boundary. Linear stability analysis suggests that there is a competition between the processes of viscous
relaxation and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary.
The effect of Deborah number, Darcy-Prandtl number, normalized porosity, and the Frank-Kamenetskii number
on the stability of the system is investigated. Using a weighted residual method we calculate numerically the
convective thresholds for both stationary and oscillatory instability. The effects of viscoelasticity and chemical
reaction on the instability are emphasized. Some existing results are reproduced as the particular cases of the
present study.
International Journal of Computational Engineering Research(IJCER) ijceronline
nternational Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Effect of Radiation on Mixed Convection Flow of a Non-Newtonian Nan fluid ove...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
1) The document describes a study of the flow of electrically conducting, chemically reacting and radiating viscoelastic fluid through a porous medium in a vertical porous channel with constant suction.
2) Governing equations for the flow velocity, temperature and concentration profiles are derived based on assumptions like uniform magnetic field, Boussinesq approximation, and negligible induced magnetic field.
3) Closed form solutions for the governing equations are obtained subject to boundary conditions of no slip at the channel walls and oscillating temperature and concentration profiles.
LINEAR STABILITY ANALYSIS ON THE ONSET OF DDC IN A DPM SATURATED WITH CSF WIT...IRJET Journal
This document presents a linear stability analysis of double diffusive convection in a rotating couple stress fluid saturated porous layer with an internal heat source. The analysis is based on a classical normal mode technique using a modified Darcy model that includes a time derivative term and Coriolis term. Expressions are obtained for the stationary and oscillatory Rayleigh numbers as functions of governing parameters like the internal Rayleigh number, couple stress parameter, Taylor number, normalized porosity, and Lewis number. The effects of these parameters on the stability of the system are shown graphically.
Magneto-Thermosolutal Convection in Maxwellian Dusty Viscoelastic Fluid throu...IJERA Editor
Thermosolutal convection in Maxwellian viscoelastic fluid under the simultaneous effects of uniform vertical magnetic field and suspended particles through porous medium is mathematically investigated. A sufficient condition for the invalidity of the ‘principle of exchange of stabilities’ is derived, in the context, which states that the exchange principle is not valid provided the thermal Rayleigh number R , solutal Rayleigh number S R , the medium permeability 1 P and the suspended particles parameter B are restricted by the inequality 1 2 1 S R R BP .
Linear and Weakly Non-Linear Stability Analyses of Double-Diffusive ElectroCo...iosrjce
The linear and weakly non-linear stability analyses of double diffusive electro-convention in a micropolar fluid
layer heated and saluted from below and cooled from above is studied. The linear and non-linear analyses are, respectively
based on normal mode technique and truncated representation of Fourier series. The influence of various parameters on the
onset of convection has been analyzed in the linear case. The resulting autonomous Lorenz model obtained in non-linear
analysis is solved numerically to quantify the heat and mass transforms through Nusselt and Sherwood number. It is
observed that the increase in concentration of suspended particles and electric field and electric Rayleigh number increases
the heat and mass transfer
IRJET- Numerical Analysis of Natural Convection of Nano Fluids on Square ...IRJET Journal
This document summarizes a numerical analysis of natural convection of nanofluids in square enclosures. Aluminum oxide and titanium oxide nanoparticles dispersed in water were analyzed at low concentrations from 0.1-0.4% and Rayleigh numbers from 2x10^7 to 8x10^7. The study found that aluminum oxide provided up to 33% enhancement in heat transfer properties over the base fluid, while titanium oxide provided up to 22% enhancement. Both nanofluids showed increased heat transfer with higher Rayleigh numbers but decreased transfer with higher concentrations. The analysis was conducted using ANSYS Fluent software to solve governing equations for the 3D, steady flow using the SIMPLE pressure-linking scheme.
Heat and Mass Transfer Hydromagnetic Radiative Casson Fluid Flow over an Expo...inventionjournals
The present paper examines the hydromagnetic two-dimensional boundary layer flow of a nonNewtonian fluid accompanied by heat and mass transfer towards an exponentially stretching sheet in the presence of chemical reaction and thermal radiation. Casson model is used to characterize non-Newtonian fluid behavior. Using the similarity transformations, the governing partial differential equations are transformed into self-similar ordinary differential equations is solved by using Matlab bvp4c package. The effects of pertinent parameters are presented in both graphical and tabular form.
Heat and Mass Transfer Hydromagnetic Radiative Casson Fluid Flow over an Expo...inventionjournals
The present paper examines the hydromagnetic two-dimensional boundary layer flow of a nonNewtonian fluid accompanied by heat and mass transfer towards an exponentially stretching sheet in the presence of chemical reaction and thermal radiation. Casson model is used to characterize non-Newtonian fluid behavior. Using the similarity transformations, the governing partial differential equations are transformed into self-similar ordinary differential equations is solved by using Matlab bvp4c package. The effects of pertinent parameters are presented in both graphical and tabular form.
Thermal Energy on Water and Oil placed Squeezed Carreau Nanofluids FlowMOHAMMED FAYYADH
this research work is focused on the numerical study regarding Carreau nanofluids’ squeezed flow via a permeable sensor surface. The nanofluids’ thermal conductivity is considered to be dependent on temperature. A convenient transformation is employed to reorganize governing equations into ordinary differential equations. The Runge–Kutta method and shooting technique are employed to accurately solve the boundary layer momentum as well as heat equations. Graphical and tabular aids are used to evaluate the solutions of applicable parameter with regards to temperature as well as the rate of heat transfer. In this work, a comparison is done from three nanofluids, i.e. copper, oxide aluminum and SWCNTs (nanoparticles) based fluids (water, crude oil and ethylene glycol) to improve heat transfer. It is found that the temperature dimensionless was dropped and dominated with the squeezed flow parameter and nanoparticle volume fraction parameter. That is for all nanomaterials. When compared with water and ethylene glycol, crude oil is cooler and a thinner thermal boundary layer is presented. For the rate of heat transfer (Nusselt number) was higher in: Ethylene glycol- SWCNT with high permeable velocity parameter 0.2, Ethylene glycol- SWCNT with low squeeze flow parameter 0.1 and Ethylene glycol- oxide aluminum with low nanoparticle volume fraction 0.05
Solar Radiation Energy Issues on Nanoparticle Shapes in the Potentiality of W...MOHAMMED FAYYADH
Energy is an extensive view for industrial advancement. Solar thermal energy is designed by light and heat which is radiated
by the sun, in the form of electromagnetic radiation. Solar energy is the highest promptly and sufficiently applicable authority of
green energy. Impact of nanoparticle shapes on the Hiemenz nano fluid (water based Cu, Al2O3 and SWCNTs) flow over a porous
wedge surface in view of solar radiation energy has been analyzed. The three classical form of nanoparticle shapes are registered
into report, i.e. sphere , cylinder and laminar . Nanoparticles in the water based Cu, Al2O3 and SWCNTs have been advanced as a
means to boost solar collector energy through explicit absorption of the entering solar energy. The controlling partial differential
equations (PDEs) are remodeled into ordinary differential equations (ODEs) by applying dependable accordance alteration and it is
determined numerically by executing Runge Kutta Fehlberg method with shooting technique. It is anticipated that the lamina shape
SWCNTs have dynamic heat transfer attainments in the flow improvement over a porous wedge surface as compared with the other nanoparticle shapes in different nano fluid flow regime.
The document presents an analytical approach to study the influence of dissipative heat energy on water-based nanofluid flow past a semi-infinite vertical plate embedded in a permeable medium. The nanofluid contains copper, aluminum oxide or titanium oxide nanoparticles suspended in water. Governing equations are derived that account for magnetic field effects, heat source/sink effects, and variable thermal properties. Both numerical and analytical solution techniques are employed to solve the equations, with results presented graphically and tabularly. The homotopy perturbation method is used as the analytical solution technique.
Lattice boltzmann simulation of non newtonian fluid flow in a lid driven cavitIAEME Publication
This document summarizes a study that uses Lattice Boltzmann Method (LBM) to simulate non-Newtonian fluid flow in a lid driven cavity. The study explores the mechanism of non-Newtonian fluid flow using the power law model to represent shear-thinning and shear-thickening fluids. It investigates the influence of power law index and Reynolds number on velocity profiles and streamlines. The LBM code is validated against published results and shows agreement with established theory and fluid rheological behavior.
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.
Unsteady MHD Flow Past A Semi-Infinite Vertical Plate With Heat Source/ Sink:...IJERA Editor
In the present paper a numerical attempt is made to study the combined effects of heat source and sink on unsteady laminar boundary layer flow of a viscous, incompressible, electrically conducting fluid along a semiinfinite vertical plate. A magnetic field of uniform strength is applied normal to the flow. The governing boundary layer equations are solved numerically, using Crank-Nicolson method. Graphical results of velocity and temperature fields, tabular values of Skin-friction and Nusselt are presented and discussed at various parametric conditions. From this study, it is found that the velocity and temperature of the fluid increase in the presence of heat source but they decrease in the presence of heat absorption parameter.
International Journal of 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.
Abstract— A scientific investigation is created to examine the nonlinear unfaltering blended convection limit
layer stream and warmth exchange of an incompressible digression hyperbolicnon-Newtonian liquid from a
non-isothermal wedge in the nearness of attractive field. The changed preservation conditions are understood
numerically subject to physically fitting limit conditions utilizing a second-arrange precise verifiable limited
distinction Keller Box method. The numerical code is accepted with past studies. The impact of various rising
non-dimensional parameters, to be specific Weissenberg number (We), power law record (n), blended
convection parameter, weight angle parameter (m), Prandtl number (Pr), Biot number, attractive parameter
(M)and dimensionless extraneous direction on speed and temperature development in the limit layer
administration are inspected in subtle element. Moreover, the impacts of these parameters on surface warmth
exchange rate and nearby skin erosion are additionally examined. Approval with prior Newtonian studies is
introduced and amazing relationship accomplished. It is found that speed is lessened with expanding We,
while, temperature is increased. Expanding n improves speed yet diminishes temperature, a comparable
pattern was seen. An expanding M is found to decline speed however temperature increments.
Keywords— Magnetic parameter, Mixed Convection parameter, Non-Newtonian digression hyperbolic liquid,
power law index, Weissenberg number, Weight inclination parameter.
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
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.
Similar to A semi circle theorem in magneto-rotatory (20)
Hudhud cyclone caused extensive damage in Visakhapatnam, India in October 2014, especially to tree cover. This will likely impact the local environment in several ways: increased air pollution as trees absorb less; higher temperatures without tree canopy; increased erosion and landslides. It also created large amounts of waste from destroyed trees. Proper management of solid waste is needed to prevent disease spread. Suggested measures include restoring damaged plants, building fountains to reduce heat, mandating light-colored buildings, improving waste management, and educating public on health risks. Overall, changes are needed to water, land, and waste practices to rebuild the environment after the cyclone removed green cover.
Impact of flood disaster in a drought prone area – case study of alampur vill...eSAT Publishing House
1) In September-October 2009, unprecedented heavy rainfall and dam releases caused widespread flooding in Alampur village in Mahabub Nagar district, a historically drought-prone area.
2) The flood damaged or destroyed homes, buildings, infrastructure, crops, and documents. It displaced many residents and cut off the village.
3) The socioeconomic conditions and mud-based construction of homes in the village exacerbated the flood's impacts, making damage more severe and recovery more difficult.
The document summarizes the Hudhud cyclone that struck Visakhapatnam, India in October 2014. It describes the cyclone's formation, rapid intensification to winds of 175 km/h, and landfall near Visakhapatnam. The cyclone caused extensive damage estimated at over $1 billion and at least 109 deaths in India and Nepal. Infrastructure like buildings, bridges, and power lines were destroyed. Crops and fishing boats were also damaged. The document then discusses coping strategies and improvements needed to disaster management plans to better prepare for future cyclones.
Groundwater investigation using geophysical methods a case study of pydibhim...eSAT Publishing House
This document summarizes the results of a geophysical investigation using vertical electrical sounding (VES) methods at 13 locations around an industrial area in India. The VES data was interpreted to generate geo-electric sections and pseudo-sections showing subsurface resistivity variations. Three main layers were typically identified - a high resistivity topsoil, a weathered middle layer, and a basement rock. Pseudo-sections revealed relatively more weathered areas in the northwest and southwest. Resistivity sections helped identify zones of possible high groundwater potential based on low resistivity anomalies sandwiched between more resistive layers. The study concluded the electrical resistivity method was useful for understanding subsurface geology and identifying areas prospective for groundwater exploration.
Flood related disasters concerned to urban flooding in bangalore, indiaeSAT Publishing House
1. The document discusses urban flooding in Bangalore, India. It describes how factors like heavy rainfall, population growth, and improper land use have contributed to increased flooding in the city.
2. Flooding events in 2013 are analyzed in detail. A November rainfall caused runoff six times higher than the drainage capacity, inundating low-lying residential areas.
3. Impacts of urban flooding include disrupted daily life, damaged infrastructure, and decreased economic activity in affected areas. The document calls for improved flood management strategies to better mitigate urban flooding risks in Bangalore.
Enhancing post disaster recovery by optimal infrastructure capacity buildingeSAT Publishing House
This document discusses enhancing post-disaster recovery through optimal infrastructure capacity building. It presents a model to minimize the cost of meeting demand using auxiliary capacities when disaster damages infrastructure. The model uses genetic algorithms to select optimal capacity combinations. The document reviews how infrastructure provides vital services supporting recovery activities and discusses classifying infrastructure into six types. When disaster reduces infrastructure services, a gap forms between community demands and available support, hindering recovery. The proposed research aims to identify this gap and optimize capacity selection to fill it cost-effectively.
Effect of lintel and lintel band on the global performance of reinforced conc...eSAT Publishing House
This document analyzes the effect of lintels and lintel bands on the seismic performance of reinforced concrete masonry infilled frames through non-linear static pushover analysis. Four frame models are considered: a frame with a full masonry infill wall; a frame with a central opening but no lintel/band; a frame with a lintel above the opening; and a frame with a lintel band above the opening. The results show that the full infill wall model has 27% higher stiffness and 32% higher strength than the model with just an opening. Models with lintels or lintel bands have slightly higher strength and stiffness than the model with just an opening. The document concludes lintels and lintel
Wind damage to trees in the gitam university campus at visakhapatnam by cyclo...eSAT Publishing House
1) A cyclone with wind speeds of 175-200 kph caused massive damage to the green cover of Gitam University campus in Visakhapatnam, India. Thousands of trees were uprooted or damaged.
2) A study assessed different types of damage to trees from the cyclone, including defoliation, salt spray damage, damage to stems/branches, and uprooting. Certain tree species were more vulnerable than others.
3) The results of the study can help in selecting more wind-resistant tree species for future planting and reducing damage from future storms.
Wind damage to buildings, infrastrucuture and landscape elements along the be...eSAT Publishing House
1) A visual study was conducted to assess wind damage from Cyclone Hudhud along the 27km Visakha-Bheemli Beach road in Visakhapatnam, India.
2) Residential and commercial buildings suffered extensive roof damage, while glass facades on hotels and restaurants were shattered. Infrastructure like electricity poles and bus shelters were destroyed.
3) Landscape elements faced damage, including collapsed trees that damaged pavements, and debris in parks. The cyclone wiped out over half the city's green cover and caused beach erosion around protected areas.
1) The document reviews factors that influence the shear strength of reinforced concrete deep beams, including compressive strength of concrete, percentage of tension reinforcement, vertical and horizontal web reinforcement, aggregate interlock, shear span-to-depth ratio, loading distribution, side cover, and beam depth.
2) It finds that compressive strength of concrete, tension reinforcement percentage, and web reinforcement all increase shear strength, while shear strength decreases as shear span-to-depth ratio increases.
3) The distribution and amount of vertical and horizontal web reinforcement also affects shear strength, but closely spaced stirrups do not necessarily enhance capacity or performance.
Role of voluntary teams of professional engineers in dissater management – ex...eSAT Publishing House
1) A team of 17 professional engineers from various disciplines called the "Griha Seva" team volunteered after the 2001 Gujarat earthquake to provide technical assistance.
2) The team conducted site visits, assessments, testing and recommended retrofitting strategies for damaged structures in Bhuj and Ahmedabad. They were able to fully assess and retrofit 20 buildings in Ahmedabad.
3) Factors observed that exacerbated the earthquake's impacts included unplanned construction, non-engineered buildings, improper prior retrofitting, and defective materials and workmanship. The professional engineers' technical expertise was crucial for effective post-disaster management.
This document discusses risk analysis and environmental hazard management. It begins by defining risk, hazard, and toxicity. It then outlines the steps involved in hazard identification, including HAZID, HAZOP, and HAZAN. The document presents a case study of a hypothetical gas collecting station, identifying potential accidents and hazards. It discusses quantitative and qualitative approaches to risk analysis, including calculating a fire and explosion index. The document concludes by discussing hazard management strategies like preventative measures, control measures, fire protection, relief operations, and the importance of training personnel on safety.
Review study on performance of seismically tested repaired shear wallseSAT Publishing House
This document summarizes research on the performance of reinforced concrete shear walls that have been repaired after damage. It begins with an introduction to shear walls and their failure modes. The literature review then discusses the behavior of original shear walls as well as different repair techniques tested by other researchers, including conventional repair with new concrete, jacketing with steel plates or concrete, and use of fiber reinforced polymers. The document focuses on evaluating the strength retention of shear walls after being repaired with various methods.
Monitoring and assessment of air quality with reference to dust particles (pm...eSAT Publishing House
This document summarizes a study on monitoring and assessing air quality with respect to dust particles (PM10 and PM2.5) in the urban environment of Visakhapatnam, India. Sampling was conducted in residential, commercial, and industrial areas from October 2013 to August 2014. The average PM2.5 and PM10 concentrations were within limits in residential areas but moderate to high in commercial and industrial areas. Exceedance factor levels indicated moderate pollution for residential areas and moderate to high pollution for commercial and industrial areas. There is a need for management measures like improved public transport and green spaces to combat particulate air pollution in the study areas.
Low cost wireless sensor networks and smartphone applications for disaster ma...eSAT Publishing House
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#Abstract:
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- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
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- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
1. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 229
A SEMI-CIRCLE THEOREM IN MAGNETO-ROTATORY
THERMOSOLUTAL CONVECTION IN RIVLIN-ERICKSEN
VISCOELASTIC FLUID IN A POROUS MEDIUM
Ajaib S. Banyal
Department of Mathematics, Govt. College Nadaun (Hamirpur), (HP) INDIA 177033
ajaibbanyal@rediffmail.com, Fax No. : 01972232688
Abstract
Thermosolutal convection in a layer of Rivlin-Ericksen viscoelastic fluid of Veronis (1965) type is considered in the presence of
uniform vertical magnetic field and rotation in a porous medium. Following the linearized stability theory and normal mode analysis,
the paper through mathematical analysis of the governing equations of Rivlin-Ericksen viscoelastic fluid convection in the presence of
uniform vertical magnetic field and rotation, for any combination of perfectly conducting free and rigid boundaries of infinite
horizontal extension at the top and bottom of the fluid, established that the complex growth rate of oscillatory perturbations,
neutral or unstable for all wave numbers, must lie inside right half of the a semi-circle
FP
P
pE
R
TQQ
FP
P
Maximumof
l
ls
A
l
l
ir
3
'
2
2
2
22
,4
4
1
,
in the ir -plane, where sR is the thermosolutal Rayleigh number, AT is the Taylor number, F is the viscoelasticity parameter, 3p
is the thermosolutal prandtl number, is the porosity and lP is the medium permeability. This prescribes the bounds to the complex
growth rate of arbitrary oscillatory motions of growing amplitude in the Rivlin-Ericksen viscoelastic fluid in Veronis (1965) type
configuration in the presence of uniform vertical magnetic field and rotation in a porous medium. A similar result is also proved for
Stern (1960) type of configuration. The result is important since the result hold for any arbitrary combinations of dynamically free and
rigid boundaries.
Keywords: Thermosolutal convection; Rivlin-Ericksen Fluid; Magnetic field; Rotation; PES; Rayleigh number;
Chandrasekhar number; Taylor number.
MSC 2000 No.: 76A05, 76E06, 76E15; 76E07; 76U05.
--------------------------------------------------------------------------------------------------------------------------------------------------
1. INTRODUCTION
Chandrasekhar 1 in his celebrated monograph presented a
comprehensive account of the theoretical and experimental
study of the onset of Bénard Convection in Newtonian fluids,
under varying assumptions in hydrodynamics and
hydromagnetics. The use of Boussinesq approximation has
been made throughout, which states that the density changes
are disregarded in all other terms in the equation of motion
except the external force term. The problem of thermohaline
convection in a layer of fluid heated from below (above) and
subjected to a stable (destabilizing) salinity gradient has been
considered by Veronis 2 and Stren 3 respectively. The
physics is quite similar in the stellar case, in that helium acts
like in raising the density and in diffusing more slowly than
heat. The problem is of great importance because of its
applications to atmospheric physics and astrophysics, especially
in the case of the ionosphere and the outer layer of the
atmosphere. The thermosolutal convection problems also arise
in oceanography, limnology and engineering. Bhatia and
Steiner 4 have considered the effect of uniform rotation on
the thermal instability of a viscoelastic (Maxwell) fluid and
found that rotation has a destabilizing influence in contrast to
the stabilizing effect on Newtonian fluid. Sharma 5 has
studied the thermal instability of a layer of viscoelastic
(Oldroydian) fluid acted upon by a uniform rotation and found
that rotation has destabilizing as well as stabilizing effects
under certain conditions in contrast to that of a Maxwell fluid
where it has a destabilizing effect. There are many elastico-
2. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 230
viscous fluids that cannot be characterized by Maxwell’s
constitutive relations or Oldroyd’s 6 constitutive relations.
Two such classes of fluids are Rivlin-Ericksen’s and Walter’s
(model B’) fluids. Rivlin-Ericksen 7 has proposed a
theoretical model for such one class of elastico-viscous fluids.
Sharma and kumar 8 have studied the effect of rotation on
thermal instability in Rivlin-Ericksen elastico-viscous fluid and
found that rotation has a stabilizing effect and introduces
oscillatory modes in the system. Kumar et al. 9 considered
effect of rotation and magnetic field on Rivlin-Ericksen
elastico-viscous fluid and found that rotation has stabilizing
effect; where as magnetic field has both stabilizing and
destabilizing effects. A layer of such fluid heated from below or
under the action of magnetic field or rotation or both may find
applications in geophysics, interior of the Earth, Oceanography,
and the atmospheric physics. With the growing importance of
non-Newtonian fluids in modern technology and industries, the
investigations on such fluids are desirable.
In all above studies, the medium has been considered to be
non-porous with free boundaries only, in general. In recent
years, the investigation of flow of fluids through porous media
has become an important topic due to the recovery of crude oil
from the pores of reservoir rocks. When a fluid permeates a
porous material, the gross effect is represented by the Darcy’s
law. As a result of this macroscopic law, the usual viscous term
in the equation of Rivlin-Ericksen fluid motion is replaced by
the resistance term
q
tk
'
1
1
, where and
'
are the viscosity and viscoelasticity of the Rivlin-Ericksen
fluid, 1k is the medium permeability and q is the Darcian
(filter) velocity of the fluid. The problem of thermosolutal
convection in fluids in a porous medium is of great importance
in geophysics, soil sciences, ground water hydrology and
astrophysics. Generally, it is accepted that comets consist of a
dusty ‘snowball’ of a mixture of frozen gases which, in the
process of their journey, changes from solid to gas and vice-
versa. The physical properties of the comets, meteorites and
interplanetary dust strongly suggest the importance of non-
Newtonian fluids in chemical technology, industry and
geophysical fluid dynamics. Thermal convection in porous
medium is also of interest in geophysical system,
electrochemistry and metallurgy. A comprehensive review of
the literature concerning thermal convection in a fluid-saturated
porous medium may be found in the book by Nield and Bejan
10 .
Pellow and Southwell 11 proved the validity of PES for the
classical Rayleigh-Bénard convection problem. Banerjee et al
12 gave a new scheme for combining the governing
equations of thermohaline convection, which is shown to lead
to the bounds for the complex growth rate of the arbitrary
oscillatory perturbations, neutral or unstable for all
combinations of dynamically rigid or free boundaries and,
Banerjee and Banerjee 13 established a criterion on
characterization of non-oscillatory motions in hydrodynamics
which was further extended by Gupta et al. 14 . However no
such result existed for non-Newtonian fluid configurations in
general and in particular, for Rivlin-Ericksen viscoelastic fluid
configurations. Banyal 15 have characterized the oscillatory
motions in Rivlin-Ericksen fluid in the presence of rotation.
Keeping in mind the importance of non-Newtonian fluids, as
stated above, the present paper is an attempt to prescribe the
bounds to the complex growth rate of arbitrary oscillatory
motions of growing amplitude, in a thermosolutal convection of
a layer of incompressible Rivlin-Ericksen fluid configuration of
Veronis 2 type in the presence of uniform vertical magnetic
field and rotation in a porous medium, when the bounding
surfaces are of infinite horizontal extension, at the top and
bottom of the fluid and are with any arbitrary combination of
perfectly conducting dynamically free and rigid boundaries. A
similar result is also proved for Stern 3 type of configuration.
The result is important since the result hold for any arbitrary
combinations of dynamically free and rigid boundaries.
2. FORMULATION OF THE PROBLEM AND
PERTURBATION EQUATIONS
Here we Consider an infinite, horizontal, incompressible
electrically conducting Rivlin-Ericksen viscoelastic fluid layer,
of thickness d, heated from below so that, the temperature,
density and solute concentrations at the bottom surface z = 0
are 0T , 0 and 0C and at the upper surface z = d are dT ,
d and dC respectively, and that a uniform adverse
temperature gradient
dz
dT
and a uniform solute gradient
dz
dC'
is maintained. The gravity field
gg
,0,0 ,
uniform vertical rotation
,0,0 and a uniform vertical
magnetic field pervade on the system
HH ,0,0
.This fluid
layer is assumed to be flowing through an isotropic and
homogeneous porous medium of porosity and medium
permeability 1k .
3. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 231
Let p , , T, C , ,
'
, g , , e and wvuq ,,
denote
respectively the fluid pressure, fluid density temperature, solute
concentration, thermal coefficient of expansion, an analogous
solvent coefficientof expansion, gravitational acceleration,
resistivity, magnetic permeability and filter velocity of the
fluid. Then the momentum balance, mass balance, and energy
balance equation of Rivlin-Ericksen fluid and Maxwell’s
equations through porous medium, governing the flow of
Rivlin-Ericksen fluid in the presence of uniform vertical
magnetic field and uniform vertical rotation (Rivlin and
Ericksen 7 ; Chandrasekhar 1 and Sharma et al 16 ) are
given by
q
tk
gpqq
t
q '
100
1
1
1
.
11
,
2
)(
4
qHH
o
e
(1)
0.
q , (2)
TTq
t
T
E 2
).(
, (3)
CCq
t
C
E 2''
).(
(4)
HqH
dt
Hd 2
).( , (5)
0.
H , (6)
Where
.1
q
tdt
d
, stands for the convective
derivatives. Here
i
ss
c
c
E
0
)1(
, is a constant and
'
E is a
constant analogous to E but corresponding to solute rather
than heat, while s , sc and 0 , ic , stands for the density
and heat capacity of the solid (porous matrix) material and the
fluid, respectively, is the medium porosity
and ),,( zyxr
.
The equation of state is
)(1 0
'
00 CCTT , (7)
Where the suffix zero refer to the values at the reference level z
= 0. In writing the equation (1), we made use of the Boussinesq
approximation, which states that the density variations are
ignored in all terms in the equation of motion except the
external force term. The kinematic viscosity , kinematic
viscoelasticity
'
, magnetic permeability e , thermal
diffusivity , the solute diffusivity
'
, and electrical
resistivity , and the coefficient of thermal expansion are
all assumed to be constants.
The steady state solution is
0,0,0
q , )1( ''
0 zz ,
0TzT , 0
'
CzC (8)
Here we use the linearized stability theory and the normal
mode analysis method. Consider a small perturbations on the
steady state solution, and let , p , , , wvuq ,,
and
zyx hhhh ,,
denote respectively the perturbations in
density , pressure p, temperature T, solute concentration C,
velocity )0,0,0(
q and the magnetic field HH ,0,0
. The
change in density , caused mainly by the perturbation and
in temperature and concentration, is given by
)( '
0 . (9)
Then the linearized perturbation equations of the Rinlin-
Ericksen fluid reduces to
qHhq
tk
gp
t
q e
2
4
1
)()(
11
0
'
1
'
0
(10)
0.
q , (11)
4. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 232
2
w
t
E , (12)
2'''
w
t
E , (13)
hqH
t
h 2
. . (14)
And 0.
h , (15)
Where 2
2
2
2
2
2
2
zyx
.
3. NORMAL MODE ANALYSIS
Analyzing the disturbances into two-dimensional waves, and
considering disturbances characterized by a particular wave
number, we assume that the Perturbation quantities are of the
form
)(,),(),(,,,,,, , zXzZzzKzzWhw z e
xp ntyikxik yx , (16)
Where yx kk , are the wave numbers along the x- and y-
directions, respectively, 2
1
22
yx kkk , is the resultant
wave number, n is the growth rate which is, in general, a
complex constant and
y
u
x
v
and
y
h
x
h xy
denote the z-component of vorticity and
current density respectively and
)(),(),(),(),( zZzzzKzW and )(zX are the functions
of z only.
Using (16), equations (10)-(15), within the framework of
Boussinesq approximations, in the non-dimensional form
transform to
DKaDQ
DZTaRRaWaDF
P
As
l
22
2222
)1(
1
(17)
QDXDWZF
Pl
)1(
1
, (18)
DWKpaD 2
22
, (19)
DZXpaD 2
22
, (20)
WEpaD 1
22
, (21)
and
WpEaD 3
'22
, (22)
Where we have introduced new coordinates ',',' zyx = (x/d,
y/d, z/d) in new units of length d and '/ dzdD . For
convenience, the dashes are dropped hereafter. Also we have
substituted ,,
2
nd
kda
1p is the thermal
Prandtl number; '3
p is the thermosolutal Prandtl
number;
2p is the magnetic Prandtl number; 2
1
d
k
Pl
is the dimensionless medium permeability, 2
'
d
F
is the
dimensionless viscoelasticity parameter of the Rivlin-Ericksen
fluid;
4
dg
R is the thermal Rayleigh number;
''
4''
dg
Rs is the thermosolutal Rayleigh number;
0
22
4
dH
Q e
is the Chandrasekhar number and
22
42
4
d
TA
is the Taylor number. Also we have
Substituted WW ,
2
d
,
2'
d
,
Z
d
Z
2
, K
Hd
K
,
5. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 233
X
dHd
X
2
and dDD , dropped for
convenience.
We now consider the cases where the boundaries are rigid-rigid
or rigid-free or free-rigid or free-free at z = 0 and z = 1
respectively, as the case may be, are perfectly conducting and
maintained at constant temperature and solute concentration.
Then the perturbations in the temperature and solute
concentration are zero at the boundaries. The appropriate
boundary conditions with respect to which equations (13)--(16),
must possess a solution are
W = 0 =K= , on both the horizontal boundaries,
DW = 0=Z=DX, on a rigid boundary,
DZWD 02
=X, on a dynamically free
boundary, (23)
Equations (17)--(22), along with boundary conditions (23),
pose an eigenvalue problem for and we wish to
characterize i , when 0r .
4. MATHEMATICAL ANALYSIS
We prove the following lemma:
Lemma 1: For any arbitrary oscillatory perturbation, neutral or
unstable
1
0
2
22
3
2'
1
0
2 1
dzW
pE
dz
Proof: Further, multiplying equation (22) and its complex
conjugate, and integrating by parts each term on right hand side
of the resulting equation for an appropriate number of times
and making use of boundary conditions on namely
)1(0)0( along with (22), we get
1
0
1
0
2222
3
2'
1
0
222
3
'
1
0
222
2
dzWdzpE
dzaDpE
dzaD
r
(24)
Since 0r therefore the equation (24) gives,
1
0
2
22
3
2'
1
0
2 1
dzW
pE
dz
(25)
This completes the proof of lemma.
Lemma 2: For any arbitrary oscillatory perturbation, neutral or
unstable
1
0
2
22
1
2
1
0
2 1
dzW
pE
dz
Proof: Further, multiplying equation (21) and its complex
conjugate, and integrating by parts each term on right hand side
of the resulting equation for an appropriate number of times
and making use of boundary conditions on namely
)1(0)0( along with (21), we get
1
0
1
0
2222
3
2'
1
0
222
3
'
1
0
222
2
dzWdzpE
dzaDpEdzaD r
(26)
Since
0r therefore the equation (26) gives,
1
0
2
22
1
2
1
0
2 1
dzW
pE
dz
(27)
This completes the proof of lemma.
Lemma 3: For any arbitrary oscillatory perturbation, neutral or
unstable
1
0
2
2
2
1
0
2
1
1
dzDW
P
F
dzZ
l
Proof: Further, multiplying equation (20) with its complex
conjugate, and integrating by parts each term on both sides of
the resulting equation for an appropriate number of times and
making use of appropriate boundary conditions (23), we get
1
0
2
2
2
2 1211
dzZ
P
F
PPP
F
ll
r
ll
6. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 234
1
0
1
0
*22
1
0
2
.Re2 dzDWDXPartofQdzDXQdzDW
(28)
Multiplying the complex conjugate of equation (16) by DX
and integrating over the range of z, we get
1
0
2
1
0
*
1
0
**
*
)1(
1
dzDXQdzDXDW
DXdzZF
Pl
, (29)
Utilizing the equation (18) and appropriate boundary condition
DX (0) = 0 = DX (1), we get
1
0
1
0
1
0
**
2
22**
dzXpaDXdzXDZdzDXZ
1
0
2
1
0
*
2
222
dzXpdzXaDX
(30)
Substituting (30) in (29), we get
1
0
2
1
0
*
.Re dzDXQdzDXDWPartof
1
0
1
0
2
2
222
)1(
1
dzXpdzXaDXF
P
rr
l
r
, (31)
Substituting (31) in (28), we get
1
0
22
1
0
2
2
2
2 1211
dzDXQdzZ
P
F
PPP
F
ll
r
ll
1
0
2
1
0
1
0
2
2
222
)1(
1
2
dzDW
dzXpdzXaDX
F
P
Q
r
r
l
r
..........................(32)
Now F 0, 0Q ,
02 p and
0r , therefore the equation
(32), gives
1
0
2
2
2
1
0
2
1
1
dzDW
P
F
dzZ
l
, (33)
This completes the proof of lemma.
Lemma 4: For any arbitrary oscillatory perturbation, neutral or
unstable
1
0
2
2
1
0
222 1
dzDW
p
dzKaDK
Proof: Multiplying equation (19) and its complex conjugate,
and integrating by parts each term on both sides of the
resulting equation for an appropriate number of times and
making use of boundary conditions on K namely
)1(0)0( KK , we get
1
0
1
0
2222
2
1
0
222
2
1
0
222
2
dzDWdzKp
dzKaDKpdzKaD r
(34)
Since
02 p and
0r , therefore the equation (34) give,
1
0
2
1
0
222
dzDWdzKaD
( 35)
And
1
0
2
22
2
1
0
2 1
dzDW
p
dzK
, (36)
It is easily seen upon using the boundary conditions (23) that
1
0
222
dzKaDK
Real part of
1
0
22
KdzaDK
1
0
22
KdzaDK
,
7. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
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Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 235
dzKaDK
1
0
22
dzKaDK
1
0
22
,
dzKaDK
1
0
22
2
1
1
0
222
2
1
1
0
2
dzKaDdzK
,
(Utilizing
Cauchy-Schwartz-inequality)
Upon utilizing the inequality (35) and (36), we get
1
0
2
2
1
0
222 1
dzDW
p
dzKaDK
, (37)
This completes the proof of lemma.
We prove the following theorem:
Theorem 1: If R 0 ,
0sR , F 0, 0Q ,
AT 0,
0lP ,
01p ,
02 p ,
03 p ,
0r and
0i then the
necessary condition for the existence of non-trivial solution
XZKW ,,,,, of equations (17) – (22), together with
boundary conditions (23) is that
2
22
3
'
4
1
14
Qp
PT
P
F
pE
R lA
l
s
.
Proof: Multiplying equation (17) by
W (the complex
conjugate of W) throughout and integrating the resulting
equation over the vertical range of z, we get
1
0
*2
1
0
22
)1(
1
dzWRaWdzaDWF
Pl
KdzaDDWQDZdzWTdzWaR As
22*
1
0
1
0
1
0
2
, (38)
Taking complex conjugate on both sides of equation (21), we
get
WEpaD 1
22
, (39)
Therefore, using (39), we get
1
0
1
0
1
22
dzEpaDdzW
, (40)
Taking complex conjugate on both sides of equation (22), we
get
WpEaD 3
'22
, (41)
Therefore, using (41), we get
1
0
1
0
3
'22
dzpEaDdzW
, (42)
Also taking complex conjugate on both sides of equation (18),
we get
DWQDXZF
Pl
**
*
)1(
1
, (43)
Therefore, using (43), we get
1
0
*
1
0
1
0
**
*1
0
)1(
1
dzZDXQZdzZF
P
ZdzDWDZdzW
l
,
(44)
Integrating by parts the third term on left hand side of equation
(44) and using equation (20), and appropriate boundary
condition (23), we get
1
0
*
2
22
1
0
1
0
*
*
)1(
1
dzXpaDXQZdzZF
P
DZdzW
l
, (45)
Also taking complex conjugate on both sides of equation (19),
we get
DWKpaD 2
22
, (46)
Therefore, utilizing equation (45), and appropriate boundary
condition (23), we get
1
0
1
0
1
0
2
22222222
dzKpaDaDKKdzaDDWKdzaDDW
, (47)
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Substituting (40), (42), (45) and (47), in the right hand side of
equation (38), we get
dzpEaDaRdzEpaDRaWdzaDWF
P
s
l
1
0
*
3
'22*2
1
0
**
1
22222
1
0
)1(
1
1
0
*
2
22
1
0
**
*
)1(
1
dzXpaDXQTZdzZF
P
T A
l
A
KdzaDKQpKdzaDKQ 22
1
0
1
0
*
2
222
, (48)
Integrating the terms on both sides of equation (48) for an
appropriate number of times and making use of the appropriate
boundary conditions (23), we get
1
0
2*
1
2222
1
0
222
)1(
1
dzEpaDRadzWaDWF
Pl
1
0
2
2
222
1
0
2*
*1
0
2*
3
'2222
)1(
1
dzXpXaDXQTdzZF
P
TdzpEaDaR A
l
As
1
0
1
0
222*
2
242222
2 dzKaDKQpdzKaDKaKDQ
, (49)
Now equating imaginary parts on both sides of equation (49),
and cancelling
)0(i , we get
1
0
2
3
'2
1
0
2
1
2
1
0
2221
dzpEaRdzEpRadzWaDW
P
F
s
l
1
0
222
2
1
0
2
2
1
0
21
dzKaDKQpdzXQpTdzZ
P
F
T A
l
A
,(50)
Now R 0,
0sR ,
0Q , 0 and
AT 0, utilizing the
inequalities (25) ,(33) and (37), the equation (50) gives,
0
1
1
0
1
0
2
1
22
2
3
'
2
1
0
2
2
dzEpRadzW
pE
R
P
F
adzDW
Q
FP
PT
P
FP
s
l
l
lA
l
l
(51)
Therefore, we must have
FP
P
pE
R
TQQ
FP
P
Maximumof
l
ls
A
l
l
3
'
2
2
2
2
,
4
4
1
(52)
Hence, if
0r and
0i , then
FP
P
pE
R
TQQ
FP
P
Maximumof
l
ls
A
l
l
3
'
2
2
2
2
,
4
4
1
, (53)
And this completes the proof of the theorem.
Theorem 2: If R 0 ,
0sR , F 0,
0lP ,
01p ,
03 p ,
0r and
0i then the necessary
condition for the existence of non-trivial solution
,,,,, XKZW of equations (17) – (20), together with
boundary conditions (21) is that
FP
P
Ep
R
TQQ
FP
P
Maximumof
l
l
A
l
l
1
2
2
2
2
,
4
4
1
(54)
Proof: Replacing R and sR
by
R
and sR
,
respectively in equations (17) – (22) and proceeding exactly as
9. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
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in Theorem 1 and utilizing the inequality (27), we get the
desired result.
CONCLUSIONS
The inequality (53) for 0r and
0i , can be written as
FP
P
pE
R
TQQ
FP
P
Maximumof
l
ls
A
l
l
ir
3
'
2
2
2
22
,
4
4
1
The essential content of the theorem, from the point of view of
linear stability theory is that for the thermosolutal Veronis
(1965) type configuration of Rivlin-Ericksen viscoelastic fluid
in the presence of uniform vertical rotation in a porous medium,
having top and bottom bounding surfaces of infinite horizontal
extension, with any arbitrary combination of dynamically free
and rigid boundaries in a porous medium, the complex growth
rate of an arbitrary oscillatory motions of growing amplitude,
lies inside a semi-circle in the right half of the r i - plane
whose centre is at the origin and radius is equal to
FP
P
pE
R
TQQ
FP
P
Maximumof
l
ls
A
l
l
3
'
2
2
2
,
4
4
1
where sR
is the thermosolutal Rayleigh number, Q is the
Chandrasekhar number, AT is the Taylor number, F is the
viscoelasticity parameter, 3p
is the thermosolutal prandtl
number, is the porosity and lP is the medium permeability.
The result is important since it hold for any arbitrary
combinations of dynamically free and rigid boundaries. The
similar conclusions are drawn for the thermosolutal
configuration of Stern (1960) type of Rivlin-Ericksen
viscoelastic fluid of infinite horizontal extension in the
presence of uniform vertical magnetic field in a porous
medium, for any arbitrary combination of free and rigid
boundaries at the top and bottom of the fluid from Theorem 2.
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