This document presents a numerical study of micropolar fluid flow and heat transfer through a vertical plate in a porous medium. Governing equations for momentum, angular momentum, and energy are derived and made dimensionless using transformations. An explicit finite difference method is used to solve the dimensionless equations numerically. Results are obtained for velocity, microrotation, and temperature for varying values of parameters like the microrotational parameter, spin gradient viscosity parameter, magnetic parameter, vortex viscosity parameter, Grashof number, and Prandtl number. Graphs of these variables are generated using tecplot software to illustrate the effects of the different parameters on the flow behavior.
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...IOSRJM
This paper deals with the influence of magnetic field on peristaltic flow of an incompressible Williamson fluid in a symmetric channel with heat and mass transfer. Convective conditions of heat and mass transfer are employed. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
Effect of Mass Transfer and Hall Current on Unsteady MHD Flow with Thermal Di...IJERA Editor
The paper investigated the effect of mass transfer and Hall current on unsteady MHD flow with Thermal Diffusivity of a viscoelastic fluid in a porous medium. The resultant equations have been solved analytically. The velocity, temperature and concentration distributions are derived, and their profiles for various physical parameters are shown through graphs. The coefficient of Skin friction, Nusselt number and Sherwood number at the plate are derived and their numerical values for various physical parameters are presented through tables. The influence of various parameters such as the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter, Hartmann number, Hall parameter, and the frequency of oscillation on the flow field are discussed. It is seen that, the velocity increases with the increase in Gc, Gr, m and K, and it decreases with increase in Sc,M, n and Pr, temperature decreases with increase in Pr and n, Also, the concentration decreases with the increase in Sc and n.
In this study, nonlinear singularly perturbed problems with nonlocal condition are evaluated by finite
difference method. The exact solution 푢(푥) has boundary layers at 푥 = 0 and 푥 = 1. We present some
properties of the exact solution of the multi-point boundary value problem (1)-(3). According to the perturbation
parameter, by the method of integral identities with the use exponential basis functions and interpolating
quadrature rules with the weight and remainder terms in integral form uniformly convergent finite difference
scheme on Bakhvalov mesh is established. The error analysis for the difference scheme is performed. 휀 −
uniform convergence for approximate solution in the discrete maximum norm is provided, which is the firstorder
(O(h)). This theoretical process is applied on the sample. By Thomas Algorithm, it has been shown to be
consistent with the theoretical results of numerical results. The results were embodied in table and graphs. The
relationship between the approximate solution with the exact solution are obtained by Maple 10 computer
program.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Numerical simulation on laminar convection flow and heat transfer over an iso...eSAT Journals
Abstract A numerical algorithm is presented for studying laminar convection flow and heat transfer over an isothermal vertical horizontal plate embedded in a saturated porous medium. By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a pair of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab environment. Velocity and temperature profiles are illustrated graphically. Heat transfer parameters are derived. Keywords
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...IOSRJM
This paper deals with the influence of magnetic field on peristaltic flow of an incompressible Williamson fluid in a symmetric channel with heat and mass transfer. Convective conditions of heat and mass transfer are employed. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
Effect of Mass Transfer and Hall Current on Unsteady MHD Flow with Thermal Di...IJERA Editor
The paper investigated the effect of mass transfer and Hall current on unsteady MHD flow with Thermal Diffusivity of a viscoelastic fluid in a porous medium. The resultant equations have been solved analytically. The velocity, temperature and concentration distributions are derived, and their profiles for various physical parameters are shown through graphs. The coefficient of Skin friction, Nusselt number and Sherwood number at the plate are derived and their numerical values for various physical parameters are presented through tables. The influence of various parameters such as the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter, Hartmann number, Hall parameter, and the frequency of oscillation on the flow field are discussed. It is seen that, the velocity increases with the increase in Gc, Gr, m and K, and it decreases with increase in Sc,M, n and Pr, temperature decreases with increase in Pr and n, Also, the concentration decreases with the increase in Sc and n.
In this study, nonlinear singularly perturbed problems with nonlocal condition are evaluated by finite
difference method. The exact solution 푢(푥) has boundary layers at 푥 = 0 and 푥 = 1. We present some
properties of the exact solution of the multi-point boundary value problem (1)-(3). According to the perturbation
parameter, by the method of integral identities with the use exponential basis functions and interpolating
quadrature rules with the weight and remainder terms in integral form uniformly convergent finite difference
scheme on Bakhvalov mesh is established. The error analysis for the difference scheme is performed. 휀 −
uniform convergence for approximate solution in the discrete maximum norm is provided, which is the firstorder
(O(h)). This theoretical process is applied on the sample. By Thomas Algorithm, it has been shown to be
consistent with the theoretical results of numerical results. The results were embodied in table and graphs. The
relationship between the approximate solution with the exact solution are obtained by Maple 10 computer
program.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Numerical simulation on laminar convection flow and heat transfer over an iso...eSAT Journals
Abstract A numerical algorithm is presented for studying laminar convection flow and heat transfer over an isothermal vertical horizontal plate embedded in a saturated porous medium. By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a pair of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab environment. Velocity and temperature profiles are illustrated graphically. Heat transfer parameters are derived. Keywords
Effect of an Inclined Magnetic Field on Peristaltic Flow of Williamson Fluid ...QUESTJOURNAL
ABSTRACT: This paper deals with the influence ofinclined magnetic field on peristaltic flow of an incompressible Williamson fluid in an inclined channel with heat and mass transfer. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
The approximate bound state of the nonrelativistic Schrӧdinger equation was
obtained with the modified trigonometric scarf type potential in the framework of
asymptotic iteration method for any arbitrary angular momentum quantum number l
using a suitable approximate scheme to the centrifugal term. The effect of the screening
parameter and potential depth on the eigenvalue was studied numerically. Finally, the
scattering phase shift of the nonrelativistic Schrӧdinger equation with the potential
under consideration was calculated.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...IRJESJOURNAL
Abstract:- Buckling analysis of a carbon nanotube (CNT) embedded in Pasternak’s medium is investigated. Eringen’s nonlocal elasticity theory in conjunction with the first-order Donell’s shell theory is used. The governing equilibrium equations are obtained and solved for CNTs subjected to mechanical loads and embedded in Winkler-Pasternak’s medium. Effects of nonlocal parameter, radius and length of CNT, as well as the foundation parameters on buckling of CNT are investigated. Comparison with the available results is made.
Couette type mhd flow with suction and injection under constant pressure grad...eSAT Journals
Abstract The unsteady magnetohydrodynamic Couette-type flow of an electrically conducting, viscous and incompressible fluid bounded by two parallel non- conducting porous plates under the influence of a constant pressure gradient and a transversely applied uniform magnetic field is studied with heat transfer. A uniform suction on the upper plate and an injection on the lower plate are applied perpendicularly to the plates keeping the rates of suction and injection the same. The two plates are maintained at different but constant temperatures. The governing nonlinear partial differential equations are solved by both analytical as well as numerical methods. An exact solution for the velocity of the fluid has been obtained by Laplace transform method. The Crank-Nicholson implicit method is used to obtain the unsteady fluid velocity profile. The transient part of the fluid velocity tends to zero as the time t tends to infinity. The energy equation is solved by the finite difference method .The effect of the magnetic field coupled with suction and injection on the velocity and temperature distributions is examined graphically and discussed in the present work. Keywords: Magnetohydrodynamics, Transverse magnetic field, Suction and injection, Constant pressure gradient, Transient state flow, Crank-Nicholson implicit method, Finite difference method.
ANALYTICAL BENDING ANALYSIS OF A CIRCULAR SANDWICH PLATE UNDER DISTRIBUTED LOADijmech
In this paper, bending analysis of a circular sandwich plate under distributed load with simply supported and clamped boundary conditions is investigated. First, the governing equations of the circular sandwich plate are obtained and they are solved using the Bessel functions. Then in order to validate the correctness of analytical results, numerical finite element method is used and its results are presented in the forms of
contours and graphs. The results indicate that under distributed load, maximum deflection happens at 0.3
of outside radius, away from centre, and minimum deflection occurs at the outer edge of the circular sandwich plate. The results from analytical and numerical methods are compared and it shows that analytical method provides an acceptable accuracy.
Flip bifurcation and chaos control in discrete-time Prey-predator model irjes
The dynamics of discrete-time prey-predator model are investigated. The result indicates that the
model undergo a flip bifurcation which found by using center manifold theorem and bifurcation theory.
Numerical simulation not only illustrate our results, but also exhibit the complex dynamic behavior, such as the
periodic doubling in period-2, -4 -8, quasi- periodic orbits and chaotic set. Finally, the feedback control method
is used to stabilize chaotic orbits at an unstable interior point.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...IJERA Editor
This paper presents a research for magnetohydrodynamic (MHD) flow of an incompressible generalized
Burgers’ fluid including by an accelerating plate and flowing under the action of pressure gradient. Where the
no – slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is
introduced to establish the constitutive relationship of the generalized Burgers’ fluid. By using the discrete
Laplace transform of the sequential fractional derivatives, a closed form solutions for the velocity and shear
stress are obtained in terms of Fox H- function for the following two problems: (i) flow due to a constant
pressure gradient, and (ii) flow due to due to a sinusoidal pressure gradient. The solutions for no – slip condition
and no magnetic field, can be derived as special cases of our solutions. Furthermore, the effects of various
parameters on the velocity distribution characteristics are analyzed and discussed in detail. Comparison between
the two cases is also made.
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014Cristina Staicu
Cristobal Bertoglio, David Barber, Nicholas Gaddum, Israel Valverde, Marcel Rutten, et al.. Identification of artery wall stiffness: in vitro validation and in vivo results of a data assimilation procedure applied to a 3D fluid-structure interaction model. Journal of Biomechanics, Elsevier, 2014, 47 (5),
pp.1027-1034. 10.1016/j.jbiomech.2013.12.029 . hal-00925902v2
I received explicit thank you from the INRIA team for my support in the Sheffield team.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
General Solution of Equations of Motion of Axisymmetric Problem of Micro-Isot...IJERA Editor
In this paper, we obtain the general solution of equations of motion of axisymmetric problem of micro-isotropic,
micro-elastic solid in static case. The equations of motion of axisymmetric problem are converted into vector
matrix differential equations using the Hankel transform. Applying the technique of solving the eigen value
problem, the general solution of the said problem is obtained. The results of the corresponding problem in linear
micropolar elasticity are obtained as a particular case of this paper.
In this article free vibration of a nonlinear cyclic symmetry system is examined. The system is composed of six identical beams which are fixed at the end. The coupling between beams is introduced via a nonlinear stiffness running between two consecutive beams. The equations of motion are as a system of second order nonlinear differential equations which are coupled by cubic nonlinear terms. To solve the equations of motion, the numerical methods are used, and the results are compared with those of the harmonic balance method. The effect of nonlinear stiffness on the backbone curves is examined. The results show that the effect of nonlinear stiffness is increased as the diametrical mode number is increased from 1 to 3, whereas there is no effect on zero-diameter mode.
Effect of an Inclined Magnetic Field on Peristaltic Flow of Williamson Fluid ...QUESTJOURNAL
ABSTRACT: This paper deals with the influence ofinclined magnetic field on peristaltic flow of an incompressible Williamson fluid in an inclined channel with heat and mass transfer. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
The approximate bound state of the nonrelativistic Schrӧdinger equation was
obtained with the modified trigonometric scarf type potential in the framework of
asymptotic iteration method for any arbitrary angular momentum quantum number l
using a suitable approximate scheme to the centrifugal term. The effect of the screening
parameter and potential depth on the eigenvalue was studied numerically. Finally, the
scattering phase shift of the nonrelativistic Schrӧdinger equation with the potential
under consideration was calculated.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...IRJESJOURNAL
Abstract:- Buckling analysis of a carbon nanotube (CNT) embedded in Pasternak’s medium is investigated. Eringen’s nonlocal elasticity theory in conjunction with the first-order Donell’s shell theory is used. The governing equilibrium equations are obtained and solved for CNTs subjected to mechanical loads and embedded in Winkler-Pasternak’s medium. Effects of nonlocal parameter, radius and length of CNT, as well as the foundation parameters on buckling of CNT are investigated. Comparison with the available results is made.
Couette type mhd flow with suction and injection under constant pressure grad...eSAT Journals
Abstract The unsteady magnetohydrodynamic Couette-type flow of an electrically conducting, viscous and incompressible fluid bounded by two parallel non- conducting porous plates under the influence of a constant pressure gradient and a transversely applied uniform magnetic field is studied with heat transfer. A uniform suction on the upper plate and an injection on the lower plate are applied perpendicularly to the plates keeping the rates of suction and injection the same. The two plates are maintained at different but constant temperatures. The governing nonlinear partial differential equations are solved by both analytical as well as numerical methods. An exact solution for the velocity of the fluid has been obtained by Laplace transform method. The Crank-Nicholson implicit method is used to obtain the unsteady fluid velocity profile. The transient part of the fluid velocity tends to zero as the time t tends to infinity. The energy equation is solved by the finite difference method .The effect of the magnetic field coupled with suction and injection on the velocity and temperature distributions is examined graphically and discussed in the present work. Keywords: Magnetohydrodynamics, Transverse magnetic field, Suction and injection, Constant pressure gradient, Transient state flow, Crank-Nicholson implicit method, Finite difference method.
ANALYTICAL BENDING ANALYSIS OF A CIRCULAR SANDWICH PLATE UNDER DISTRIBUTED LOADijmech
In this paper, bending analysis of a circular sandwich plate under distributed load with simply supported and clamped boundary conditions is investigated. First, the governing equations of the circular sandwich plate are obtained and they are solved using the Bessel functions. Then in order to validate the correctness of analytical results, numerical finite element method is used and its results are presented in the forms of
contours and graphs. The results indicate that under distributed load, maximum deflection happens at 0.3
of outside radius, away from centre, and minimum deflection occurs at the outer edge of the circular sandwich plate. The results from analytical and numerical methods are compared and it shows that analytical method provides an acceptable accuracy.
Flip bifurcation and chaos control in discrete-time Prey-predator model irjes
The dynamics of discrete-time prey-predator model are investigated. The result indicates that the
model undergo a flip bifurcation which found by using center manifold theorem and bifurcation theory.
Numerical simulation not only illustrate our results, but also exhibit the complex dynamic behavior, such as the
periodic doubling in period-2, -4 -8, quasi- periodic orbits and chaotic set. Finally, the feedback control method
is used to stabilize chaotic orbits at an unstable interior point.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...IJERA Editor
This paper presents a research for magnetohydrodynamic (MHD) flow of an incompressible generalized
Burgers’ fluid including by an accelerating plate and flowing under the action of pressure gradient. Where the
no – slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is
introduced to establish the constitutive relationship of the generalized Burgers’ fluid. By using the discrete
Laplace transform of the sequential fractional derivatives, a closed form solutions for the velocity and shear
stress are obtained in terms of Fox H- function for the following two problems: (i) flow due to a constant
pressure gradient, and (ii) flow due to due to a sinusoidal pressure gradient. The solutions for no – slip condition
and no magnetic field, can be derived as special cases of our solutions. Furthermore, the effects of various
parameters on the velocity distribution characteristics are analyzed and discussed in detail. Comparison between
the two cases is also made.
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014Cristina Staicu
Cristobal Bertoglio, David Barber, Nicholas Gaddum, Israel Valverde, Marcel Rutten, et al.. Identification of artery wall stiffness: in vitro validation and in vivo results of a data assimilation procedure applied to a 3D fluid-structure interaction model. Journal of Biomechanics, Elsevier, 2014, 47 (5),
pp.1027-1034. 10.1016/j.jbiomech.2013.12.029 . hal-00925902v2
I received explicit thank you from the INRIA team for my support in the Sheffield team.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
General Solution of Equations of Motion of Axisymmetric Problem of Micro-Isot...IJERA Editor
In this paper, we obtain the general solution of equations of motion of axisymmetric problem of micro-isotropic,
micro-elastic solid in static case. The equations of motion of axisymmetric problem are converted into vector
matrix differential equations using the Hankel transform. Applying the technique of solving the eigen value
problem, the general solution of the said problem is obtained. The results of the corresponding problem in linear
micropolar elasticity are obtained as a particular case of this paper.
In this article free vibration of a nonlinear cyclic symmetry system is examined. The system is composed of six identical beams which are fixed at the end. The coupling between beams is introduced via a nonlinear stiffness running between two consecutive beams. The equations of motion are as a system of second order nonlinear differential equations which are coupled by cubic nonlinear terms. To solve the equations of motion, the numerical methods are used, and the results are compared with those of the harmonic balance method. The effect of nonlinear stiffness on the backbone curves is examined. The results show that the effect of nonlinear stiffness is increased as the diametrical mode number is increased from 1 to 3, whereas there is no effect on zero-diameter mode.
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Modeling And Analytical Simulation Of Heat And Mass Transfer In The Flow Of A...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Chemical Reaction on Heat and Mass TransferFlow through an Infinite Inclined ...iosrjce
The numerical studies are performed to examine the mass transfer flow with thermal diffusion and
diffusion thermo effect past an infinite, inclined vertical plate in a porous medium in the presence of chemical
reaction. First of all, the governing equations are transformed to a system of dimensionless coupled partial
equations. Explicit finite difference method has been used to solve these dimensionless equations for momentum,
concentration and energy equations. During the course of discussion, it is found that various parameters related
to the problem influence the calculated result. Finally, the profiles of velocity, concentration and temperature
are analyzed and illustrated with graphs.
Unsteady Mhd free Convective flow in a Rotating System with Dufour and Soret ...IOSRJM
Numerical analysis is used to examine the unsteady MHD free convection and mass transfer fluid flow through a porous medium in a rotating system. Impulsively started plate moving its individual plane is considered. Similarity equations of the corresponding momentum, energy, and concentration equations are derived by introducing a time dependent length scale which infect plays the role of a resemblance parameter. The velocity component is taken to be inversely proportional to this parameter. The effects on the velocity, temperature, concentration, local skin-friction coefficients, Nusselt number, Prandl number, Dufour, Soret number and the Sherwood number of the various important parameters entering into the problem separately are discussed with the help of graphs.
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.
Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow...iosrjce
The unsteady hydromagnetic boundary layer flow of an incompressible and electrically conducting
fluid through a porous medium bounded by a moving surface has been considered. It is assumed that the moving
surface has a velocity profile with respect to time and fluid flow is taken under the influence of a transverse
magnetic field. The similarity solution is used to transform the system of partial differential equations,
describing the problem under consideration, into a boundary value problem of coupled ordinary differential
equations and an efficient numerical technique is implemented to solve the reduced system. The effects of the
parameters such as Magnetic parameter, Prandtl number and Eckert number are discussed graphically on
velocity and temperature distributions
Numerical simulation on laminar convection flow and heat transfer over a non ...eSAT Journals
Abstract
A numerical algorithm is presented for studying laminar convection flow and heat transfer over a non-isothermal horizontal plate.
plate temperature Tw varies with x in the following prescribed manner:
T T Cx w
n 1
where C and n are constants. By means of similarity transformation, the original nonlinear partial differential equations of flow
are transformed to a pair of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and
integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical
analysis in Matlab environment. Velocity, and temperature profiles for various Prandtl number and n are illustrated graphically.
Flow and heat transfer parameters are derived. The results of the present simulation are then compared with experimental data in
literature with good agreement.
Keywords: Free Convection, Heat Transfer, Non-isothermal Horizontal Plate, Matlab, Numerical Simulation.
The numerical solution of helmholtz equation via multivariate padé approximationeSAT Journals
Abstract
In this study, we consider the numerical solution of the Helmholtz equation, arising from numerous physical phenomena and
engineering applications including energy systems. We make use of a Padé Approximation method for the solution of the
Helmholtz equation. Firstly, Helmholtz partial differential equation had been converted to power series by two-dimensional
differential transformation, then the numerical solution of the equation was put into Padé series form for accelerating
convergence and decreasing computational time. Hereby, we obtained numerical solution of Helmholtz type partial differential
equation.
Key Words: Helmholtz equation, Two-dimensional differential transformation, Multivariate Padé approximation,
power series
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 Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...IJMER
This paper presents an analytical study for the magnetohydrodynamic (MHD) flow of a
generalized Burgers’ fluid in an annular pipe. Closed from solutions for velocity is obtained by using finite
Hankel transform and discrete Laplace transform of the sequential fractional derivatives. Finally, the
figures are plotted to show the effects of different parameters on the velocity profile.
Numerical simulation of marangoni driven boundary layer flow over a flat plat...eSAT Journals
Abstract
A numerical algorithm is presented for studying Marangoni convection flow over a flat plate with an imposed temperature
distribution. Plate temperature varies with x in the following prescribed manner: where A and k are constants.
By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a pair of
nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton
Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab
environment. Velocity profiles for various values of k, and temperature profiles for various Prandtl number and k are illustrated
graphically. The results of the present simulation are then compared with the previous works available in literature with good
agreement.
Keywords: Matlab, Marangoni Convection, Numerical Simulation, Surface Tension, Flat Plate.
analysis of unsteady squeezing flow between two porous plates with variable m...IJAEMSJORNAL
Analysis will be made for the non-isothermal Newtonian fuid flow between two unsteady squeezing porous plates under the infuence of variable magnetic feld. The similarity transformations will be used to transform the partial differential equations into nonlinear coupled ordinary differential equations. The modeled nonlinear differential equations representing the flow behavior in the geometry under consideration will be investigated using analytical and numerical method. Comparison of the solutions will be made. Convergence of solution will also be discussed. Flow behavior under the infuence of non-dimensional parameters will be discussed with the help of graphical aids.
MHD convection flow of viscous incompressible fluid over a stretched vertical...IJERA Editor
The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
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C05421827
1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 05, Issue 04 (April. 2015), ||V2|| PP 18-27
International organization of Scientific Research 18 | P a g e
Numerical Study on MHD Flow And Heat Transfer With The
Effect Of Microrotational Parameter In The Porous Medium
R.K.Mondal**
B. M. Jewel Rana, R. Ahmed *
Mathematics Discipline Khulna University, Bangladesh.
Mathematics Discipline Khulna University, Bangladesh.
Mathematics Discipline Khulna University, Bangladesh.
Abstract: The numerical studies are performed to examine the Micropolar fluid flow past an infinite vertical
plate. Finite difference technique is used as a tool for the numerical approach. The micropolar fluid behaviour
on two- dimensional unsteady flow have been considered and its nonsimilar solution have been obtained.
Nonsimilar equations of the corresponding momentum, angular momentum and continuity equations are
derived by employing the usual transformation. The dimensionless nonsimilar equations for momentum, angular
momentum and continuity equations are solved numerically by finite difference technique. The effects on the
velocity, micro rotation and the spin gradient viscosity of the various important parameters entering into the
problem separately are discussed with the help of graphs.
Keywords: Micropolar fluid, unsteady, Explicit finite difference method, Porous medium.
I. INTRODUCTION
Because of the increasing importance of materials flow in industrial processing and elsewhere and the
fact shear behavior cannot be characterized by Newtonian relationships, a new stage in the evaluation of fluid
dynamic theory is in the progress. Eringen(1966) proposed a theory of molecular fluids taking into account the
internal characteristics of the subtractive particles, which are allowed to undergo rotation. Physically, the
micropolar fluid can consists of a suspension of small, rigid cylindrical elements such as large dumbbell-shaped
molecules. The theory of micropolar fluids is generating a very much increased interest and many classical
flows are being re-examined to determine the effects of the fluid microstructure. The concept of micropolar fluid
deals with a class of fluids which exhibit certain microscopic effects arising from the local structure and
micromotions of the fluids elements. These fluid contain dilute suspension of rigid macromolecules with
individual motions that support stress and body moments and are influenced by spin inertia. Micropolar fluids
are those which contain micro-constituents that can undergo rotation, the presence of which can affect the
hydrodynamics of the flow so that it can be distinctly non-Newtonian. It has many practical applications, for
example analyzing the behavior of exotic lubricants, the flow of colloidal suspensions, polymetric fluids, liquid
crystals, additive suspensions, human and animal blood, turbulent shear flow and so forth.
Peddision and McNitt(1970) derived boundary layer theory for micropolar fluid which is important in a number
of technical process and applied this equations to the problems of steady stagnation point flow, steady flow past
a semi-infinite flat plate. [1] Eringen (1972) developed the theory of thermo micropolar fluids by extending the
theory of micropolar fluids.The above mentioned work they have extended the work of El-Arabawy (2003) to a
MHD flow taking into account the effect of free convection and micro rotation inertia term which has been
neglected by El-Arabawy (2003) [3] . However, most of the previous works assume that the plate is at rest.
Free-convection flow with thermal radiation and mass transfer past a moving vertical porous plate have
analyzed by Makinde, O. D [9]. Unsteady MHD free convection flow of a compressible fluid past a moving
vertical plate in the presence of radioactive heat transfer have been discussed by Mbeledogu, I. U, Amakiri,
A.R.C and Ogulu, A, [10]. Numerical Study on MHD free convection and mass transfer flow past a vertical flat
plate has been discussed by S. F. Ahmmed [11].
In our present work, we have studied about numerical study on Numerical study on micropolar fluid
flow through vertical plate.The governing equations for the unsteady case are also studied. Then these governing
equations are transformed into dimensionless momentum, energy and concentration equations are solved
numerically by using explicit finite difference technique with the help of a computer programming language
Compaq visual FORTRAN 6.6. The obtained results of this problem have been discussed for the different
values of well-known parameters with different time steps. The tecplot is used to draw graph of the flow.
II. MATHEMATICALFORMULATION
Let us consider the micropolar fluid flow through vertical plate. We also consider the x-axis be directed upward
along the plate and y-axis normal to the plate. Again let u and v be the velocity components along the x-axis and y -
2. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 19 | P a g e
axis respectively,
0
u v
x y
(1)
' 22
0
2
( )
uBu u u u
u v g T T
t x y y y
(2)
2
2
u
u v
t x y j y j y
(3)
2
2
p
T T T k T
u v
t x y C y
(4)
The boundary conditions for the problem are:
at 0; 0; 0; 0; 0u v T every where
at 0
0; 0; 0; 0 0
0; 0; 1; 1 0
0; 0; 0; 0
u v T at x
u v T at y
u v T at y
(5)
It is required to make the given equations dimensionless. For this intention we introduce the following
dimensionless quantities
2 2
0 0 0 0
0 0
, , , , , , w
U U U Uu v
X x Y y U V t T T T T
U U
Finally, we obtain
0
U V
X Y
(6)
' 22
0
2
1 r
UBU U V U
U V G T
X Y Y Y
(7)
2
2
U
U V
X Y Y Y
(8)
2
2
1
r
T T T T
U V
X Y P Y
(9)
The corresponding boundary conditions for the problem are reduced in the following from
at, 0, 0, , 0, 0U V O T everywhere
0, 0, 0, 0 0
0 0, 0, 1, 1 0
0, 0, 0, 0
U V T at X
U V T at Y
U V T at Y
(10)
III. NUMERICAL SOLUTIONS
Many physical phenomena in applied science and engineering when formulated into mathematical models fall
into a category of systems known as non-linear coupled partial differential equations. Most of these problems
can be formulated as second order partial differential equations. A system of non-linear coupled partial
differential equations with the boundary conditions is very difficult to solve analytically. For obtaining the
at,
3. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 20 | P a g e
solution of such problems we adopt advanced numerical methods. The governing equations of our problem
contain a system of partial differential equations which are transformed by usual transformations into a non-
dimensional system of non-linear coupled partial differential equations with initial and boundary conditions.
Hence the solution of the problem would be based on advanced numerical methods. The finite difference
Method will be used for solving our obtained non-similar coupled partial differential equations.
From the concept of the above discussion, for simplicity the explicit finite difference method has been used to
solve from equations (6) to (9) subject to the conditions given by (10).To obtain the difference equations the
region of the flow is divided into a grid or mesh of lines parallel to X and Y axis is taken along the plate and Y-
axis is normal to the plate.
Here the plate of height max ( 20)X i.e. X varies from 0 to 20 and regard max ( 50)Y as corresponding to
Y i.e. Y varies from 0 to 50. There are m=100 and n=200 grid spacing in the X and Y directions
respectively.
It is assumed that ΔX and ΔY are constant mesh sizes along X and Y directions respectively and taken as follows,
0.20 0 20X x
0.25 0 50Y x
with the smaller time-step, Δt=0.005.
Now using the finite difference method we convert our governing equations in the following form
, 1, , , 1
0
i j i j i j i jU U V V
X Y
(11)
'
, , , 1, , 1 , , 1 , , 1
, , 2
, 1 ,
,
2
1
( )
i j i j i j i j i j i j i j i j i j
i j i j
i j i j
r i j
U U U U U U U U U
U V
X Y Y
G T MU
Y
(12)
'
, , 1, , 1 , , 1 , , 1,
, , 2
, 1 ,
2
( )
i j i j i j i j i j i j i j i ji j
i j i j
i j i j
U V
X Y Y
U U
Y
(13)
'
, , 1, , 1 , , 1 , , 1,
, , 2
1 2
( )
i j i j i j i j i j i j i j i ji j
i j i j
r
T T T T T T T T T
U V
X Y P Y
(14)
And the initial and boundary conditions with the finite difference scheme are
0 00 0
, ,, ,
0, 0,0, 0,
,0 ,0,0 ,0
, ,, ,
0, 0, 0, 0
0, 0, 0, 0
0, 0, 1, 1
0, 0, 0, 0
i j i ji j i j
n nn n
j jj j
n nn n
i ii i
n nn n
i L i Li L i L
U V T
U V T
U V T
U V T
(15)
Here the subscripts i and j designate the grid points with x and y coordinates respectively.
IV. RESULTS AND DISCUSSION
The effects of unsteady micropolar fluid behavior on a heated plate have been investigated using the
finite difference technique. To study the physical situation of this problem, we have computed the numerical
values by finite difference technique of velocity, micrirotation and temperature effect at the plate. It can be seen
that the solutions are affected by the parameters namely, Microrotation parameter (∆), Spin gradient viscosity
parameter (𝛬), Magnetic parameter (M), the vortex viscosity parameter (), Grashof Number (𝐺𝑟 ) and Prandtl
number (𝑃𝑟 ). The main goal of the computation is to obtain the steady state solutions for the non-dimensional
4. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 21 | P a g e
velocity U, microrotation Γ′
and temperature 𝑇′
for different values of Microrotation parameter (∆), Spin
gradient viscosity parameter (𝛬), the vortex viscosity parameter (), Grashof Number (𝐺𝑟 ) Prandtl number
(𝑃𝑟). For these computations the results have been calculated and presented graphically by dimensionless time
= 10 up to = 80. The results of the computations show little changes for = 10to = 60. But while
arising at = 70 and 80 the results remain approximately same but microrotation. Thus the solution for
= 80 are become steady-state. Moreover, the steady state solutions for transient values of U, Γ′
and 𝑇′
are
shown in figures 1- 24), for time = 10, 20, 30, 40, 50, 60, 70, 80 respectively. The values of Microrotation
parameter (∆ = 0.01) is fixed. Whereas, figures (1-8) show the velocity profile for different values of Grashof
Number (𝐺𝑟 = 0.2,0.4,0.6) at time = 10, 20, 30,40, 50, 60, 70, 80 respectively. From this figures it is
observed that the velocity profile increase with the increase of Grashof Number(𝐺𝑟), and the velocity profiles
are going upward direction. While arising at = 70 and 80 the solutions become steady-state. Other important
effects of microrotations are shown in figures (9-16) for different values of Spin gradient viscosity parameter
(𝛬) at time = 10, 20, 30, 40, 50,60, 70, 80 respectively. It is seen from this figures that the microrotaion
increases with the increase of Spin gradient viscosity parameter (𝛬) and is going to the upward direction from
the horizontal wall with the increase of time. While arising at = 70 and 80 the results also increasing with
time. Other effects of temperature are shown in figures (17-24) for different values of Prandtl number (𝑃𝑟 =
0.71, 1.0, 7.0) at time = 10, 20, 30, 40, 50, 60, 70, 80 respectively.
One the other hand, at salt water the Prandtl number is(𝑃𝑟 = 1.0 ). It is seen from this figures that the
temperature distribution is decreases with the increase of Prandtl number and the flow pattern is directed to the
outer wall with the increase of time. While arising at = 70 and 80 the flow becomes steady state
Fig. 1: Velocity Profile for different values of
Grashoff Number (𝐺𝑟) and ∆ = 0.01, M = 0.02 at
time 𝜏 = 10
Fig. 2: Velocity Profile for different values of
Grashoff Number (𝐺𝑟) and ∆ = 0.01, M = 0.02 at
time 𝜏 = 20
0 10 20
0
0.1
0.2
0.3
0.4
0.5
0 10 20
0
0.2
0.4
0.6
0.8𝑮 𝒓=0.2,0.4,0.6
𝑮 𝒓=0.2,0.4,0.6
U U
Y
Y
5. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 22 | P a g e
Fig. 3: Velocity Profile for different values of
Grashoff Number (𝐺𝑟) and ∆ = 0.01, M = 0.02 at
time 𝜏 = 30
Fig. 4: Velocity Profile for different values of
Grashoff Number (𝐺𝑟) and ∆ = 0.01, M = 0.02 at
time 𝜏 = 40
Fig. 5: Velocity Profile for different values of
Grashoff Number (𝐺𝑟) and ∆ = 0.01,M = 0.02 at
time 𝜏 = 50
Fig. 6: Velocity Profile for different values of
Grashoff Number (𝐺𝑟) and ∆ = 0.01, M = 0.02
at time 𝜏 = 60
0 10 20
0
0.5
1
0 10 20
0
0.5
1
1.5
0 10 20
0
1
2
0 10 20
0
1
2
𝑮 𝒓=0.2,0.4,0.6 𝑮 𝒓=0.2,0.4,0.6
𝑮 𝒓=0.2,0.4,0.6 𝑮 𝒓=0.2,0.4,0.6
U U
U U
Y Y
Y
Y
6. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 23 | P a g e
Fig. 7: Velocity Profile for different values of
Grashoff Number (𝐺𝑟) and ∆ = 0.01, M = 0.02 at
time 𝜏 = 70
Fig. 8: Velocity Profile for different values of
Grashoff Number (𝐺𝑟) and ∆ = 0.01, M = 0.02
at time 𝜏=80
Fig. 9: Microrotation Profile for different values
of Spin Gradient viscosity parameter(Λ) and ∆ =
0.01, M = 0.02 at time 𝜏 = 10
Fig. 10: Microrotation Profile for different
values of Spin Gradient viscosity parameter(Λ)
and ∆ = 0.01, M = 0.02 at time 𝜏 = 20
0 10 20
0
1
2
0 10 20
0
1
2
0 10 20
0
0.2
0.4
0.6
0.8
1
0 10 20
0
0.2
0.4
0.6
0.8
1
𝑮 𝒓=0.2,0.4,0.6 𝑮 𝒓=0.2,0.4,0.6
Λ=0.2,0.4,0.6 Λ=0.2,0.4,0.6
U U
Y Y
Г
Г
YY
7. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 24 | P a g e
Fig. 11: Microrotation Profile for different values
of Spin Gradient viscosity parameter(Λ) and ∆ =
0.01, M = 0.02 at time 𝜏 = 30
Fig. 12: Microrotation Profile for different values
of Spin Gradient viscosity parameter(Λ) and ∆ =
0.01, M = 0.02 at time 𝜏 = 40
Fig. 13: Microrotation Profile for different values
of Spin Gradient viscosity parameter(Λ) and ∆ =
0.01, M = 0.02 at time 𝜏 = 50
Fig. 14: Microrotation Profile for different
values of Spin Gradient viscosity parameter(Λ)
and ∆ = 0.01, M = 0.02 at time 𝜏 = 60
0 10 20
0
0.2
0.4
0.6
0.8
1
0 10 20
0
0.5
1
0 10 20
0
0.5
1
1.5
0 10 20
0
1
2
Λ=0.2,0.4,0.6
Λ=0.2,0.4,0.6
Λ=0.2,0.4,0.6Λ=0.2,0.4,0.6
Г
Г
Г
Г
YY
Y
Y
8. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 25 | P a g e
Fig. 15: Microrotation Profile for different values
of Spin Gradient viscosity parameter(Λ) and ∆ =
0.01, M = 0.02 at time 𝜏 = 70
Fig. 16: Microrotation Profile for different values
of Spin Gradient viscosity parameter(Λ) and ∆ =
0.01, M = 0.02 at time 𝜏 = 80
Fig. 17: Temperature Profile for different values
of Prandtl Number (𝑃𝑟) and M = 0.02 at time 𝜏 =
10
Fig. 18: Temperature Profile for different values
of Prandtl Number (𝑃𝑟) and M = 0.02 at time 𝜏 =
20
0 10 20
0
1
2
0 10 20
0
1
2
3
0 10 20
0
0.2
0.4
0.6
0.8
1
0 10 20
0
0.2
0.4
0.6
0.8
1
Λ=0.2,0.4,0.6Λ=0.2,0.4,0.6
Pr=0.71,1.0,7.0Pr=0.71,1.0,7.0
Г Г
T
T
YY
YY
9. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 26 | P a g e
Fig. 19: Temperature Profile for different values
of Prandtl Number (𝑃𝑟) and M = 0.02 at time 𝜏 =
30
Fig. 20: Temperature Profile for different values
of Prandtl Number (𝑃𝑟) and M = 0.02 at time 𝜏 =
40
Fig. 21: Temperature Profile for different values
of Prandtl Number (𝑃𝑟) and M = 0.02 at time 𝜏 =
50
Fig.22: Temperature Profile for different values
of Prandtl Number (𝑃𝑟) and M = 0.02 at time 𝜏 =
60
0 10 20
0
0.2
0.4
0.6
0.8
1
0 10 20
0
0.2
0.4
0.6
0.8
1
0 10 20
0
0.2
0.4
0.6
0.8
1
0 10 20
0
0.2
0.4
0.6
0.8
1
Pr=0.71,1.0,7.0Pr=0.71,1.0,7.0
Pr=0.71,1.0,7.0Pr=0.71,1.0,7.0
T T
T T
Y Y
YY
10. Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter
International organization of Scientific Research 27 | P a g e
Fig. 23: Temperature Profile for different values
of Prandtl Number (𝑃𝑟) and M = 0.02 at time 𝜏 =
70
Fig. 24: Temperature Profile for different values
of Prandtl Number (𝑃𝑟) and M = 0.02 at time 𝜏 =
80
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0 10 20
0
0.2
0.4
0.6
0.8
1
0 10 20
0
0.2
0.4
0.6
0.8
1
Pr=0.71,1.0,7.0Pr=0.71,1.0,7.0
T
T
Y Y