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Analytic Solution for Fluid Flow over an Exponentially Stretching Porous Sheet with Surface Heat Flux in Porous Medium by Means of Homotopy Analysis Method  [PDF]
Azhar Ali, H. Zaman, M. Z. Abidin, S. I. A. Shah
American Journal of Computational Mathematics (AJCM) , 2015, DOI: 10.4236/ajcm.2015.52019
Abstract: In this paper, the analytical solution of a viscous and incompressible fluid towards an exponentially stretching porous sheet with surface heat flux in porous medium, for the boundary layer and heat transfer flow, is presented. The equations of continuity, momentum and the energy are transformed into non-linear ordinary differential by using similarity transformation. The solutions of these highly non-linear ordinary differential equations are found analytically by means of Homotopy Analysis Method (HAM). The result obtained by HAM is compared with numerical results presented in the literature. The accuracy of the HAM is indicated by close agreement of the two sets of results. By this method, an expression is obtained which is admissible for all values of effective parameters. This method has the ability to control the convergence of the solution.
On a Homotopy Perturbation Treatment of Steady Laminar Forced Convection Flow over a Nonlinearly Stretching Porous Sheet  [PDF]
Nemat Dalir,Salman Nourazar
Chinese Journal of Engineering , 2014, DOI: 10.1155/2014/297163
Abstract: The steady two-dimensional laminar forced convection boundary layer flow of an incompressible viscous Newtonian fluid over a nonlinearly stretching porous (permeable) sheet with suction is considered. The sheet’s permeability is also considered to be nonlinear. The boundary layer equations are transformed by similarity transformations to a nonlinear ordinary differential equation (ODE). Then the homotopy perturbation method (HPM) is used to solve the resultant nonlinear ODE. The dimensionless entrainment parameter and the dimensionless sheet surface shear stress are obtained for various values of the suction parameter and the nonlinearity factor of sheet stretching and permeability. The results indicate that the dimensionless sheet surface shear stress decreases with the increase of suction parameter. The results of present HPM solution are compared to the values obtained in a previous study by the homotopy analysis method (HAM). The HPM results show that they are in good agreement with the HAM results within 2% error. 1. Introduction Boundary-layer flow of an incompressible fluid over a stretching sheet has many applications in engineering such as in liquid film condensation process, aerodynamic extrusion of plastic sheets, cooling process of metallic plate in a cooling bath, and glass and polymer industries. In the last decade, many semianalytical methods have been used to solve the boundary layer flow problems. For example, He [1] proposed a new perturbation technique coupled with the homotopy technique, which requires no small parameters in the equations and can readily eliminate the limitations of the traditional perturbation techniques. He named this method as the homotopy perturbation method (HPM). Esmaeilpour and Ganji [2] presented the problem of forced convection over a horizontal flat plate and employed the HPM to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. Xu [3] obtained an approximate solution of a boundary layer equation in unbounded domain by means of He’s homotopy perturbation method (HPM). Fathizadeh and Rashidi [4] solved the convective heat transfer equations of boundary layer flow with pressure gradient over a flat plate using the HPM. They studied the effects of Prandtl number and pressure gradient on both temperature and velocity profiles in the boundary layer. Raftari and Yildirim [5] obtained by means of the HPM an approximate analytical solution of the magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a permeable
Magnetohydrodynamic Boundary Layer Flow of Nanofluid over an Exponentially Stretching Permeable Sheet  [PDF]
Krishnendu Bhattacharyya,G. C. Layek
Physics Research International , 2014, DOI: 10.1155/2014/592536
Abstract: A mathematical model of the steady boundary layer flow of nanofluid due to an exponentially permeable stretching sheet with external magnetic field is presented. In the model, the effects of Brownian motion and thermophoresis on heat transfer and nanoparticle volume friction are considered. Using shooting technique with fourth-order Runge-Kutta method the transformed equations are solved. The study reveals that the governing parameters, namely, the magnetic parameter, the wall mass transfer parameter, the Prandtl number, the Lewis number, Brownian motion parameter, and thermophoresis parameter, have major effects on the flow field, the heat transfer, and the nanoparticle volume fraction. The magnetic field makes enhancement in temperature and nanoparticle volume fraction, whereas the wall mass transfer through the porous sheet causes reduction of both. For the Brownian motion, the temperature increases and the nanoparticle volume fraction decreases. Heat transfer rate becomes low with increase of Lewis number. For thermophoresis effect, the thermal boundary layer thickness becomes larger. 1. Introduction The term “nanofluid” was proposed by Choi [1], referring to dispersions of nanoparticles in the base fluids such as water, ethylene glycol, and propylene glycol. The thermal conductivity enhancement characteristic of nanofluid was observed by Masuda et al. [2]. Buongiorno [3] discussed the reasons behind the enhancement in heat transfer for nanofluid and he found that Brownian diffusion and thermophoresis are the main causes. Later, Nield and Kuznetsov [4] and Kuznetsov and Nield [5] investigated the natural convective boundary layer flow of a nanofluid employing Buongiorno model. The study of boundary layer flow and heat transfer due to stretching surface has numerous applications in industry and technology, such as in polymer extrusion, drawing of copper wires, artificial fibers, paper production, hot rolling, wire drawing, glass fiber, metal extrusion and metal spinning, and continuous stretching of plastic films. Crane [6] first studied the boundary layer flow due to linearly stretching sheet. Many researchers [7–17] extended the work of Crane, whereas Magyari and Keller [18] considered the boundary layer flow and heat transfer due to an exponentially stretching sheet. The flow and heat transfer over an exponentially stretching surface were investigated by Elbashbeshy [19] taking wall mass suction. Khan and Sanjayanand [20] presented the boundary layer flow of viscoelastic fluid and heat transfer over an exponentially stretching sheet with viscous
Numerical and Series Solutions for Stagnation-Point Flow of Nanofluid over an Exponentially Stretching Sheet  [PDF]
Meraj Mustafa, Muhammad A. Farooq, Tasawar Hayat, Ahmed Alsaedi
PLOS ONE , 2013, DOI: 10.1371/journal.pone.0061859
Abstract: This investigation is concerned with the stagnation-point flow of nanofluid past an exponentially stretching sheet. The presence of Brownian motion and thermophoretic effects yields a coupled nonlinear boundary-value problem (BVP). Similarity transformations are invoked to reduce the partial differential equations into ordinary ones. Local similarity solutions are obtained by homotopy analysis method (HAM), which enables us to investigate the effects of parameters at a fixed location above the sheet. The numerical solutions are also derived using the built-in solver bvp4c of the software MATLAB. The results indicate that temperature and the thermal boundary layer thickness appreciably increase when the Brownian motion and thermophoresis effects are strengthened. Moreover the nanoparticles volume fraction is found to increase when the thermophoretic effect intensifies.
Effects of Hall Current on Flow of Unsteady MHD Axisymmetric Second-Grade Fluid with Suction and Blowing over an Exponentially Stretching Sheet  [PDF]
Haider Zaman, Arif Sohail, Azhar Ali, Tarique Abbas
Open Journal of Modelling and Simulation (OJMSi) , 2014, DOI: 10.4236/ojmsi.2014.22005
Abstract: This paper investigates effects of Hall current on flow of unsteady magnetohydrodynamic (MHD) axisymmetric second-grade fluid with suction and blowing over a sheet stretching exponentially with radius. The governing non-linear partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformations. The complex analytical solution is found by using the homotopy analysis method (HAM). The existing literature on the topic shows that it is the first study regarding the effects of Hall current on flow over an exponentially stretching sheet in cylindrical coordinates. The convergence of the obtained complex series solutions is carefully analyzed. The effects of dimensionless parameters on the radial and axial components of the velocity are illustrated through plots. Also the effects of the pertinent parameters on the shear stress at the wall are presented numerically in tabular form.
Tau-Homotopy Analysis Method for Solving Micropolar Flow due to a Linearly Stretching of Porous Sheet  [PDF]
S. Kazem,M. Shaban
Communications in Numerical Analysis , 2012, DOI: 10.5899/2012/cna-00114
Abstract: A modification of the homotopy analysis method (HAM) for solving a system of nonlinear boundary value problems (BVPs) in semi-infinite domain, micropolar flow due to a linearly stretching of porous sheet, is proposed. This method is based on operational matrix of exponential Chebyshev functions to construct the derivative and product of the unknown function in the matrix form. In addition, by using Tau method the problem converts to algebraic equations to obtain the solution iteratively. In whole we can say, the computer oriented of this method is the most important aspect of it. During comparison between our methods and those previously reported, significant consequences are demonstrated.
Effect of Internal Heat Generation/Absorption on Dusty Fluid Flow over an Exponentially Stretching Sheet with Viscous Dissipation  [PDF]
G. M. Pavithra,B. J. Gireesha
Journal of Mathematics , 2013, DOI: 10.1155/2013/583615
Abstract: A numerical analysis has been carried out to describe the boundary layer flow and heat transfer of a dusty fluid over an exponentially stretching surface in the presence of viscous dissipation and internal heat generation/absorption. The governing partial differential equations are reduced to nonlinear ordinary differential equations by a similarity transformation, before being solved numerically by Runge-Kutta-Fehlberg 45 method. The heat transfer analysis has been carried out for both PEST and PEHF cases. The numerical results are compared with the earlier study and found to be in excellent agreement. Some important features of the flow and heat transfer in terms of velocities and temperature distributions for different values of the governing parameters like fluid-particle interaction parameter, Prandtl number, Eckert number, Number density, heat source/sink parameter, and suction parameter which are of physical and engineering interests are analyzed, discussed, and presented through tables and graphs. 1. Introduction An investigation on boundary layer flow and heat transfer of viscous fluids over a moving continuous stretching surface has considerable practical applications in industries and engineering, since the study of heat transfer has become important industrially for determining the quality of final product which greatly depends on the rate of cooling. Many researchers inspired by Sakiadis [1, 2] who initiated the boundary layer behavior studied the stretching flow problem in various aspects. Extension to that, an exact solution was given by Crane [3] for a boundary layer flow caused by stretching surface. A new dimension to the boundary layer flow was given by Magyari and Keller [4] by considering the nonstandard stretching flow known as exponentially stretching surface. They described the mass and heat transfer characteristics of the boundary layer. After that, Elbashbeshy [5] was the first who considered the heat transfer problem over an exponentially stretching sheet with suction parameter. The effect of viscous dissipation on the boundary layer flow along vertical exponential stretching sheet was explained by Partha et al. [6]. Sanjayanand and Khan [7] discussed heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet. The numerical solution for the boundary layer flow with thermal radiation over an exponentially stretching sheet was given by Bidin and Nazar [8]. Samad and Mohebujjaman [9] worked on both heat and mass transfer of free convective flow along a vertical stretching sheet in presence
MHD flow and heat transfer over an exponentially stretching sheet with viscous dissipation and radiation effects
R. N. Jat,G. Chand
Applied Mathematical Sciences , 2013,
Abstract: The steady two-dimensional laminar flow of a viscous incompressible electricallyconducting fluid over an exponentially stretching sheet in the presence of auniform transverse magnetic field with viscous dissipation and radiative heat fluxis studied. By suitable similarity transformations, the governing boundary layerequations are transformed to ordinary differential equations and solvednumerically by standard techniques. The effects of various parameters like,Magnetic and Radiation parameters, Prandtl number and Eckert number forvelocity and temperature distributions have been discussed in detail withgraphical representation.
MHD Mixed Convective Boundary Layer Flow of a Nanofluid through a Porous Medium due to an Exponentially Stretching Sheet
M. Ferdows,Md. Shakhaoath Khan,Md. Mahmud Alam,Shuyu Sun
Mathematical Problems in Engineering , 2012, DOI: 10.1155/2012/408528
Abstract: Magnetohydrodynamic (MHD) boundary layer flow of a nanofluid over an exponentially stretching sheet was studied. The governing boundary layer equations are reduced into ordinary differential equations by a similarity transformation. The transformed equations are solved numerically using the Nactsheim-Swigert shooting technique together with Runge-Kutta six-order iteration schemes. The effects of the governing parameters on the flow field and heat transfer characteristics were obtained and discussed. The numerical solutions for the wall skin friction coefficient, the heat and mass transfer coefficient, and the velocity, temperature, and concentration profiles are computed, analyzed, and discussed graphically. Comparison with previously published work is performed and excellent agreement is observed.
Stagnation Point Flow of a Nanofluid toward an Exponentially Stretching Sheet with Nonuniform Heat Generation/Absorption  [PDF]
A. Malvandi,F. Hedayati,G. Domairry
Journal of Thermodynamics , 2013, DOI: 10.1155/2013/764827
Abstract: This paper deals with the steady two-dimensional stagnation point flow of nanofluid toward an exponentially stretching sheet with nonuniform heat generation/absorption. The employed model for nanofluid includes two-component four-equation nonhomogeneous equilibrium model that incorporates the effects of Brownian diffusion and thermophoresis simultaneously. The basic partial boundary layer equations have been reduced to a two-point boundary value problem via similarity variables and solved analytically via HAM. Effects of governing parameters such as heat generation/absorption λ, stretching parameter ε, thermophoresis , Lewis number Le, Brownian motion , and Prandtl number Pr on heat transfer and concentration rates are investigated. The obtained results indicate that in contrast with heat transfer rate, concentration rate is very sensitive to the abovementioned parameters. Also, in the case of heat generation , despite concentration rate, heat transfer rate decreases. Moreover, increasing in stretching parameter leads to a gentle rise in both heat transfer and concentration rates. 1. Introduction For years, many researchers have paid much attention to viscous fluid motion near the stagnation region of a solid body, where “body” corresponds to either fixed or moving surfaces in a fluid. This multidisciplinary flow has frequent applications in high speed flows, thrust bearings, and thermal oil recovery. Hiemenz [1] developed the first investigation in this field. He applied similarity transformation to collapse two-dimensional Navier-Stokes equations to a nonlinear ordinary differential one and then presented its exact solution. Extension of this study was carried out with a similarity solution by Homann [2] to the case of axisymmetric three-dimensional stagnation point flow. After these original studies, many researchers have put their attention on this subject [3–9]. Besides stagnation point flow, stretching surfaces have a wide range of applications in engineering and several technical purposes particularly in metallurgy and polymer industry, for instance, gradual cooling of continuous stretched metal or plastic strips which have multiple applications in mass production. Crane [10] was the first to present a similarity solution in the closed analytical form for steady two-dimensional incompressible boundary layer flow caused by the stretching plate whose velocity varies linearly with the distance from a fixed point on the sheet. The combination of stretching surface and stagnation point flow was analyzed by Yao et al. [11]. Different types fluids such
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