Showing papers on "Homotopy analysis method published in 2012"
Book•
10 Jul 2012
TL;DR: In this paper, a convergence series for Divergent Taylor Series is proposed to solve nonlinear initial value problems and nonlinear Eigenvalue problems with free or moving boundary in heat transfer.
Abstract: Basic Ideas.- Systematic Descriptions.- Advanced Approaches.- Convergent Series For Divergent Taylor Series.- Nonlinear Initial Value Problems.- Nonlinear Eigenvalue Problems.- Nonlinear Problems In Heat Transfer.- Nonlinear Problems With Free Or Moving Boundary.- Steady-State Similarity Boundary-Layer Flows.- Unsteady Similarity Boundary-Layer Flows.- Non-Similarity Boundary-Layer Flows.- Applications In Numerical Methods.
852 citations
TL;DR: The fully developed flow of an incompressible, thermodynamically compatible non-Newtonian third-grade nanofluid in coaxial cylinders is studied and the role of pertinent parameters is illustrated graphically.
218 citations
TL;DR: In this article, a new base function and an auxiliary linear operator to form a better homotopy is proposed, and a mathematical proof of the convergence is provided for the convergence.
215 citations
TL;DR: In this paper, the heat and mass transfer characteristics in a viscous fluid which is squeezed between parallel plates are reported, and the governing partial differential equations for unsteady two-dimensional flow with heat-and mass-transfer characteristics are reduced to ordinary differential equations by similarity transformations.
Abstract: This paper reports the heat and mass transfer characteristics in a viscous fluid which is squeezed between parallel plates. The governing partial differential equations for unsteady two-dimensional flow with heat and mass transfer of a viscous fluid are reduced to ordinary differential equations by similarity transformations. Homotopy analysis method (HAM) is employed to construct the series solution of the problem. Physical interpretation to various embedding parameters is assigned through graphs for temperature and concentration profiles and tables for skin friction coefficient, local Nusselt number and local Sherwood number.
186 citations
TL;DR: In this paper, the effects of emerging fluid parameters (ϵ, (δ) and Prandtl number (Pr) on the velocity and temperature are illustrated through graphs and tables for different values of λ.
176 citations
TL;DR: The steady boundary layer flow of nanofluid over an exponential stretching surface is investigated analytically and expressions for velocity, temperature and nanoparticle volume fraction are computed for some values of the parameters.
Abstract: The steady boundary layer flow of nanofluid over an exponential stretching surface is investigated analytically. The transport equations include the effects of Brownian motion parameter and thermophoresis parameter. The highly nonlinear coupled partial differential equations are simplified with the help of suitable similarity transformations. The reduced equations are then solved analytically with the help of homotopy analysis method (HAM). The convergence of HAM solutions are obtained by plotting h-curve. The expressions for velocity, temperature and nanoparticle volume fraction are computed for some values of the parameters namely, suction injection parameter α, Lewis number Le, the Brownian motion parameter Nb and thermophoresis parameter Nt.
167 citations
TL;DR: In this article, the homotopy analysis method was applied to obtain the approximate analytical solutions of the steady flow over a rotating disk in porous medium with heat transfer, and the convergence of the obtained series solutions was carefully analyzed.
145 citations
TL;DR: In this article, the peristaltic transport of nanofluid in a channel with complaint walls has been investigated by utilizing long wavelength and low Reynolds number assumptions, and the coupled nonlinear boundary value problem has been solved numerically by using shooting technique through software Mathematica.
135 citations
TL;DR: In this paper, the mixed convection stagnation-point flow of an incompressible non-Newtonian fluid over a stretching sheet under convective boundary conditions is investigated, and the resulting partial differential equations are converted into the ordinary differential equations by the suitable transformations.
Abstract: The mixed convection stagnation-point flow of an incompressible non-Newtonian fluid over a stretching sheet under convective boundary conditions is investigated. Mathematical formulation is presented for a Casson fluid. The resulting partial differential equations are converted into the ordinary differential equations by the suitable transformations. The velocity and temperature profiles are computed by employing the homotopy analysis method. The plotted graphs illustrate the flow and heat transfer characteristics and their dependence upon the embedded parameters. Numerical values of skin-friction coefficient and Nusselt number are given and examined. Comparison of the present results with the existing solution is also given.
133 citations
TL;DR: It is found that for both suction and injection, the heat transfer rate at the surface increases with increasing the nanoparticle volume fraction, Reynolds number, and injection/suction parameter and it decreases with power of rotation parameter.
Abstract: The aim of the present paper is to study the flow of nanofluid and heat transfer characteristics between two horizontal plates in a rotating system. The lower plate is a stretching sheet and the upper one is a solid porous plate. Copper (Cu) as nanoparticle and water as its base fluid have been considered. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions using similarity transformation, which is then solved analytically using the homotopy analysis method (HAM). Comparison between HAM and numerical solutions results showed an excellent agreement. The results for the flow and heat transfer characteristics are obtained for various values of the nanoparticle volume fraction, suction/injection parameter, rotation parameter, and Reynolds number. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable of causing change in the flow pattern. It is found that for both suction and injection, the heat transfer rate at the surface increases with increasing the nanoparticle volume fraction, Reynolds number, and injection/suction parameter and it decreases with power of rotation parameter.
121 citations
TL;DR: In this article, a similarity transformation is applied to reduce the governing partial differential equations for mass, momentum, and energy conservation to dimensionless, nonlinear, coupled, ordinary differential equations, and the homotopy analysis method (HAM) is employed to generate approximate analytical solutions for the transformed nonlinear equations under the prescribed boundary conditions.
Abstract: The thermoconvective boundary layer flow of a generalized third-grade viscoelastic power-law non-Newtonian fluid over a porous wedge is studied theoretically. The free stream velocity, the surface temperature variations, and the injection velocity at the surface are assumed variables. A similarity transformation is applied to reduce the governing partial differential equations for mass, momentum, and energy conservation to dimensionless, nonlinear, coupled, ordinary differential equations. The homotopy analysis method (HAM) is employed to generate approximate analytical solutions for the transformed nonlinear equations under the prescribed boundary conditions. The HAM solutions, in comparison with numerical solutions (fourth-order Runge-Kutta shooting quadrature), admit excellent accuracy. The residual errors for dimensionless velocity and dimensionless temperature are also computed. The influence of the “power-law” index on flow characteristics is also studied. The mathematical model finds important appl...
TL;DR: In this article, an alternative approach to construct the homotopy equation with an auxiliary term was proposed, where the auxiliary term is used as an example to illustrate the solution procedure.
Abstract: The two most important steps in application of the homotopy perturbation method are to construct a suitable homotopy equation and to choose a suitable initial guess. The homotopy equation should be such constructed that when the homotopy parameter is zero, it can approximately describe the solution property, and the initial solution can be chosen with an unknown parameter, which is determined after one or two iterations. This paper suggests an alternative approach to construction of the homotopy equation with an auxiliary term; Dufing equation is used as an example to illustrate the solution procedure.
TL;DR: The proposed HATM technique solves the nonlinear problems without using Adomian polynomials and He’s polynomes which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method (HPTM).
TL;DR: In this paper, the continuity and momentum equations governing the unsteady two-dimensional flow of a second-grade fluid are reduced to a single differential equation through similarity transformations, and the resulting differential system is computed by a homotopy analysis method.
Abstract: SUMMARY
This paper examines the unsteady two-dimensional flow of a second-grade fluid between parallel disks in the presence of an applied magnetic field. The continuity and momentum equations governing the unsteady two-dimensional flow of a second-grade fluid are reduced to a single differential equation through similarity transformations. The resulting differential system is computed by a homotopy analysis method. Graphical results are discussed for both suction and blowing cases. In addition, the derived results are compared with the homotopy perturbation solution in a viscous fluid (Math. Probl. Eng., DOI: 10.1155/2009/603916). Copyright © 2011 John Wiley & Sons, Ltd.
01 Jan 2012
TL;DR: In this article, a more general method of homotopy analysis method (HAM) is introduced to solve non-linear differential equations, it is called (q-HAM) and the interval of convergence of HAM, if exists, is increased when using q-HAM.
Abstract: In this paper, a more general method of homotopy analysis method (HAM) is introduced to solve non-linear differential equations, it is called (q-HAM). The interval of convergence of HAM, if exists, is increased when using q-HAM. The analysis shows that the series solution in the case of q-HAM is more likely to converge than that on HAM. The new method is applied to some nonlinear differential equations to illustrate the method of analysis.
TL;DR: In this paper, the boundary-layer flow and heat transfer of a magnetohydrodynamic viscous fluid over a nonlinear radially porous stretching sheet within a porous medium was investigated.
Abstract: We investigate the boundary-layer flow and heat transfer of a magnetohydrodynamic viscous fluid over a nonlinear radially porous stretching sheet within a porous medium. The flow is generated due to a nonlinear stretching sheet and influenced by a continuous suction/blowing of the fluid through the porous sheet. The governing momentum and thermal boundary layer equations are converted into ordinary differential equations by appropriate similarity transformations. The exact solution for the velocity and the temperature fields are derived in the form of an incomplete Gamma function. Also analytic solutions are found by the homotopy analysis method. The graphical results for velocity and temperature fields are presented and discussed. Further, the numerical values of the skin friction coefficient and the Nusselt number are calculated and discussed.
Journal Article•
TL;DR: In this paper, the Optimal Homotopy Asymptotic Method (OHAM) is employed to approximate the solution of the system of nonlinear differential equations governing the problem.
Abstract: In this paper, the problem of laminar viscous flow in a semi-porous channel in the presence of transverse magnetic field is studied. The Optimal Homotopy Asymptotic Method (OHAM) is employed to approximate the solution of the system of nonlinear differential equations governing the problem. The influence of the Hartmann number (Ha) and the Reynolds number (Re) on the flow was investigated. The results of the OHAM were compared with homotopy analysis method (HAM) and variation iteration method (VIM) results.
TL;DR: In this article, the transient squeezing flow of a magneto-micropolar biofluid in a noncompressible porous medium intercalated between two parallel plates in the presence of a uniform strength transverse magnetic field is investigated.
Abstract: The transient squeezing flow of a magneto-micropolar biofluid in a noncompressible porous medium intercalated between two parallel plates in the presence of a uniform strength transverse magnetic field is investigated. The partial differential equations describing the two-dimensional flow regime are transformed into nondimensional, nonlinear coupled ordinary differential equations for linear and angular momentum (micro-inertia). These equations are solved using the robust Homotopy Analysis Method (HAM) and also numerical shooting quadrature. Excellent correlation is achieved. The influence of magnetic field parameter (Ha), micropolar spin gradient viscosity parameter (Γ) and unsteadiness parameter (S) on linear and angular velocity (micro-rotation) are presented graphically, for specified values of the micropolar vortex viscosity parameter (R), Darcy number (Da i.e. permeability parameter) and medium porosity parameter (e). Increasing magnetic field (Ha) serves to decelerate both the linear and angular velocity i.e. enhances lubrication. The excellent potential of HAM in bio-lubrication flows is highlighted.
TL;DR: In this paper, the quintic non-linear equation of motion is derived based on Hamilton's principle and solved by means of an analytical technique, namely the Homotopy analysis method.
Abstract: non-linear vibration analysis of beam used in steel structures is of particular importance in mechanical and industrial applications. To achieve a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. Equation of transversal vibration of hinged–hinged flexible beam subjected to constant excitation at its free end is identified as a non-linear differential equation. The quintic non-linear equation of motion is derived based on Hamilton’s principle and solved by means of an analytical technique, namely the Homotopy analysis method. To verify the soundness of the results, a comparison between analytical and numerical solutions is developed. Finally, to express the impact of the quintic nonlinearity, the non-linear responses obtained by HAM are compared with the results from usual beam theory.
TL;DR: In this article, the magnetohydrodynamic squeezing flow of nanofluid between parallel disks is investigated and the convergence analysis is performed and optimal values of the convergence control parameters are determined.
Abstract: This article investigates the magnetohydrodynamic squeezing flow of nanofluid between parallel disks. Governing partial differential equations are converted into ordinary differential system via similarity transformations. We employ homotopy analysis method (HAM) to construct analytic expressions of velocity, temperature and nanoparticles volume fraction. Convergence analysis is performed and optimal values of the convergence-control parameters are determined. The computations are validated with the built in routine for solving nonlinear boundary value problems via shooting technique through software Mathematica 8.0. The behaviors of key parameters such as suction/blowing parameter (A), squeeze parameter (S), Hartman number (M), Brownian motion parameter (Nb) and thermophoresis parameter (Nt) are thoroughly examined. It is seen that the parameters have a great impact on the concentration field for the suction flow when compared with the blowing case. An intensification in the Brownian motion and thermophoresis effects results in the appreciable increase in the temperature and nanoparticles concentration.
TL;DR: In this study, a steady, incompressible, and laminar-free convective flow of a two-dimensional electrically conducting viscoelastic fluid over a moving stretching surface through a porous medium is considered and the nonlinear ordinary differential equations for the momentum and energy equations are obtained and solved analytically.
Abstract: In this study, a steady, incompressible, and laminar-free convective flow of a two-dimensional electrically conducting viscoelastic fluid over a moving stretching surface through a porous medium is considered. The boundary-layer equations are derived by considering Boussinesq and boundary-layer approximations. The nonlinear ordinary differential equations for the momentum and energy equations are obtained and solved analytically by using homotopy analysis method �
TL;DR: In this article, the three-dimensional flow of Jeffrey fluid over a linearly stretching surface has been reported, where transformation method has been utilized for the reduction of partial differential equations into the ordinary differential equations.
TL;DR: In this paper, it is shown that the employed homotopy method yields uniformly convergent solutions, and the obtained explicit analytical expressions for the solution generate results that compare excellently with the numerically computed ones, which are further confirmed analytically by the absolute error formula.
Abstract: This paper deals with the purely analytic solutions to the damped Duffing equation. It is shown that the employed homotopy method yields uniformly convergent solutions. Optimum values of the convergence control parameter of the computed homotopy series are calculated from the square residual error. The obtained explicit analytical expressions for the solution generate results that compare excellently with the numerically computed ones, which are further confirmed analytically by the absolute error formula.
TL;DR: In this article, the homotopy analysis method is used to obtain the approximate analytical solutions of the non-linear Swift Hohenberg equation with fractional time derivative, described in Caputo sense.
TL;DR: In this article, the steady laminar boundary layer flow and heat transfer past a stretching sheet is considered, and the resulting nonlinear differential system is solved by homotopy analysis method (HAM).
Abstract: The steady laminar boundary layer flow and heat transfer past a stretching sheet arre considered. Upper-convected Maxwell (UCM) fluid is treated as a rheological model. The resulting nonlinear differential system is solved by homotopy analysis method (HAM). The influence of melting parameter (M), Prandtl number (Pr), Deborah number (β) and stretching ratio (A = a/c) on the velocity and temperature profiles is thoroughly examined. It is noticed that fields are effected appreciably with the variation of parameters. Furthermore, it is seen that the local Nusselt number is a decreasing function of melting parameter. Copyright © 2011 John Wiley & Sons, Ltd.
Book•
05 Jan 2012
TL;DR: In this paper, the optimal homotopy perturbation method and the optimal variational iteration method are presented. And the optimal parametric iteration method is also discussed, as well as the optimal asymptotic method.
Abstract: Introduction.- Perturbation method (Lindstedt-Poincare).- The method of harmonic balance.- The method of Krylov and Bogolyubov.- The method of multiple scales.- The optimal homotopy asymptotic method.- The optimal homotopy perturbation method.- The optimal variational iteration method.- Optimal parametric iteration method.
TL;DR: In this article, the authors applied homotopy perturbation method (HPM) to solve delay differential equations, and the results reveal that the method is very effective and simple.
TL;DR: The present study deals with the three-dimensional flow in a rotating channel of lower permeable stretching wall with unsteady squeezing flow in the presence of transverse magnetic flux mathematically modeled with the help of Navier-Stokes equations.
Abstract: The present study deals with the three-dimensional flow in a rotating channel of lower permeable stretching wall. The unsteady squeezing flow in the presence of transverse magnetic flux is mathematically modeled with the help of Navier-Stokes equations. The governing equations are normalized with the help of suitable similarity transformations and the analysis is based on a numerical technique. The numerical results are validated with the analytic solution by homotopy analysis method. The flow characteristics are investigated by a comprehensive parametric study. Various aspects of squeezing flow are focused and examined by plotting graphs and tables of stream lines, velocity profiles, pressure gradient and shear stresses. The vertical motion of upper plate interrupts the velocity in the channel remarkably and the pressure variations are significant near the boundaries of the channel. The downward motion of upper plate augments the forward flow and viscous drag on lower plate, whereas, upward motion enhances the reverse flow. However, a suitable choice of squeezing velocity can minimize the viscous drag on lower plate.
TL;DR: In this paper, the authors investigated the problem of pulsatile flow in a porous annulus for small Reynolds number, where the similarity transformation for the governing equations gives a system of nonlinear ordinary differential equations which are analytically solved by the homotopy analysis method (HAM).
Abstract: Purpose – The purpose of this study is to investigate the problem of pulsatile flow in a porous annulus for small Reynolds number.Design/methodology/approach – The similarity transformation for the governing equations gives a system of nonlinear ordinary differential equations which are analytically solved by the homotopy analysis method (HAM). The analytic solutions of non‐linear differential equation are constructed in the series form. The convergence of the series solutions is carefully analyzed.Findings – Graphical results are presented to investigate the influence of different parameters on the flow behavior. Comparison between the solutions obtained by the HAM and the numerical solution shows good agreement.Originality/value – An analysis for study on flow of an incompressible viscous fluid in the region lying between two concentric porous cylinders, under the assumption that a periodic pressure gradient is imposed across the annulus and that there is a uniform small transfer across two walls is pre...
TL;DR: In this article, the homotopy perturbation transform (HPTM) was used to give analytical solutions of the time fractional diffusion equation, where the HPTM is a combined form of the Laplace transform and HPTM methods.