Showing papers on "Homotopy analysis method published in 2014"
TL;DR: The effects of elasticity number, slip coefficient, the relaxation time of the heat flux and the Prandtl number on velocity and temperature fields are analyzed and a comparison of Fourier's Law and the Cattaneo–Christov heat flux model is presented.
354 citations
TL;DR: In this paper, the authors proposed a new and simple algorithm for space-fractional telegraph equation, namely new fractional homotopy analysis transform method (HATM), which is an innovative adjustment in Laplace transform algorithm and makes the calculation much simpler.
231 citations
TL;DR: In this article, the entropy analysis in an unsteady magneto-hydrodynamic nano-fluid regime adjacent to an accelerating stretching permeable surface with the water as the base fluid and four different types of nanoparticles; copper (Cu), copper oxide (CuO), aluminum oxide (Al 2 O 3 ) and titanium dioxide (TiO 2 ).
219 citations
TL;DR: In this article, a homotopy analysis method is employed to examine free convective heat and mass transfer in a steady two-dimensional magnetohydrodynamic fluid flow over a stretching vertical surface in porous medium.
215 citations
TL;DR: In this article, the authors derived semi-analytical/numerical solutions for transport phenomena (momentum, heat and mass transfer) in a nanofluid regime adjacent to a nonlinearly porous stretching sheet by means of the Homotopy analysis method (HAM).
Abstract: In this article we derive semi-analytical/numerical solutions for transport phenomena (momentum, heat and mass transfer) in a nanofluid regime adjacent to a nonlinearly porous stretching sheet by means of the Homotopy analysis method (HAM). The governing equations are reduced to a nonlinear, coupled, non-similar, ordinary differential equation system via appropriate similarity transformations. This system is solved under physically realistic boundary conditions to compute stream function, velocity, temperature and concentration function distributions. The results of the present study are compared with numerical quadrature solutions employing a shooting technique with excellent correlation. Furthermore the current HAM solutions demonstrate very good correlation with the non-transpiring finite element solutions of Rana and Bhargava (Commun. Nonlinear Sci. Numer. Simul. 17:212–226, 2012). The influence of stretching parameter, transpiration (wall suction/injection) Prandtl number, Brownian motion parameter, thermophoresis parameter and Lewis number on velocity, temperature and concentration functions is illustrated graphically. Transpiration is shown to exert a substantial influence on flow characteristics. Applications of the study include industrial nanotechnological fabrication processes.
200 citations
TL;DR: In this article, the authors used homotopy analysis method and numerical Keller-box method to investigate the Falkner-Skan flow of magnetohydrodynamic (MHD) Oldroyd-B fluid.
Abstract: Purpose – In this paper, analysis is presented to investigate the Falkner-Skan flow of magnetohydrodynamic (MHD) Oldroyd-B fluid. The paper aims to discuss these issues. Design/methodology/approach – In this paper, the authors used two methods: homotopy analysis method and numerical Keller-box method. Findings – It is observed that skin friction coefficient in Oldroyd-B fluid is larger when compared with viscous fluid. Further, the relaxation and retardation times have opposite effects on the velocity components. Practical implications – A comparative study between the series and numerical solutions for the skin friction is shown in the paper. The results indicated that both solutions are in well agreement. Originality/value – This model is investigated for the first time, as the authors know.
167 citations
01 Jan 2014
TL;DR: A Short Review of Homotopy Analysis Method: Change and Challenge Predictor Homometopy analysis method Spectral homotopy analysis Method Stability of Auxiliary Linear Operator and Convergence - Control Parameter on the Convergence of the Homometopies Analysis Method Homhotopy Analysis method for Some Boundary Layer Flows of Nanofluids, Homotopic Analysis Method for Fractional Swift - Hohenberg Equation Homotopies, NOPH for Periodic Oscillations, and BVPh 20 for Nonlinear BVPs as mentioned in this paper.
Abstract: A Short Review of Homotopy Analysis Method: Change and Challenge Predictor Homotopy Analysis Method Spectral Homotopy Analysis Method Stability of Auxiliary Linear Operator and Convergence - Control Parameter On the Convergence of the Homotopy Analysis Method Homotopy Analysis Method for Some Boundary Layer Flows of Nanofluids Homotopy Analysis Method for Fractional Swift - Hohenberg Equation Homotopy Analysis Method-based Package NOPH for Periodic Oscillations Homotopy Analysis Method-based Package BVPh 20 for Nonlinear BVPs
165 citations
TL;DR: It can be observed that the auxiliary parameter ℏ, which controls the convergence of the HATM approximate series solutions, also can be used in predicting and calculating multiple solutions.
136 citations
TL;DR: In this article, the steady stagnation point flow of a Casson nano fluid in the presence of convective boundary conditions is investigated. And the resulting ordinary differential equations are successfully solved analytically using Optimal Homotopy analysis method (OHAM) via BVPh2.0.
127 citations
TL;DR: This article addresses the steady three-dimensional flow of an Oldroyd-B nanofluid over a bidirectional stretching surface with heat generation/absorption effects and sees that the increasing values of the Brownian motion parameter and thermophoresis parameter leads to an increase in the temperature field and thermal boundary layer thickness.
Abstract: This article addresses the steady three-dimensional flow of an Oldroyd-B nanofluid over a bidirectional stretching surface with heat generation/absorption effects Suitable similarity transformations are employed to reduce the governing partial differential equations into coupled nonlinear ordinary differential equations These nonlinear ordinary differential equations are then solved analytically by using the homotpy analysis method (HAM) Graphically results are presented and discussed for various parameters, namely, Deborah numbers β1 and β2, heat generation/absorption parameter λ, Prandtl parameter Pr, Brownian motion parameters Nb, thermophoresis parameter Nt and Lewis number Le We have seen that the increasing values of the Brownian motion parameter Nt and thermophoresis parameter Nt leads to an increase in the temperature field and thermal boundary layer thickness while the opposite behavior is observed for concentration field and concentration boundary layer thickness To see the validity of the present work, the numerical results are compared with the analytical solutions obtained by Homotopy analysis method and noted an excellent agreement for the limiting cases
115 citations
TL;DR: In this paper, a homotopy perturbation method with two expanding parameters is proposed for a nonlinear equation with two nonlinear terms, which might have different effects on the solution.
Abstract: A homotopy perturbation method with two expanding parameters is suggested. The method is especially effective for a nonlinear equation with two nonlinear terms, which might have different effects on the solution. A nonlinear oscillator is used as an example to elucidate the solution procedure.
TL;DR: In this article, the optimal homotopy analysis method (OHAM) is employed to investigate the steady laminar incompressible free convective flow of a nanofluid past a chemically reacting upward facing horizontal plate in a porous medium taking into account heat generation/absorption and the thermal slip boundary condition.
Abstract: The optimal homotopy analysis method (OHAM) is employed to investigate the steady laminar incompressible free convective flow of a nanofluid past a chemically reacting upward facing horizontal plate in a porous medium taking into account heat generation/absorption and the thermal slip boundary condition. Using similarity transformations developed by Lie group analysis, the continuity, momentum, energy, and nanoparticle volume fraction equations are transformed into a set of coupled similarity equations. The OHAM solutions are obtained and verified by numerical results using a Runge-Kutta-Fehlberg fourth-fifth order method. The effect of the emerging flow controlling parameters on the dimensionless velocity, temperature, and nanoparticle volume fraction have been presented graphically and discussed. Good agreement is found between analytical and numerical results of the present paper with published results. This close agreement supports our analysis and the accuracy of the numerical computations. This paper also includes a representative set of numerical results for reduced Nusselt and Sherwood numbers in a table for various values of the parameters. It is concluded that the reduced Nusselt number increases with the Lewis number and reaction parameter whist it decreases with the order of the chemical reaction, thermal slip, and generation parameters.
TL;DR: In this article, the natural convection boundary layer flow along with inverted cone, magnetic and heat generation on water and ethylene glycol based nanofluids is considered by means of variable wall temperature.
Abstract: In this study, the natural convection boundary layer flow along with inverted cone, magnetic and heat generation on water and ethylene glycol based nanofluids is considered by means of variable wall temperature. Porous medium is also taken into account. The physical problem is first modeled and then the governing equations are transformed into nonlinear ordinary differential equations under the assumptions of the Boussinesq approximation. Analytical solutions of nonlinear coupled equations are obtained by the homotopy analysis method. Correlation of skin friction and heat transfer rate corresponding to active parameters is also presented. Obtained results are illustrated by graphs and tables in order to see the effects of physical parameters.
TL;DR: In this paper, an analytical solution for the size dependent nonlinear vibration behavior of micro-pipes conveying fluid made of functionally graded materials (FGMs) is presented for the Euler-Bernoulli beam model, the strain gradient theory and von Karman geometric nonlinearity.
TL;DR: In this article, a detailed analysis is given for the mixed convection flow of a nanofluid over a stretching surface with uniform free stream in the presence of both nanoparticles and gyrotactic microorganisms.
TL;DR: The heat and mass transfer of two-layer flows of non-Newtonian (third-grade) fluid in a vertical channel is investigated in details, when one layer of water is nano-fluid and the other is clear, with the viscous dissipation.
TL;DR: In this paper, the authors presented a numerical algorithm for time-fractional Black-Scholes equation with boundary condition for a European option problem by using homotopy perturbation method.
Abstract: The aim of present paper is to present a numerical algorithm for time-fractional Black–Scholes equation with boundary condition for a European option problem by using homotopy perturbation method and homotopy analysis method. The fractional derivative is described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The methods show improvements over existing analytical techniques. Two examples are given and show that the homotopy perturbation method and homotopy analysis method are very effective and convenient overcomes the difficulty of traditional methods. The numerical results show that the approaches are easy to implement and accurate when applied to time-fractional Black–Scholes equation.
TL;DR: Two-dimensional boundary layer flow of Carreau fluid past a permeable stretching sheet is studied and heat transfer subject to convective boundary conditions is analyzed.
TL;DR: In this article, a comprehensive analysis of megnetohydrodynamic (MHD) steady boundary layer flow has been discussed through the linear stretching surface, where the effect of thermal radiation are also taken into account while fluid consists of nanoparticles with convective boundary conditions.
Abstract: In the present article, a comprehensive analysis of megnetohydrodynamic (MHD) steady boundary layer flow have been discussed through the linear stretching surface. The effect of thermal radiation are also taken into account while fluid consists of nanoparticles with convective boundary conditions. The resulting expressions for temperature and nanoparticle equations are coupled. The simplified non-linear equations are tackled with the help of homotopy analysis method (HAM). Main objective of this article is to inspect the effects of emerging parameters which appears in solution. Interesting results are shown graphically. The dimensionless heat transfer rates and dimensionless concentration rate are also plotted against flow control parameters.
TL;DR: In this article, the steady boundary layer flow of an Eyring-Powell model fluid due to an exponentially shrinking sheet was examined and the heat transfer process in the presence of thermal radiation was considered.
TL;DR: Pressure in the case of fractional models (fractional Oldroyd-B model and fractional Maxwell model) of viscoelastic fluids is considerably more substantial than that in the corresponding classical visCoelastic models (OldroyD-B and Maxwell models).
Abstract: This investigation deals with the peristaltic flow of generalised Oldroyd-B fluids (with the fractional model) through a cylindrical tube under the influence of wall slip conditions. The analysis is carried out under the assumptions of long wavelength and low Reynolds number. Analytical approximate solutions are obtained by using the highly versatile and rigorous semi-numerical procedure known as the homotopy analysis method. It is assumed that the cross section of the tube varies sinusoidally along the length of the tube. The effects of the dominant hydromechanical parameters, i.e. fractional parameters, material constants, slip parameter, time and amplitude on the pressure difference across one wavelength, are studied. Graphical plots reveal that the influence of both fractional parameters on pressure is opposite to each other. Interesting responses to a variation in the constants are obtained. Pressure is shown to be reduced by increasing the slip parameter. Furthermore, the pressure in the case of fra...
TL;DR: In this paper, a derivation of the nonlinear equation of motion is presented based on Hamilton's principle and a modified couple stress theory, which captures the micro-structure dependent size effects when the microtubes are at micron and submicron scales.
TL;DR: In this article, the authors studied the flow of an incompressible fluid of modified second grade past an infinite porous plate subject to either suction or blowing at the plate, and derived equations of motion in dimensionless form.
Abstract: The flow of an incompressible fluid of modified second grade past an infinite porous plate subject to either suction or blowing at the plate is studied. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. Equations of motion in dimensionless form are derived. Analytical solutions of the outcoming non-linear differential equations are found by using the homotopy analysis method (HAM), which is a powerful semi-analytical method. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.
TL;DR: This article studies the viscous flow and heat transfer over a plane horizontal surface stretched non-linearly in two lateral directions by employing Appropriate wall conditions characterizing the non-linear variation in the velocity and temperature of the sheet.
Abstract: This article studies the viscous flow and heat transfer over a plane horizontal surface stretched non-linearly in two lateral directions Appropriate wall conditions characterizing the non-linear variation in the velocity and temperature of the sheet are employed for the first time A new set of similarity variables is introduced to reduce the boundary layer equations into self-similar forms The velocity and temperature distributions are determined by two methods, namely (i) optimal homotopy analysis method (OHAM) and (ii) fourth-fifth-order Runge-Kutta integration based shooting technique The analytic and numerical solutions are compared and these are found in excellent agreement Influences of embedded parameters on momentum and thermal boundary layers are sketched and discussed
TL;DR: In this paper, a comparative study between homotopy perturbation method (HPM) and OASM method (OHAM) is presented, where the approximate solutions of the Boussinesq-Burger equation are compared with the optimal HPM as well as with the exact solutions.
05 Aug 2014
TL;DR: The Hom4PS-3 software package has a new fully modular design which allows it to be easily extended and includes interfaces to a variety of existing mathematical software and programming languages such as Python, Ruby, Octave, Sage and Matlab.
Abstract: Homotopy continuation methods have been proved to be an efficient and reliable class of numerical methods for solving systems of polynomial equations which occur frequently in various fields of mathematics, science, and engineering. Based on the successful software package Hom4PS-2.0 for solving such polynomial systems, Hom4PS-3 has a new fully modular design which allows it to be easily extended. It implements many different numerical homotopy methods including the Polyhedral Homotopy continuation method. Furthermore, it is capable of carrying out computation in parallel on a wide range of hardware architectures including multi-core systems, computer clusters, distributed environments, and GPUs with great efficiency and scalability. Designed to be user-friendly, it includes interfaces to a variety of existing mathematical software and programming languages such as Python, Ruby, Octave, Sage and Matlab.
TL;DR: In this article, the effects of heat transfer on the Williamson fluid over a porous exponentially stretching surface were explored, and the boundary layer equations of Williamson fluid model for two dimensional flow with heat transfer were presented.
Abstract: This study explores the effects of heat transfer on the Williamson fluid over a porous exponentially stretching surface. The boundary layer equations of the Williamson fluid model for two dimensional flow with heat transfer are presented. Two cases of heat transfer are considered, i.e., the prescribed exponential order surface temperature (PEST) case and the prescribed exponential order heat flux (PEHF) case. The highly nonlinear partial differential equations are simplified with suitable similar and non-similar variables, and finally are solved analytically with the help of the optimal homotopy analysis method (OHAM). The optimal convergence control parameters are obtained, and the physical features of the flow parameters are analyzed through graphs and tables. The skin friction and wall temperature gradient are calculated.
TL;DR: This research examines the Falkne–Skan flow of magnetohydrodynamic (MHD) Maxwell fluid and establishes both analytic and numerical solutions for the governing problem.
TL;DR: In this article, the homotopy analysis method with different numbers of unknown convergence-control parameters has been used to derive accurate analytic solution for micropolar flow in a porous channel with mass injection.
Abstract: Purpose – The paper aims to find an accurate analytic solution (series solution) for the micropolar flow in a porous channel with mass injection for different values of Reynolds number. Design/methodology/approach – In this paper, the homotopy analysis method (HAM) with different numbers of unknown convergence-control parameters has been used to derive accurate analytic solution for micropolar flow in a porous channel with mass injection. The possible optimal value of convergence-control parameter determined by minimizing the averaged residual error. Findings – The results obtained from HAM solution with two parameters are compared with numerical results and that obtained from HAM solution with only one parameter. The results show that this method gives an analytical solution with high order of accuracy with a few iterations. As shown in this paper, by minimizing the averaged residual error, the authors can get the possible optimal value of the convergence-control parameters which may give the fastest con...
TL;DR: This study investigates the unsteady flow of Powell-Eyring fluid past an inclined stretching sheet, finding unsteadiness in the flow is due to the time-dependence of the stretching velocity and wall temperature.
Abstract: This study investigates the unsteady flow of Powell-Eyring fluid past an inclined stretching sheet. Unsteadiness in the flow is due to the time-dependence of the stretching velocity and wall temperature. Mathematical analysis is performed in the presence of thermal radiation and non-uniform heat source/sink. The relevant boundary layer equations are reduced into self-similar forms by suitable transformations. The analytic solutions are constructed in a series form by homotopy analysis method (HAM). The convergence interval of the auxiliary parameter is obtained. Graphical results displaying the influence of interesting parameters are given. Numerical values of skin friction coefficient and local Nusselt number are computed and analyzed.