Showing papers on "Homotopy analysis method published in 2015"
TL;DR: In this article, the authors provided an analytical investigation of the fluid flow, heat and mass transfer and entropy generation for the steady laminar non-Newtonian nano-fluid flow induced by a stretching sheet in the presence of velocity slip and convective surface boundary conditions using Optimal Homotopy Analysis Method (OHAM).
214 citations
TL;DR: In this article, the natural convection boundary layer flow along a vertical cone with variable wall temperature under the presence of magnetohydrodynamics is investigated, and the mathematical model based on nanolayer single and multi-wall carbon nanotubes in salt-water solutions and interfacial layers is considered.
Abstract: The natural convection boundary layer flow along a vertical cone with variable wall temperature under the presence of magnetohydrodynamics is investigated. The mathematical model based on nanolayer single and multi-wall carbon nanotubes in salt-water solutions and interfacial layers is considered. The nonlinear partial differential equations are analytically solved by means of Mathematica package BVPh 2.0 based on the homotopy analysis method. Flow behavior under altering involving physical parameters such as: Eckert number, Prandtl number, power law index, and nanoparticle volume fraction on the velocity and the temperature profiles is discussed and explained through graphs and tables. The values of Skin friction and Nusselt number are computed and examined.
211 citations
TL;DR: In this paper, the authors have considered the three dimensional heat and mass transfer with magnetic effects for the flow of a nanofluid between two parallel plates in a rotating system and applied Homotopy Analysis Method (HAM) to see the validity of analytical solution, a numerical solution using RK-4 method coupled with shooting method has also been sought Both the solutions are found to be in excellent agreement to capture the effects of involved physical parameters graphical representation of the flow are included with comprehensive discussions.
147 citations
TL;DR: In this paper, the BVPh 2.0 is successfully applied to solve magnetohydrodynamic (MHD) Falkner-Skan flow of nano-fluid past a fixed wedge in a semi-infinite domain, and the influence of physical parameters on the considered flows is investigated in details.
142 citations
TL;DR: In this article, the authors focused on the study of combined heat and mass transfer by MHD stagnation point flow toward a permeable stretching surface in the presence of a first order chemical reaction.
103 citations
TL;DR: The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media in the sense of the local fractional differential operator.
Abstract: In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media.
102 citations
TL;DR: In this article, the analytic solution of steady boundary layer axisymmetric flow of third-grade fluid over a continuously stretching cylinder in the presence of magnetic field is focused on the analytic solutions are considered to reduce the partial differential equation into the ordinary differential equation.
Abstract: This work is focused on the analytic solution of steady boundary layer axisymmetric flow of third-grade fluid over a continuously stretching cylinder in the presence of magnetic field. Suitable transformations are considered to reduce the partial differential equation into the ordinary differential equation. The obtained non-linear differential system is solved by homotopy analysis method (HAM). The effects of the emerging parameters such as third-grade parameter, second-grade parameters and Reynolds number on the velocity are displayed and discussed. The expression of skin-friction coefficient is computed and presented. It is found that velocity and momentum boundary layer thickness are increasing functions of curvature parameter. Velocity profile is higher for third-grade fluid when compared with Newtonian and second-grade fluids with and without MHD effects for the cases (i) stretching cylinder and (ii) flat plate.
102 citations
TL;DR: In this paper, the laminar axisymmetric flow of nanofluid over a non-linearly stretching sheet is reported, which contains the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles.
Abstract: This article reports the laminar axisymmetric flow of nanofluid over a non-linearly stretching sheet. The model used for nanofluid contains the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The recently proposed boundary condition is considered which requires the mass flux of nanoparticles at the wall to be zero. Analytic solutions of the arising boundary value problem are obtained by optimal homotopy analysis method. Moreover the numerical solutions are computed by Keller–Box method. Both the solutions are found in excellent agreement. The behavior of Brownian motion on the fluid temperature and wall heat transfer rate is insignificant. Further the nanoparticle volume fraction distribution is found to be negative near the vicinity of the stretching sheet.
95 citations
TL;DR: The combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface is investigated.
Abstract: In this paper, we have investigated the combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface. The governing non-linear analysis of partial differential equations is reduced into the ordinary differential equations using similarity transformations. The resulting problems are computed for both series and numerical solutions. Series solution is constructed using homotopy analysis method (HAM) whereas numerical solution is presented by two different techniques namely shooting method and bvp4c. A comparison of homotopy solution with numerical solution is also tabulated. Both solutions are found in an excellent agreement. Dimensionless velocity and temperature profiles are plotted and discussed for various emerging physical parameters.
70 citations
TL;DR: In this article, the exact solution of a time fractional Burgers' equation, where the derivative is conformable fractional derivative, with dirichlet and initial conditions is found byHopf-Cole transform.
Abstract: Abstract Firstly in this article, the exact solution of a time fractional Burgers’ equation, where the derivative is conformable fractional derivative, with dirichlet and initial conditions is found byHopf-Cole transform. Thereafter the approximate analytical solution of the time conformable fractional Burger’s equation is determined by using a Homotopy Analysis Method(HAM). This solution involves an auxiliary parameter ~ which we also determine. The numerical solution of Burgers’ equation with the analytical solution obtained by using the Hopf-Cole transform is compared.
68 citations
TL;DR: In this paper, the authors studied the Joules heating effects on stagnation point flow of Newtonian and non-Newtonian fluids over a stretching cylinder by means of GA and found the analytical and numerical solutions for the said mathematical model.
Abstract: Purpose – The purpose of this paper is to study the Joules heating effects on stagnation point flow of Newtonian and non-Newtonian fluids over a stretching cylinder by means of genetic algorithm (GA). The main emphasis is to find the analytical and numerical solutions for the said mathematical model. The work undertaken is a blend of numerical and analytical studies. Effects of active parameters such as: Hartmann number, Prandtl number, Eckert number, Nusselt number, Skin friction and dimensionless fluids parameters on the flow and heat transfer characteristics have been examined by graphs and tables. Compression is also made with the existing benchmark results. Design/methodology/approach – Analytical solutions of non-linear coupled equations are developed by optimal homotopy analysis method (OHAM). A very effective and higher order numerical scheme hybrid GA and Nelder-Mead optimization Algorithms are used for numerical investigations. Findings – An excellent agreement with the existing results in limit...
TL;DR: In this article, the convergence of homotopy perturbation method has been discussed and elaborated briefly, but only a few works have considered the problem of convergence of the method.
TL;DR: In this article, the authors presented a reliable algorithm based on the new homotopy perturbation transform method (HPTM) to solve a time-fractional Navier-Stokes equation in a tube.
TL;DR: In this article, a homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomial coefficients are introduced, which makes the solution procedure more straightforward and more effective.
Abstract: In this paper, a new adaption of homotopy analysis method is presented to handle nonlinear problems. The proposed approach is capable of reducing the size of calculations and easily overcome the difficulty arising in calculating complicated integrals. Furthermore, the homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomials are introduced. Then, an algorithm of calculating such polynomials, which makes the solution procedure more straightforward and more effective, is constructed. Numerical examples are examined to highlight the significant features of the developed techniques. The algorithms described in this paper are expected to be further employed to solve nonlinear problems in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.
TL;DR: In this paper, the two-dimensional forced convective flow of a generalized Burgers fluid over a linearly stretched sheet under the impacts of nano-sized material particles is reported.
Abstract: This article reports the two-dimensional forced convective flow of a generalized Burgers fluid over a linearly stretched sheet under the impacts of nano-sized material particles. Utilizing appropriate similarity transformations the coupled nonlinear partial differential equations are converted into a set of coupled nonlinear ordinary differential equations. The analytic results are carried out through the homotopy analysis method (HAM) to investigate the impact of various pertinent parameters for the velocity, temperature and concentration fields. The obtained results are presented in tabular form as well as graphically and discussed in detail. The presented results show that the rate of heat transfer at the wall and rate of nanoparticle volume fraction diminish with each increment of the thermophoresis parameter. While incremented values of the Brownian motion parameter lead to a quite opposite effect on the rates of heat transfer and nanoparticle volume fraction at the wall.
TL;DR: The aim of present paper is to study the series solution of time dependent MHD second grade incompressible nanofluid towards a stretching sheet and it is observed that temperature and concentration profiles show similar behavior for thermophoresis parameter Μ but opposite tendency is noted in case of Brownian motion parameter Νb.
Abstract: The aim of present paper is to study the series solution of time dependent MHD second grade incompressible nanofluid towards a stretching sheet. The effects of mixed convection and thermal radiation are also taken into account. Because of nanofluid model, effects Brownian motion and thermophoresis are encountered. The resulting nonlinear momentum, heat and concentration equations are simplified using appropriate transformations. Series solutions have been obtained for velocity, temperature and nanoparticle fraction profiles using Homotopy Analysis Method (HAM). Convergence of the acquired solution is discussed critically. Behavior of velocity, temperature and concentration profiles on the prominent parameters is depicted and argued graphically. It is observed that temperature and concentration profiles show similar behavior for thermophoresis parameter Νt but opposite tendency is noted in case of Brownian motion parameter Νb. It is further analyzed that suction parameter S and Hartman number Μ depict decreasing behavior on velocity profile.
TL;DR: In this paper, the authors deal with the study of third grade fluid flow over a rotating vertical cone in the presence of nanoparticles i.e. thermophoresis and Brownian motion.
TL;DR: In this article, a study of the Marangoni boundary layer flow and heat transfer of copper-water nanofluid over a porous medium disk is presented, assuming that the base fluid water and the nanoparticles copper are in thermal equilibrium and that no slippage occurs between them.
Abstract: In this paper we present a study of the Marangoni boundary layer flow and heat transfer of copper-water nanofluid over a porous medium disk. It is assumed that the base fluid water and the nanoparticles copper are in thermal equilibrium and that no slippage occurs between them. The governing partial differential equations are transformed into a set of ordinary differential equations by generalized Karman transformation. The corresponding nonlinear two-point boundary value problem is solved by the Homotopy analysis method and the shooting method. The effects of the solid volume fraction, the permeability parameter and the Marangoni parameter on the velocity and temperature fields are presented graphically and analyzed in detail.
TL;DR: In this paper, a modified analytical approach is introduced to obtain quick and accurate solution of wave-like fractional physical models using an innovative adjustment in Laplace transform algorithm and homotopy analysis method for fractional partial differential equations.
Abstract: The aim of this article is to introduce a modified analytical approach to obtain quick and accurate solution of wave-like fractional physical models. This modified analytical approach is an innovative adjustment in Laplace transform algorithm and homotopy analysis method for fractional partial differential equations. The proposed technique solves the problems using Adomian’s polynomials. The homotopy analysis transform method utilizes a simple and powerful method to adjust and control the convergence region of the infinite series solution using an auxiliary parameter. The numerical solutions obtained by this modified proposed method indicate that the approach is easy to implement, highly accurate, and computationally very attractive. A good agreement between the obtained solutions and some well-known results has been obtained.
TL;DR: In this paper, a new method based on the Chebyshev wavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is proposed to solve time-fractional fifth-order Sawada-Kotera (SK) equation.
TL;DR: The present exploration discusses the influence of Newtonian heating on the magnetohydrodynamic three dimensional couple stress nanofluid past a stretching surface and it is seen that an increase in conjugate heating parameter results in considerable increase in the temperature profile of the stretching wall.
Abstract: The present exploration discusses the influence of Newtonian heating on the magnetohydrodynamic (MHD) three dimensional couple stress nanofluid past a stretching surface. Viscous dissipation and Joule heating effects are also considered. Moreover, the nanofluid model includes the combined effects of thermophoresis and Brownian motion. Using an appropriate transformation, the governing non linear partial differential equations are converted into nonlinear ordinary differential equations. Series solutions using Homotopy Analysis method (HAM) are computed. Plots are presented to portrait the arising parameters in the problem. It is seen that an increase in conjugate heating parameter results in considerable increase in the temperature profile of the stretching wall. Skin friction coefficient, local Nusselt and local Sherwood numbers tabulated and analyzed. Higher values of conjugate parameter, Thermophoresis parameter and Brownian motion parameter result in enhancement of temperature distribution.
TL;DR: In this paper, the authors describe the unsteady flow of liquid containing nanoparticles and motile gyrotactic microorganisms between two parallel plates while keeping one moving and other fixed.
TL;DR: The numerical results rendering that the applied OHAM method is explicit, effective and easy to use, for handling more general fractional order heat- and wave-like models.
Abstract: Optimal homotopy asymptotic method (OHAM) is prolifically implemented to find the optimal solutions of fractional order heat- and wave-like equations. We inspect the competence of the method by examining fractional order time dependent partial differential equations. It is observed that OHAM is a prevailing and convergent method for the solutions of linear and nonlinear fractional order time dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective and easy to use, for handling more general fractional order heat- and wave-like models.
TL;DR: In this paper, the mutual effects of viscous dissipation and slip effects on a rotating vertical cone in a viscous fluid were investigated and the results showed that the slip parameter γ enhances the primary velocity while the secondary velocity reduces.
Abstract: This paper is concerned with the mutual effects of viscous dissipation and slip effects on a rotating vertical cone in a viscous fluid. Similarity solutions for rotating cone with wall temperature boundary conditions provides a system of nonlinear ordinary differential equations which have been treated by optimal homotopy analysis method (OHAM). The obtained analytical results in comparison with the numerical ones show a noteworthy accuracy for a special case. Effects for the velocities and temperature are revealed graphically and the tabulated values of the surface shear stresses and the heat transfer rate are entered in tables. From the study it is seen that the slip parameter γ enhances the primary velocity while the secondary velocity reduces. Further it is observed that the heat transfer rate NuRe
x
−½ increases with Eckert number Ec and Prandtl number Pr.
TL;DR: In this article, the authors focus on the stratified phenomenon through vertical stretching cylinder in the region of stagnation point with slip effects and find that velocity profile decreases with an increase in stratified parameters due to temperature and concentration.
Abstract: Purpose – The purpose of this paper is to focus on the stratified phenomenon through vertical stretching cylinder in the region of stagnation point with slip effects. Design/methodology/approach – Homotopy analysis method is used to find the series solutions of the governing equations. Findings – Velocity profile decreases with an increase in stratified parameters due to temperature and concentration. Velocity and thermal slips cause a reduction in the velocity profile. Thermally stratified and thermal slip parameters reduce the temperature field. Originality/value – The present analysis has not been existed in the literature yet.
TL;DR: In this article, the second-order velocity slip and temperature jump boundary conditions on the magnetohydrodynamic (MHD) flow and heat transfer in the presence of nanoparticle fractions are investigated.
Abstract: The effects of the second-order velocity slip and temperature jump boundary conditions on the magnetohydrodynamic (MHD) flow and heat transfer in the presence of nanoparticle fractions are investigated. In the modeling of the water-based nanofluids containing Cu and Al2O3, the effects of the Brownian motion, thermophoresis, and thermal radiation are considered. The governing boundary layer equations are transformed into a system of nonlinear differential equations, and the analytical approximations of the solutions are derived by the homotopy analysis method (HAM). The reliability and efficiency of the HAM solutions are verified by the residual errors and the numerical results in the literature. Moreover, the effects of the physical factors on the flow and heat transfer are discussed graphically.
TL;DR: A reduced order model obtained from first mode Galerkin’s decomposition method is used for numerical and analytical investigations and the obtained analytical expressions are able to capture nonlinear behaviors including softening type vibrations and dynamic snap-through.
TL;DR: In this article, the authors presented a user friendly numerical algorithm based on homotopy perturbation transform method for solving various linear and nonlinear convection-diffusion problems arising in physical phenomena where particles, energy and other physical quantities are transferred inside a physical system due to two processes: diffusion and convection.
Abstract: The aim of this paper was to present a user friendly numerical algorithm based on homotopy perturbation transform method for solving various linear and nonlinear convection-diffusion problems arising in physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. The homotopy perturbation transform method is a combined form of the homotopy perturbation method and Laplace transform method. The nonlinear terms can be easily obtained by the use of He’s polynomials. The technique presents an accurate methodology to solve many types of partial differential equations The approximate solutions obtained by proposed scheme in a wide range of the problem’s domain were compared with those results obtained from the actual solutions. The comparison shows a precise agreement between the results.
TL;DR: In this paper, the boundary layer flow of an Oldroyd-B fluid with Newtonian heating was analyzed by homotopy analysis method and series solutions were found by homotropic analysis method.
Abstract: Purpose – The purpose of this paper is to analyze the boundary layer flow of an Oldroyd-B fluid with Newtonian heating. Design/methodology/approach – Series solutions are found by homotopy analysis method. Findings – Temperature profile increases with an increase in conjugate parameter. Increase in parameter β and Prandtl number Pr decreases the temperature profile. Originality/value – This work does not currently exist in the literature.
TL;DR: In this article, the effects of convective heat and concentration conditions in magnetohydrodynamic flow of non-Newtonian fluid with nanoparticles were investigated, where the nonlinear partial differential equations were reduced into nonlinear ordinary differential equations by suitable similarity variables.
Abstract: This article investigates the effects of convective heat and concentration conditions in magnetohydrodynamic flow of non-Newtonian fluid with nanoparticles. Convective type boundary conditions are utilized for heat and nanoparticles concentration. The nonlinear partial differential equations are reduced into the nonlinear ordinary differential equations by suitable similarity variables. Homotopy analysis method is employed to obtain the dimensionless velocity, temperature and nanoparticles concentration expressions. Graphical results for temperature and nanoparticles concentration are plotted and examined. Numerical values of skin friction coefficient are computed and discussed. Heat transfer and nanoparticles concentration transfer rates at the wall are examined by plotting the graphs of different governing physical parameters. We noticed that the temperature and nanoparticles concentration profiles are enhanced when the values of Biot numbers are increased. An increase in thermophoresis parameter leads to an enhancement in the temperature and nanoparticles concentration. On the other hand the increasing values of Brownian motion parameter has reverse effects on the temperature and nanoparticles concentration fields.