Analytic solution for axisymmetric flow over a nonlinearly stretching sheet

An analysis is performed for the boundary-layer flow of a viscous fluid over a nonlinear axisymmetric stretching sheet. By introducing new nonlinear similarity transformations, the partial differential equations governing the flow are reduced to an ordinary differential equation. The resulting ordinary differential equation is solved using the homotopy analysis method (HAM). Analytic solution is given in the form of an infinite series. Convergence of the obtained series solution is explicitly established. The solution for an axisymmetric linear stretching sheet is obtained as a special case.     

Extended homotopy perturbation method and the axisymmetric flow past a porous stretching sheet

The extended homotopy perturbation method, which is an extension of the celebrated homotopy perturbation method (HPM), is applied to obtain a solution to the problem of the steady, laminar, axisymmetric flow of a viscous, incompressible fluid past a porous stretching sheet. The solution so obtained is totally analytical and is expressible in terms of the cross‐flow velocity of the fluid past the stretching sheet. Its hallmark is that it does not depend upon computation of any auxiliary parameter for enlarging the convergence region of the solution. Rather, it calculates the solution automatically adjusting the scaling factor of the independent similarity variable normal to the sheet. The results obtained by the extended HPM are in excellent agreement with the exact numerical solution. Also, an asymptotic solution valid for large suction parameter is developed, which matches well with the exact solution even for moderate values of the suction parameter. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical solution to stagnation-point flow and heat transfer over a stretching sheet based on homotopy analysis

This paper is concerned with two-dimensional stagnation-point steady flow of an incompressible viscous fluid towards a stretching sheet whose velocity is proportional to the distance from the slit. The governing system of partial differential equations is first transformed into a system of dimensionless ordinary differential equations. Analytical solutions of the velocity distribution and dimensionless temperature profiles are obtained for different ratios of free stream velocity and stretching velocity, Prandtl number, Eckert number and dimensionality index in series forms using homotopy analysis method（HAM）. It is shown that a boundary layer is formed when the free stream velocity exceeds the stretching velocity, and an inverted boundary layer is formed when the free stream velocity is less than the stretching velocity. Graphs are presented to show the effects of different parameters.     

Accurate solutions for viscoelastic boundary layer flow and heat transfer over stretching sheet

In this article, we present accurate analytical solutions for boundary layer flow and heat transfer of an incompressible and electrically conducting viscoelastic fluid over a linearly stretching surface subject to a transverse uniform magnetic field using the homotopy analysis method （HAM） for two general types of non-isothermal boundary conditions. In addition, we demonstrate that the previously reported analytical solutions for the temperature field given in terms of Kummer＇s function do not converge at the boundary. We provide a graphical and numerical demonstration of the convergence of the HAM solutions and tabulate the effects of various parameters on the skin friction coefficient and wall heat transfer.     

Note on similarity solutions for viscous flow over an impermeable and non-linearly (quadratic) stretching sheet

Similarity solutions for flow over an impermeable, non-linearly (quadratic) stretching sheet were studied recently by Raptis and Perdikis (Int. J. Non-linear Mech. 41 (2006) 527-529) using a stream function of the form ψ=αxf(η)+βx²g(η). A fundamental error in their problem formulation is pointed out. On correction, it is shown that similarity solutions do not exist for this choice of ψ.     

Similarity solutions to viscous flow and heat transfer of nanofluid over nonlinearly stretching sheet

The boundary-layer flow and heat transfer in a viscous fluid containing metallic nanoparticles over a nonlinear stretching sheet are analyzed. The stretching velocity is assumed to vary as a power function of the distance from the origin. The governing partial differential equation and auxiliary conditions are reduced to coupled nonlinear ordinary differential equations with the appropriate corresponding auxiliary conditions. The resulting nonlinear ordinary differential equations (ODEs) are solved numerically. The effects of various relevant parameters, namely, the Eckert number Ec, the solid volume fraction of the nanoparticles φ, and the nonlinear stretching parameter n are discussed. The comparison with published results is also presented. Different types of nanoparticles are studied. It is shown that the behavior of the fluid flow changes with the change of the nanoparticles type.     

Boundary layer flow of Oldroyd-B fluid by exponentially stretching sheet

The present paper investigates the steady flow of an Oldroyd-B fluid. The fluid flow is induced by an exponentially stretched surface. Suitable transformations reduce a system of nonlinear partial differential equations to a system of ordinary differential equations. Convergence of series solution is discussed explicitly by a homotopy analysis method (HAM). Velocity, temperature and heat transfer rates are examined for different involved parameters through graphs. It is revealed that for a larger retardation time constant, the velocity is enhanced and the temperature is lowered. It is noted that relaxation time constant and the Prandtl number enhance the heat transfer rate.     

On the solution of MHD flow over a nonlinear stretching sheet by an efficient semi‐analytical technique

The problem of magneto‐hydrodynamic fluid flow past a nonlinear stretching sheet in the presence of a transverse magnetic field is analyzed. The governing equations are transformed into a nonlinear ordinary differential equation that is solved using a novel spectral homotopy analysis method and the Matlab in‐built numerical solverttbvp4c. The new technique removes some known limitations of the homotopy analysis method and offers a more systematic way of selecting initial approximations and the optimal auxiliary parameter ?. A comparison with the numerical solution confirms the robustness, the computational efficiency, and the accuracy of the technique. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical treatment on magnetohydrodynamic (MHD) flow and heat transfer due to a stretching hollow cylinder

This paper studied on magnetohydrodynamics flow and heat transfer outside a stretching cylinder. Momentum and energy equations are reduced using similarity transformation and converted into a system of ordinary differential equations which are solved analytically by the homotopy analysis method. The effects of the parameters involved, namely the magnetic parameter (M), Prandtl number (Pr) and Reynolds number (Re) on the velocity and temperature fields are investigated. The obtained results are valid for the whole solutions' domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical solutions to heat transfer of nanofluid flow over stretching sheet subjected to variations of nanoparticle volume fraction and wall temperature

The numerical analysis of heat transfer of laminar nanofluid flow over a fiat stretching sheet is presented. Two sets of boundary conditions （BCs） axe analyzed, i.e., a constant （Case 1） and a linear streamwise variation of nanopaxticle volume fraction and wall temperature （Case 2）. The governing equations and BCs axe reduced to a set of nonlinear ordinary differential equations （ODEs） and the corresponding BCs, respectively. The dependencies of solutions on Prandtl number Pr, Lewis number Le, Brownian motion number Nb, and thermophoresis number Nt are studied in detail. The results show that the reduced Nusselt number and the reduced Sherwood number increase for the BCs of Case 2 compared with Case 1. The increases of Nb, Nt, and Le numbers cause a decrease of the reduced Nusselt number, while the reduced Sherwood number increases with the increase of Nb and Le numbers. For low Prandtl numbers, an increase of Nt number can cause to decrease in the reduced Sherwood number, while it increases for high Prandtl numbers.     

具有热源项的驻点流动和传热问题的近似解析解

研究持续拉伸变形表面上二维平面和轴对称驻点流的动量和热量传输问题。利用同伦分析方法获得速度分布和温度分布的级数解,讨论了级数的敛散性。通过图形分析主流速度与拉伸速度的比率参数,普朗特参数,热源参数和流动类型指标对速度边界层和温度边界层的影响。结果表明,这些参数对二维平面驻点流动和传热有较大的影响。     

Cattaneo-Christov heat flux model for third-grade fluid flow towards exponentially stretching sheet

The Cattaneo-Christov heat flux in the two-dimensional (2D) flow of a third-grade fluid towards an exponentially stretching sheet is investigated. The energy equation is considered through thermal relaxation. Similarity transformations are accounted to obtain the ordinary differential systems. The converted non-dimensional equations are solved for the series solutions. The convergence analysis of the computed solutions is reported. The graphical results of the velocity and temperature profiles are plotted and elaborated in detail. The results show that the thermal relaxation enhances the temperature gradient while reduces the temperature profile.     

Viscous flow over a non-linearly stretching sheet in the presence of a chemical reaction and magnetic field

The steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field is analyzed. The equations governing the flow and concentration field are reduced to a system of coupled non-linear ordinary differential equations. These non-linear differential equations are solved numerically by using the shooting method. The numerical results for the concentration field are presented through graphs.     

MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with homotopy analysis method

The present paper investigates the magnetohydrodynamic（MHD） flow of a viscous fluid towards a nonlinear porous shrinking sheet.The governing equations are simplified by similarity transformations.The reduced problem is then solved by the homotopy analysis method.The pertinent parameters appearing in the problem are discussed graphically and presented in tables.It is found that the shrinking solutions exist in the presence of MHD.It is also observed from the tables that the solutions for f（0） with different values of parameters are convergent.     

Effects of slip condition on MHD stagnation-point flow over a power-law stretching sheet

The steady two-dimensional magnetohydrodynamic stagnation flow towards a nonlinear stretching surface is studied. The no-slip condition on the solid boundary is replaced with a partial slip condition. A scaling group transformation is used to get the invariants. Using the invariants, a third-order ordinary differential equation corresponding to the momentum is obtained. An analytical solution is obtained in a series form using a homotopy analysis method. Reliability and efficiency of series solutions are shown by the good agreement with numerical results presented in the literature. The effects of the slip parameter, the magnetic field parameter, the velocity ratio parameter, the suction velocity parameter, and the power law exponent on the flow are investigated. The results show that the velocity and shear stress profiles are greatly influenced by these parameters.     

MHD flow and heat transfer in a thin liquid film on an unsteady stretching sheet by the homotopy analysis method

The magnetohydrodynamics flow and heat transfer in a thin liquid film over an unsteady elastic stretching surface are analyzed by the homotopy analysis method. A more general surface temperature is taken into consideration. The effects of various parameters in this study are discussed and presented graphically. The good agreement between the analytic series solutions and the previous numerical results shows the effectiveness of HAM to this problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Analytic homotopy solution of generalized three-dimensional channel flow due to uniform stretching of the plate

In this communication a generalized threedimensional steady flow of a viscous fluid between two infinite parallel plates is considered. The flow is generated due to uniform stretching of the lower plate in x- and y-directions. It is assumed that the upper plate is uniformly porous and is subjected to constant injection. The governing system is fully coupled and nonlinear in nature. A complete analytic solution which is uniformly valid for all values of the dimensionless parameters β, Re and λ is obtained by using a purely analytic technique, namely the homotopy analysis method. Also the effects of the parameters β, Re and λ on the velocity field are discussed through graphs.     

Mixed convection unsteady stagnation point flow over a stretching sheet with heat transfer in the presence of variable free stream

An analysis is performed for the unsteady mixed convection flow of an incompressible viscous fluid about a stagnation point on a stretching sheet in the presence of a variable free stream. The equations of motion and energy are transformed into the ordinary differential equations by using similarity transformations. Homotopy analysis method is used for the solution of the governing problem. The results have been discussed by plots. The present values of the function are shown very close to the previous limiting solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Approximate solutions for the problem of liquid film flow over an unsteady stretching sheet with thermal radiation and magnetic field

The proposed method is based on replacement of the unknown function by a truncated series of the shifted Legendre polynomial expansion. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error for the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shifted Legendre coefficients. Thus, after solving this system of equations, the shifted Legendre coefficients are obtained. This efficient numerical method is used to solve the system of ordinary differential equations which describe the thin film flow and heat transfer with the effects of the thermal radiation, magnetic field, and slip velocity.     

MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field

The problem of the steady magnetohydrodynamic (MHD) stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.