Series solutions of annular axisymmetric stagnation flow and heat transfer on moving cylinder

The steady, laminar, incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are investigated. The governing Navier-Stokes and thermal equations of the flow are reduced to a nonlinear system of ordinary differential equations. The equations are solved analyt- ically using the homotopy analysis method （HAM）. Convergence of the HAM solutions is discussed in detail. These solutions are then compared with recently obtained numericM and perturbative solutions. Plots of the velocity and temperature profiles are provided for various values of the relevant parameters.     

Solution of boundary layer flow and heat transfer of an electrically conducting micropolar fluid in a non-Darcian porous medium

In this paper, we have studied the effects of radiation on the boundary layer flow and heat transfer of an electrically conducting micropolar fluid over a continuously moving stretching surface embedded in a non-Darcian porous medium with a uniform magnetic field has been analyzed analytically. The governing fundamental equations are approximated by a system of nonlinear locally similar ordinary differential equations which are solved analytically by applying homotopy analysis method (HAM). The effects of Darcy number, heat generation parameter and inertia coefficient parameter are determined on the flow. Convergence of the obtained series solution is discussed. The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter which provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.     

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.     

Second-order slip MHD flow and heat transfer of nanofluids with thermal radiation and chemical reaction

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.     

Steady flow and heat transfer of a magnetohydrodynamic Sisko fluid through porous medium in annular pipe

In this paper, the steady flow and heat transfer of a magnetohydrodynamic fluid is studied. The fluid is assumed to be electrically conducting in the presence of a uniform magnetic field and occupies the porous space in annular pipe. The governing nonlinear equations are modeled by introducing the modified Darcy's law obeying the Sisko model. The system is solved using the homotopy analysis method (HAM), which yields analytical solutions in the form of a rapidly convergent infinite series. Also, HAM is used to obtain analytical solutions of the problem for noninteger values of the power index. The resulting problem for velocity field is then numerically solved using an iterative method to show the accuracy of the analytic solutions. The obtained solutions for the velocity and temperature fields are graphically sketched and the salient features of these solutions are discussed for various values of the power index parameter. We also present a comparison between Sisko and Newtonian fluids. Copyright © 2011 John Wiley & Sons, Ltd.     

Steady laminar mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface in the presence of a magnetic field

A similarity solution for a steady laminar mixed convection boundary layer flow of a nanofluid near the stagnation point on a vertical permeable plate with a magnetic field and a buoyancy force is obtained by solving a system of nonlinear ordinary differential equations. These equations are solved analytically by using a new kind of a powerful analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). Three different types of nanoparticles, namely, copper (Cu), alumina (Al₂O₃), and titanium oxide (TiO₂), with water as the base fluid are considered. The influence of the volume fraction of nanoparticles, permeability parameter, magnetic parameter, and mixed convection parameter on the surface shear stress and surface heat transfer, as well as on the velocity and temperature profiles, is considered. It is observed that the skin friction coefficient and the local Nusselt number increase with the nanoparticle volume fraction for all types of nanoparticles considered in this study. The greatest values of the skin friction coefficient and the local Nusselt number are obtained for Cu nanoparticles.     

Homotopy analysis solution for micropolar fluid flow through porous channel with expanding or contracting walls of different permeabilities

The flow of a micropolar fluid through a porous channel with expanding or contracting walls of different permeabilities is investigated. Two cases are considered, in which opposing walls undergo either uniform or non-uniform motion. In the first case,the homotopy analysis method (HAM) is used. to obtain the expressions for the velocity and micro-rotation fields. Graphs are sketched for some parameters. The results show that the expansion ratio and the different permeabilities have important effects on the dynamic characteristics of the fluid. Following Xu's model, in the second case which is more general, the wall expansion ratio varies with time. Under this assumption, the governing equations are transformed into nonlinear partial differential equations that can also be solved analytically by the HAM. In the process, both algebraic and exponential models are considered to describe the evolution of α(t) from the initial state α0 to the final state α1. As a result, the time-dependent solutions are found to approach the steady state very rapidly. The results show that the time-dependent variation of the wall expansion ratio can be ignored because of its limited effects.     

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.     

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.     

Thermal radiation effect on a mixed convection flow and heat transfer of the Williamson fluid past an exponentially shrinking permeable sheet with a convective boundary condition

The thermal radiation effect on a steady mixed convective flow with heat transfer of a nonlinear (non-Newtonian) Williamson fluid past an exponentially shrinking porous sheet with a convective boundary condition is investigated numerically. In this study, both an assisting flow and an opposing flow are considered. The governing equations are converted into nonlinear ordinary differential equations by using a suitable transformation. A numerical solution of the problem is obtained by using the Matlab software package for different values of the governing parameters. The results show that dual nonsimilar solutions exist for the opposing flow, whereas the solution for the assisting flow is unique. It is also observed that the dual nonsimilar solutions exist only if a certain amount of mass suction is applied through the porous sheet, which depends on the Williamson parameter, convective parameter, and radiation parameter.     

Analytic study on non-Newtonian natural convection boundary layer flow with variable wall temperature on a horizontal plate

The similarity transform for the steady free convection boundary layer flow of a non-Newtonian fluid (Casson model) with variable wall temperature on a horizontal plate gives a system of nonlinear ordinary differential equations which is solved analytically by applying a newly developed method namely the homotopy analysis method (HAM). The velocity and temperature profiles are obtained and the influence of Prandtl number and various physical parameters of the problem on these distributions are discussed in detail and are illustrated graphically through a set of graphs. The validity of our solutions is verified by the numerical results.     

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid past nonlinear stretching surface

Sufficient conditions are found for the existence of similar solutions of the mixed convection flow of a Powell-Eyring fluid over a nonlinear stretching permeable sur- face in the presence of magnetic field. To achieve this, one parameter linear group trans- formation is applied. The governing momentum and energy equations are transformed to nonlinear ordinary differential equations by use of a similarity transformation. These equations are solved by the homotopy analysis method （HAM） to obtain the approximate solutions. The effects of magnetic field, suction, and buoyancy on the Powell-Eyring fluid flow with heat transfer inside the boundary layer are analyzed. The effects of the non- Newtonian fluid （Powell-Eyring model） parameters ε and δon the skin friction and local heat transfer coefficients for the cases of aiding and opposite flows are investigated and discussed. It is observed that the momentum boundary layer thickness increases and the thermal boundary layer thickness decreases with the increase in ε whereas the momentum boundary layer thickness decreases and thermal boundary layer thickness increases with the increase in δ for both the aiding and opposing mixed convection flows.     

Heat transfer analysis of Williamson fluid over exponentially stretching surface

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 fea- tures of the flow parameters are analyzed through graphs and tables. The skin friction and wall temperature gradient are calculated.     

Influence of heat and mass transfer on a peristaltic motion of a Jeffrey-six constant fluid in an annulus

In the present investigation we have studied the combined heat and mass transfer effects on the peristaltic flow of non-Newtonian fluid. The governing equations are simplified under the assumptions of longwave length low Reynold’s number approximation. The resulting highly nonlinear equations are solved analytically and numerically. At the end the numerical results are discussed pictorially for various emerging parameters of interest.     

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.     

Melting heat transfer in steady laminar flow over a moving surface

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.     

Analysis of a laminar boundary layer flow over a flat plate with injection or suction

An analysis is performed to study a laminar boundary layer flow over a porous flat plate with injection or suction imposed at the wall. The basic equations of this problem are reduced to a system of nonlinear ordinary differential equations by means of appropriate transformations. These equations are solved analytically by the optimal homotopy asymptotic method (OHAM), and the solutions are compared with the numerical solution (NS). The effect of uniform suction/injection on the heat transfer and velocity profile is discussed. A constant surface temperature in thermal boundary conditions is used for the horizontal flat plate.     

Series solutions for steady three-dimensional stagnation point flow of a nanofluid past a circular cylinder with sinusoidal radius variation

This article deals with the study of the steady three-dimensional stagnation point flow of a nanofluid past a circular cylinder that has a sinusoidal radius variation. By means of similarity transformation, the governing partial differential equations are reduced into highly non-linear ordinary differential equations. The resulting non-linear system has been solved analytically using an efficient technique namely homotopy analysis method (HAM). Expressions for velocity and temperature fields are developed in series form. In this study, three different types of nanoparticles are considered, namely alumina (Al₂O₃), titania (TiO₂), and copper (Cu) with water as the base fluid. For alumina–water nanofluid, graphical results are presented to describe the influence of the nanoparticle volume fraction φ and the ratio of the gradient of velocities c on the velocity and temperature fields. Moreover, the features of the flow and heat transfer characteristics are analyzed and discussed for foregoing nanofluids. It is found that the skin friction coefficient and the heat transfer rate at the surface are highest for copper–water nanofluid compared to the alumina–water and titania–water nanofluids.     

MHD flow and heat transfer over a stretching surface with variable thermal conductivity and partial slip

In this paper we analyze the flow and heat transfer of an MHD fluid over an impermeable stretching surface with variable thermal conductivity and non-uniform heat source/sink in the presence of partial slip. The governing partial differential equations of the problem are reduced to nonlinear ordinary differential equations by using a similarity transformation. The temperature boundary conditions are assumed to be linear functions of the distance from the origin. Analytical solutions of the energy equations for Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF) cases are obtained in terms of a hypergeometric function, without applying the boundary-layer approximation. The effects of the governing parameters on the flow and heat transfer fields are presented through tables and graphs, and they are discussed. Furthermore, the obtained numerical results for the skin friction, wall-temperature gradient and wall temperature are analyzed and compared with the available results in the literature for special cases.     

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.