Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet

This paper investigates the magnetohydrodynamic (MHD) flow and heat transfer characteristics in the presence of a uniform applied magnetic field. The boundary layer flow of a third-order fluid is induced due to linear stretching of a non-conducting sheet. The heat transfer analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). The governing non-linear differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence of these solutions is discussed. Velocity and temperature distributions are shown graphically. The numerical values for the skin friction coefficient and the Nusselt number are entered in tabular form. Emphasis has been given to the variations of the emerging parameters such as third-order parameter, magnetic parameter, Prandtl number and the Eckert number. It is noted that the skin friction coefficient decreases as the magnetic parameter or the third grade parameter increases.     

Flow and heat transfer of a third grade fluid past an exponentially stretching sheet with partial slip boundary condition

Non-Newtonian boundary layer flow and heat transfer over an exponentially stretching sheet with partial slip boundary condition has been studied in this paper. The flow is subject to a uniform transverse magnetic field. The heat transfer analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST), and (ii) prescribed heat flux (PHF). Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. An effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip and the third grade fluid parameters on the velocity, skin-friction coefficient and the temperature boundary layer. It is found that the third grade fluid parameter β increases the momentum boundary layer thickness and decreases the thermal boundary layer thickness.     

Radiation effects on MHD flow in a porous space

This paper deals with the study of the radiation effects on the magnetohydrodynamic (MHD) flow of an incompressible viscous fluid in a porous space. The flow is induced due to a non-linear stretching sheet. Two cases of heat transfer analysis are discussed. These are: (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case). 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 a very efficient technique namely homotopy analysis method (HAM). Expressions for velocity and temperature fields are developed in series form. Convergence of the series solution is shown explicitly. The influence of various pertinent parameters is also seen on the velocity and temperature fields. The tabulated values of the wall shear stress and the Nusselt number show good agreement with the existing results.     

Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to suction and to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary-layer flow of an electrically conducting fluid of second grade subject to suction and to a transverse uniform magnetic field past a semi-infinite stretching sheet. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation and an analytical solution for this flow is utilized. The effects of viscous dissipation and work due to deformation are considered in the energy equation and the variations of dimensionless surface temperature and dimensionless surface temperature gradient with various parameters are graphed and tabulated. Two cases are studied, namely, (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case).     

Heat transfer in a second grade fluid through a porous medium from a permeable stretching sheet with non-uniform heat source/sink

In the present article an analysis is carried out to study the boundary layer flow and heat transfer characteristics of a second grade, non-Newtonian fluid through a porous medium. The stretching sheet is assumed to be permeable so that suction effects come into play. The effects of viscous dissipation, non-uniform heat source/sink on heat transfer are addressed. The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. Analytical solutions are obtained for the resulting boundary value problems. The effects of viscous dissipation and non-uniform heat source/sink, Prandtl number, Eckert number and suction/injection on heat transfer are shown in several plots for two different heating processes (CST and PST cases). Dimensionless surface temperature gradient is tabulated for various values of the governing the parameters.     

Effects of slip on sheet-driven flow and heat transfer of a third grade fluid past a stretching sheet

The entrained flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip is considered. The partial slip is controlled by a dimensionless slip factor, which varies between zero (total adhesion) and infinity (full slip). The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST case) and (ii) prescribed surface heat flux (PHF case). Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip, magnetic field and the third grade fluid parameter on the velocity, skin-friction coefficient and the temperature field. It is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.     

Numerical modelling of second‐grade fluid flow past a stretching sheet

The present numerical study reports the chemically reacting boundary layer flow of a magnetohydrodynamic second‐grade fluid past a stretching sheet under the influence of internal heat generation or absorption with work done due to deformation in the presence of a porous medium. To distinguish the non‐Newtonian behaviour of the second‐grade fluid with those of Newtonian fluids, a very popularly known second‐grade fluid flow model is used. The fourth order momentum equation with four appropriate boundary conditions along with temperature and concentration equations governing the second‐grade fluid flow are coupled and highly nonlinear in nature. Well‐established similarity transformations are efficiently used to reduce the dimensional flow equations into a set of nondimensional ordinary differential equations with the necessary conditions. The standard bvp4c MATLAB solver is effectively used to solve the fluid flow equations to get the numerical solutions in terms of velocity, temperature, and concentration fields. Numerical results are obtained for a different set of physical parameters and their behaviour is described through graphs and tables. The viscoelastic parameter enhances the velocity field whereas the magnetic and porous parameters suppress the velocity field in the flow region. The temperature field is magnified for increasing values of the heat source/sink parameter. However, from the present numerical study, it is noticed that the flow of heat occurs from sheet to the surrounding ambient fluid. Before concluding the considered problem, our results are validated with previous results and are found to be in good agreement.     

Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid over a vertical stretching sheet

We consider the steady state, viscous, incompressible two-dimensional magneto hydrodynamic flow of an electrically conducting power law fluid over a vertical stretching sheet. The stretching of the surface velocity and the prescribed surface temperature are assumed to vary linearly with the distance from the slit. The coupled partial differential equations governing the flow and heat transfer are transformed into non-linear coupled ordinary differential equations by a similarity transformation. The transformed boundary layer equations are solved numerically by Keller-Box method for several sets of values of the parameters governing the flow and heat transfer. The flow and heat transfer characteristics are analysed and discussed for different values of the parameters. We observe that the local skin friction coefficient and the local Nusselt number decrease as the magnetic parameter Mn increase for fixed value of the buoyancy parameter λ. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Heat transfer analysis for a hydromagnetic viscous fluid over a non-linear shrinking sheet

The boundary layer flow and heat transfer analysis of electrically conducting viscous fluid over a nonlinearly shrinking sheet is investigated. A similarity transformation is used to reduce the governing equations to a set of nonlinear ordinary differential equations. The system of equations is solved numerically employing an implicit finite difference scheme known as Keller-box method. It is found that dual solutions exist for this particular problem. The numerical results for the velocity, temperature, wall skin friction coefficient and local rate of heat transfer through the surface for various values of physical parameters both in case of stretching and shrinking sheet are analyzed and discussed for both the solutions. Present results in the hydrodynamic case (M = 0) are compared with existing numerical results in case of stretching flow and found in good agreement.     

Influence of thermal radiation and Joule heating on MHD flow of a Maxwell fluid in the presence of thermophoresis

This work investigates the magnetohydrodynamic (MHD) two-dimensional flow with heat and mass transfer over a stretching sheet in the presence of Joule heating and thermophoresis. The resulting partial differential equations are converted into a set of coupled ordinary differential equations. Series solutions have been derived by using homotopy analysis method (HAM). The local Nusselt and Sherwood numbers are also computed. Graphical results for the dimensionless velocity, temperature and concentration fields are reported and examined for some parameters showing the interesting aspects of the obtained solutions.     

Heat transfer over a bidirectional stretching sheet with variable thermal conditions

The heat transfer characteristics of a steady three-dimensional viscous fluid flow driven by the bidirectional stretching of an elastic surface are investigated. The hydrodynamic part of the problem is determined by the ratio between the stretching rates in the two lateral directions. The prescribed temperature or heat transfer rate varies along the surface. A heat source is included in the thermal boundary layer equation, which transforms into an ordinary differential equation by means of a similarity transformation. The numerical results show that the principal effect of the variable surface conditions is to thicken the thermal boundary layer when the temperature or heat transfer rate decreases with the distance from the center of sheet. The boundary layer thickness is correspondingly reduced if the sheet temperature or heat transfer rate increases in one or both of the lateral directions.     

Convection heat and mass transfer in a hydromagnetic flow of a second grade fluid in the presence of thermal radiation and thermal diffusion

The convection heat and mass transfer in a hydromagnetic flow of a second grade fluid past a semi-infinite stretching sheet in the presence of thermal radiation and thermal diffusion are considered. The governing coupled non-linear partial differential equations describing the flow problem are transformed into non-linear ordinary differential equations by method of similarity transformation. The resulting similarity equations are solved numerically using Runge-Kutta shooting method. The results are presented as velocity, temperature and concentration fields for different values of parameters entering into the problem. The skin friction, rate of heat transfer and mass transfer are presented numerically in tabular form. In addition, the results obtained showed that these parameters have significant influence on the flow, heat and mass transfer.     

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source

In this paper, visco-elastic boundary layer flow and heat transfer over a stretching sheet in presence of viscous dissipation and non-uniform heat source have been discussed. Analytical solutions of highly non-linear momentum equation and confluent hypergeometric similarity solution of heat transfer equations are obtained. Here two types of different heating processes are considered namely (i) prescribed surface temperature (PST) and (ii) prescribed wall heat flux (PHF). The effect of various parameters like visco-elastic parameter, Eckert number, Prandtl number, and non-uniform heat source/sink parameter on temperature distribution are analyzed and effect of all these parameters on wall temperature gradient and wall temperature are tabulated and discussed.     

Dual solutions for flow and radiative heat transfer of a micropolar fluid over stretching/shrinking sheet

A boundary layer analysis is presented for the flow and radiative heat transfer of an incompressible micropolar fluid over stretching/shrinking sheet with power-law surface velocity and temperature distributions. Dual solutions are analytically obtained firstly by homotopy analysis method (HAM). It is found that dual solutions not only exist for the shrinking flow as reported in the previous literatures, but also exist for the stretching flow. The special case of the first branch (K = 0, classical Newtonian fluid) is compared with the existing numerical results of stretching flow in good agreement. Our results show that both solutions are physically meaningful (two solutions are closely related to each other), unlike the results previously reported that only one solution is acceptable. Moreover, the effects of the material parameter K, the radiative Prandtl number Pr_n, the velocity exponent parameter m and the temperature exponent parameter λ on the flow and heat transfer characteristics are analyzed in detail.     

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet

Studies are made on the viscoelastic fluid flow and heat transfer characteristics over a stretching sheet with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for the cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The momentum equation is decoupled from the energy equation for the present incompressible boundary layer flow problem with constant physical parameters. Exact solution for the velocity field and the skin-friction are obtained. Also, the solutions for the temperature and heat transfer characteristics are obtained in terms of Kummer’s function. The work due to deformation in energy equation, which is essential while formulating the viscoelastic boundary layer flow problems, is considered. This paper examines the effect of viscoelastic parameter, Eckert number, Prandtl number and non-uniform heat source/sink parameter on temperature distribution, wall temperature gradient for PST-case and wall temperature for PHF-case.     

Micropolar fluid flow towards a stretching/shrinking sheet in a porous medium with suction

This paper presents a numerical analysis of a micropolar fluid flow towards a permeable stretching/shrinking sheet in a porous medium. The governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, before being solved numerically by a finite-difference scheme known as the Keller-box method. The effects of the governing parameters on the fluid flow and heat transfer characteristics are illustrated graphically. It is found that dual solutions exist for the shrinking case, whereas for the stretching case, the solution is unique.     

Uncertainties in physical property effects on viscous flow and heat transfer over a nonlinearly stretching sheet with nanofluids

The purpose of this paper is to investigate a numerical analysis for the flow and heat transfer in a viscous fluid over a nonlinear stretching sheet utilizing nanofluid. The governing partial differential equations are converted into highly nonlinear ordinary differential equations by a similarity transformation. Different water-based nanofluids containing Cu, Ag, CuO, Al₂O₃, and TiO₂ are considered in our problem. Furthermore, four different models of nanofluid based on different formulas for thermal conductivity and dynamic viscosity on the flow and heat transfer characteristics are discussed. The variations of dimensionless surface temperature, dimensionless surface temperature gradient as well as the flow and heat transfer characteristics with the governing parameters are graphed and tabulated. Comparison with published results for pure fluid flow is presented and it is found to be in excellent agreement.     

Heat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperature

In this paper we study the flow and heat transfer characteristics of a viscous fluid over a nonlinearly stretching sheet in the presence of non-uniform heat source and variable wall temperature. A similarity transformation is used to transform the governing partial differential equations to a system of nonlinear ordinary differential equations. An efficient numerical shooting technique with a fourth-order Runge–Kutta scheme is used to obtain the solution of the boundary value problem. The effects of various parameters (such as the power law index n, the Prandtl number Pr, the wall temperature parameter λ, the space dependent heat source parameter A¹ and the temperature dependent heat source parameter B¹) on the heat transfer characteristics are analyzed. The numerical results for the heat transfer coefficient (the Nusselt number) are presented for several sets of values of the parameters and are discussed. The results reveal many interesting behaviors that warrant further study on the effects of non-uniform heat source and the variable wall temperature on the heat transfer phenomena at the nonlinear stretching sheet.     

Unsteady Mixed Convection Flow of a Rotating Second‐Grade Fluid on a Rotating Cone

In the present article, we have investigated the unsteady mixed convection flow of a rotating second‐grade fluid in a rotating cone with time‐dependent angular velocities. Two cases of heat transfer are presented which are known as (i) prescribed wall temperature (PWT) and (ii) prescribed heat flux (PHF). The governing coupled nonlinear partial differential equations are simplified with the help of transformations and non‐dimensional similar and non‐similar variables, and solved analytically with the help of the homotopy analysis method (HAM). The effects of pertinent parameters on the velocity, temperature, concentration, skin friction coefficients, Nusselt number, and Sherwood number have been examined through graphs. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(3): 204–220, 2014; Published online 30 August 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21072     

Optimal Homotopy Asymptotic Solutions for Nonlinear Ordinary Differential Equations Arising in Flow and Heat Transfer due to Nonlinear Stretching Sheet

Optimal homotopy asymptotic method (OHAM) is used to obtain solutions for nonlinear ordinary differential equations (ODEs) arising in fluid flow and heat transfer at a nonlinear stretching sheet. The solutions for skin friction and temperature gradient for some special cases are tabulated and compared with the available numerical results in the literature. Moreover, OHAM is found to be very easy to use and the technique could be used for solving coupled nonlinear systems of ordinary differential equations arising in science and engineering.