Mixed convection in the stagnation-point flow of a Maxwell fluid towards a vertical stretching surface

In the present analysis, we study the steady mixed convection boundary layer flow of an incompressible Maxwell fluid near the two-dimensional stagnation-point flow over a vertical stretching surface. It is assumed that the stretching velocity and the surface temperature vary linearly with the distance from the stagnation-point. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. Analytical and numerical solutions of the derived system of equations are developed. The homotopy analysis method (HAM) and finite difference scheme are employed in constructing the analytical and numerical solutions, respectively. Comparison between the analytical and numerical solutions is given and found to be in excellent agreement. Both cases of assisting and opposing flows are considered. The influence of the various interesting parameters on the flow and heat transfer is analyzed and discussed through graphs in detail. The values of the local Nusselt number for different physical parameters are also tabulated. Comparison of the present results with known numerical results of viscous fluid is shown and a good agreement is observed.     

Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux

In this paper, a powerful analytical method, called homotopy analysis method (HAM) is used to obtain the analytical solution for a nonlinear ordinary deferential equation that often appear in boundary layers problems arising in heat and mass transfer which these kinds of the equations contain infinity boundary condition. The boundary layer approximations of fluid flow and heat transfer of vertical full cone embedded in porous media give us the similarity solution for full cone subjected to surface heat flux boundary conditions. Nonlinear ODE which is obtained by similarity solution has been solved through homotopy analysis method (HAM). The main objective is to propose alternative methods of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The obtained analytical solution in comparison with the numerical ones represents a remarkable accuracy. The results also indicate that HAM can provide us with a convenient way to control and adjust the convergence region.     

Cattaneo–Christov heat flux model for three-dimensional flow of a viscoelastic fluid on an exponentially stretching surface

ABSTRACT

In this article, we explore the three-dimensional boundary-layer flow over an exponentially stretching surface in two parallel ways. Constitutive equations of a second-grade fluid are used. Instead of classical Fourier’s law, Cattaneo–Christov heat flux model is employed for the formulation of the energy equation. This model can predict the effects of thermal relaxation time on the boundary layer. The resulting partial differential equations are reduced into ordinary differential equations by similarity transformations. Homotopy Analysis Method (HAM) is employed to solve the non-linear problem. Physical impact of emerging parameters on the momentum and thermal boundary-layer thickness are studied.     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets

Flow induced in a viscoelastic fluid by a linearly stretched sheet is investigated assuming that the fluid is Maxwellian and the sheet is subjected to a transverse magnetic field. The objective is to investigate the effects of parameters such as elasticity number, magnetic number, radiative heat transfer, Prandtl number, and Eckert number on the temperature field above the sheet. To do this, boundary layer theory will be used to simplify energy and momentum equations assuming that fluid physical/rheological properties remain constant. A suitable similarity transformation will be used to transform boundary layer equations from PDEs into ODEs. Homotopy analysis method (HAM) will be invoked to find an analytical solution for the temperature field above the sheet knowing the velocity profiles (see Alizadeh-Pahlavan et al. [Alizadeh-Pahlavan A, Aliakbar V, Vakili-Farahani F, Sadeghy K. MHD flows of UCM fluids above porous stretching sheets using two-auxiliary parameter homotopy analysis method. Commun. Nonlinear Sci Numer Simulat, in press]). The importance of manipulating the transverse velocity component, v, will be discussed on the temperature field above the sheet.     

Investigation of a powerful analytical method into natural convection boundary layer flow

The natural convection boundary layer flow modeled by a system of nonlinear differential equations is considered. By means of similarity transformation, the non-linear partial differential equations are reduced to a system of two coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. Finally some figures are illustrated to show the accuracy of the applied method and assessment of various prandtl numbers on the temperature and the velocity is undertaken.     

Series solutions for unsteady laminar MHD flow near forward stagnation point of an impulsively rotating and translating sphere in presence of buoyancy forces

The similarity solution for the unsteady laminar incompressible boundary layer flow of a viscous electrically conducting fluid in stagnation point region of an impulsively rotating and translating sphere with a magnetic field and a buoyancy force gives a system of non-linear partial differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the homotopy analysis method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the magnetic parameter, buoyancy parameter and rotation parameter on the surface shear stresses and surface heat transfer. It is noted that the behavior of the HAM solution for the surface shear stresses and surface heat transfer is in good agreement with the numerical solution given in reference [H. S. Takhar, A. J. Chamkha, G. Nath, Unsteady laminar MHD flow and heat transfer in the stagnation region of an impulsively spinning and translating sphere in the presence of buoyancy forces, Heat Mass Transfer 37 (2001) 397].     

Mixed convection flow of couple stress fluid between parallel vertical plates with Hall and Ion-slip effects

An analysis is presented to investigate the Hall and Ion-slip effects on fully developed electrically conducting couple stress fluid flow between vertical parallel plates in the presence of a temperature dependent heat source. The governing non-linear partial differential equations are transformed into a system of ordinary differential equations using similarity transformations and then solved using homotopy analysis method (HAM). The effects of the magnetic parameter, Hall parameter, Ion-slip parameter and couple stress fluid parameter on velocity and temperature are discussed and shown graphically.     

The influence of heat transfer on peristaltic transport of a Jeffrey fluid in a vertical porous stratum

The peristaltic flow of a Jeffrey fluid in a vertical porous stratum with heat transfer is studied under long wavelength and low Reynolds number assumptions. The nonlinear governing equations are solved using perturbation technique. The expressions for velocity, temperature and the pressure rise per one wave length are determined. The effects of different parameters on the velocity, the temperature and the pumping characteristics are discussed. It is observed that the effects of the Jeffrey number λ₁, the Grashof number Gr, the perturbation parameter N = EcPr, and the peristaltic wall deformation parameter ϕ are the strongest on the trapping bolus phenomenon. The results obtained for the flow and heat transfer characteristics reveal many interesting behaviors that warrant further study on the non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear-thinning reduces the wall shear stress.     

朝向产热或吸热伸展平面的MHD驻点流动

分析了微极流体朝向加热伸展平面的磁流体动力学(MHD)驻点流动,考虑了粘性耗散和内部产热/吸热对流动的影响.讨论了指定表面温度(PST)和指定热通量(PHF)两种情况,采用同伦分析方法(HAM)求解边界层流动和能量方程.通过图表的显示,研究了感兴趣物理量的变化.注意到高伸展参数时解的存在与外加应用磁场密切相关.     

Influence of radiation on MHD free convection from a vertical flat plate embedded in porous media with thermophoretic deposition of particles

Magneto-hydrodynamics and thermal radiation effects on heat and mass transfer in steady laminar boundary layer flow of a Newtonian, viscous fluid over a vertical flat plate embedded in a fluid saturated porous media in the presence of the thermophoresis particle deposition effect is studied in this paper. The governing equations are transformed by special transformations. Brownian motion of particles and thermophoretic transport are considered in the flow equations. The magnetic field is considered to be applied. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting similarity equations are solved numerically by the fourth-order Runge–Kutta method with shooting technique. Many results are obtained and representative set is displayed graphically to illustrate the influence of the various parameters on the wall thermophoretic deposition velocity, concentration, temperature and velocity profiles.     

Unsteady flow and heat transfer of a second grade fluid over a stretching sheet

The problem of unsteady boundary layer flow of a second grade over a stretching sheet is investigated in this paper. The governing equations of motion are reduced into a partial differential equation with two independent variables by using similarity transformations. The heat transfer analysis has been also carried out for two heating processes namely the prescribed surface temperature (PST case) and prescribed surface heat flux (PHF case). The series solutions of the problem are developed by employing homotopy analysis method (HAM). Convergence of the obtained series solutions are analyzed. It is noted that the present solutions of a second grade are valid for all dimensionless times. Finally, the results are obtained and discussed through graphs for various parameters of interest.     

Stagnation point flow of dusty Casson fluid with thermal radiation and buoyancy effects

The aim of the study is to examine the stagnation point flow of a dusty Casson fluid over a stretching sheet with thermal radiation and buoyancy effects. The governing boundary layer equations are represented by a system of partial differential equation. After applying suitable similarity transformations, the resulting boundary layer equations are solved numerically using the Runge Kutta Fehlberg fourth-fifth order method (RKF-45 method). The behaviors of velocity, temperature and concentration profiles of fluid and dusty particles with respect to change in fluid particle interaction parameter, Casson paramter, Grashof number, radiation parameter, Prandtl number, number density, thermal equilibrium time, relaxation time, specific heat of fluid and dusty particles, ratio of diffusion coefficients, Schmidt number and Eckert number are analysed graphically and discussed. Our computed results interpret that velocity distribution decays for higher estimation of Casson parameter while temperature distribution shows increasing behavior for larger radiation parameter.     

Finite element analysis of hydromagnetic flow and heat transfer of a heat generation fluid over a surface embedded in a non-Darcian porous medium in the presence of chemical reaction

An analysis is carried out to study the flow, chemical reaction and mass transfer of a steady laminar boundary layer of an electrically conducting and heat generating fluid driven by a continuously moving porous surface embedded in a non-Darcian porous medium in the presence of a transfer magnetic field. The governing partial differential equations are converted into ordinary differential equations by similarity transformation and are solved numerically by using the finite element method. The results obtained are presented graphically for velocity, temperature and concentration profiles, as well as the Sherwood number for various parameters entering into the problem.     

Channel flow of a Maxwell fluid with chemical reaction

This work is concerned with the two-dimensional boundary layer flow of an upper-convected Maxwell (UCM) fluid in a channel with chemical reaction. The walls of the channel are porous. Employing similarity transformations the governing non-linear partial differential equations are reduced into non-linear ordinary differential equations. The resulting ordinary differential equations are solved analytically using homotopy analysis method (HAM). Expressions for series solutions are derived. The convergence of the obtained series solutions are shown explicitly. The effects of Reynold’s number Re, Deborah number De, Schmidt number Sc and chemical reaction parameter γ on the velocity and the concentration fields are shown through graphs and discussed.     

Partial slip effects on the flow and heat transfer characteristics in a third grade fluid

This article looks at the slip effects on the flow and heat transfer of a third grade fluid past a porous plate. The resulting equations and boundary conditions are non-linear. The non-linear boundary condition is reduced into a linear one and a series solution of the problem is obtained using the homotopy analysis method (HAM). Variations of interesting parameters are seen on the velocity and temperature profiles.     

Effects of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over an exponentially stretching surface with suction in the presence of internal heat generation/absorption

In this paper, the problem of unsteady laminar two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid in the presence of thermal radiation, internal heat generation or absorption, and magnetic field over an exponentially stretching surface subjected to suction with an exponential temperature distribution is discussed numerically. The governing boundary layer equations are reduced to a system of ordinary differential equations. New numerical method using Mathematica has been used to solve such system after obtaining the missed initial conditions. Comparison of obtained numerical results is made with previously published results in some special cases, and found to be in a good agreement.     

Stagnation-point flow and heat transfer over an exponentially shrinking sheet

An analysis is carried out to investigate the stagnation-point flow and heat transfer over an exponentially shrinking sheet. Using the boundary layer approximation and a similarity transformation in exponential form, the governing mathematical equations are transformed into coupled, nonlinear ordinary differential equations which are then solved numerically by a shooting method with fourth order Runge-Kutta integration scheme. The analysis reveals that a solution exists only when the velocity ratio parameter satisfies the inequality −1.487068 ? c/a. Also, the numerical calculations exhibit the existence of dual solutions for the velocity and the temperature fields; and it is observed that their boundary layers are thinner for the first solution (in comparison with the second). Moreover, the heat transfer from the sheet increases with an increase in c/a for the first solution, while the heat transfer decreases with increasing c/a for the second solution, and ultimately heat absorption occurs.     

Heat transfer in a visco-elastic fluid past a stretching sheet with viscous dissipation and internal heat generation

This paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of a visco-elastic fluid over a linearly stretching continuous surface with variable wall temperature subjected to suction or blowing. The study considers the effects of frictional heating (viscous dissipation) and internal heat generation or absorption. An analysis has been carried out for two different cases of heating processes namely: (i) Prescribed surface temperature (PST) and (ii) Prescribed wall heat flux (PHF) to get the effect of visco-elastic parameter for various situations. Further increase of visco-elastic parameter is to decrease the skin friction on the sheet. The solutions for the temperature and the heat transfer characteristics are obtained in terms of Kummers function. Received: June 16, 2004; revised: February 8, 2005     

Dual solutions in mixed convection boundary layer flow of micropolar fluids

The steady mixed convection boundary layer flow over a vertical surface immersed in an incompressible micropolar fluid is considered in this paper. Employing suitable similarity transformations, the governing partial differential equations are transformed into ordinary differential equations, and the transformed equations are solved numerically by the Keller-box method. Numerical results are obtained for the skin friction coefficient and the local Nusselt number as well as the velocity, angular velocity and temperature profiles. Both cases of assisting and opposing buoyant flows are considered. It is found that dual solutions exist for the assisting flow, besides that usually reported in the literature for the opposing flow. Moreover, in contrast to the classical boundary layer theory, the separation point of the boundary layer is found to be distinct from the point of vanishing skin friction.