Steady flow of an Eyring Powell fluid over a moving surface with convective boundary conditions
Introduction
The dynamics of non-Newtonian fluids has been a popular area of research because of its applications. Examples of such fluids include coal-oil slurries, shampoo, paints, clay coating and suspensions, grease, cosmetic products, custard, animals bloods, body fluids and many others. The well known governing equations namely the Navier–Stokes equations cannot adequately describe the characteristics of non-Newtonian fluids. Hence several constitutive equations of non-Newtonian fluids have been presented by keeping in mind the fluid density in nature. The relationship between the shear stress and rate of strain in such fluids are very complicated in comparison to viscous fluids. The viscoelastic features in non-Newtonian fluids add more complexities in the resulting equations when compared with Navier–Stokes equations. Besides all these challenges, several researchers even now are engaged in the flow analysis of non-Newtonian fluids [1], [2], [3], [4], [5].
Convective heat transfer studies has received considerable attention owing to its important in processes involving high temperatures such as gas turbines, nuclear plants, thermal energy storage, etc. Recently, Ishak [7] discussed the similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition. Aziz [8] provided a similarity solution for laminar thermal boundary layer over a flat surface with a convective surface boundary condition. He also examined the hydrodynamic and thermal slip flow boundary layers over a flat surface with constant heat flux boundary condition (see [9]). The buoyancy effects on thermal boundary layer over a vertical plate subject a convective surface boundary condition has been investigated by Makinde and Olanrewaju [10]. Further one can refer the works of Makinde and Aziz [11] and Makinde [12], [13] regarding the heat and mass transfer over a vertical plate with convective boundary conditions.
The flows induced by moving boundary with a parallel free stream have many industrial applications such as heat treatment of material traveling between a feed roll and wind-up roll or an conveyer belts, extrusion of steel, melt spinning process in the extrusion of polymers continuous casting, glass blowing, cooling of a large metallic plate in a bath etc. Sakiadis [14] initiated the problem of flow and heat transfer over a moving surface. A considerable amount of research has been reported on the topic [15], [16], [17], [18]. Hayat et al. [19] examined the flow and heat transfer characteristics due to moving surface with parallel free stream. They used the constitutive equations for a second grade fluid, and computed the series solutions by homotopy analysis method. Zaman et al. [20] reported an analytical analysis of Ref. [19] by incorporating the viscous dissipation terms in the energy equations.
Up to date not much attention has been presented for the two-dimensional flow in Eyring Powell fluid. Although this fluid model has certain advantages over the other non-Newtonian fluid models. Firstly it is deduced from kinetic theory of liquids rather than the empirical relation. Secondly it correctly reduces to Newtonian behavior for low and high shear rates. Some previous attentions regarding to Eyring–Powell fluid flow have been made in the studies [21], [22], [23], [24], [25], [26], [27], [28]. Zueco and Beg [29] numerically studied the pulsatile flow of Eyring–Powell model. Homotopy perturbation analysis of slider bearing lubricated with Eyring Powell model is presented by Islam et al. [30]. Patel and Timol [31] numerically examined the flow of Eyring–Powell model past a wedge. The purpose of this communication is to put forward the analysis of Eyring Powell fluid. Hence we intend to examine the flow and heat transfer in an Eyring Powell fluid over a moving surface. The surface possesses convective boundary conditions. Homotopy analysis method (HAM) has been employed for the solutions of velocity and temperature fields. This method has been successfully applied to various interesting problems [32], [33], [34], [35], [36], [37], [38]. The well known Blasius and Sakiadis flows can be recovered as the limiting cases from the presented series solutions. Graphical results are displayed for the several values of interesting parameters.
Section snippets
Mathematical model
Consider the steady boundary layer flow of an Eyring Powell fluid over a surface moving with constant velocity uw in the same direction as that of the uniform free stream velocity u∞. The physical flow model [19] described in Fig. 1. The surface exhibit convective boundary conditions. It is assumed that the wall and free stream temperatures Tw and T∞ are constants with Tw > T∞.
The extra stress tensor in an Eyring–Powell model is [6]in which μ is the dynamic viscosity,
Zeroth-order deformation problems
We express f(η) and θ(η) through the set of base functionsin the following expressionswhere and are the coefficients. Using rule of solution expressions for f(η) and θ(η) and Eqs. (6), (7), the initial guesses f0 and θ0 are chosen aswhile the auxiliary operators and their associated properties are expressed as
Convergence of the series solutions
The convergence rate of HAM solutions strictly depends upon the values of auxiliary parameter (see [32]). For an appropriate range of auxiliary parameters and we displayed Fig. 2. This figure obviously indicates that ranges for permissible values of and are and , respectively. A convergent series solution in the whole region of η is obtained when .
Results and discussion
For the intention of discussing the results of interesting parameters on the velocity and temperature
Closing remarks
Heat transfer analysis in the flow of Eyring Powell fluid over a moving plate in the presence of a free stream velocity is examined. Analytic solutions are derived by HAM. The main points of the present analysis are described below.
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Table 1 shows that 15th order of approximations are sufficient for a convergent series solution.
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The velocity and boundary layer thickness are decreasing functions of δ for λ > 0.5 whereas velocity increases and the boundary layer thickness decreases with an increase in
Acknowledgments
We are grateful to the Higher Education Commission (HEC) of pakistan for the financial support. First author as a visiting Professor also thanks the support of Global Research Newtwork for computational Mathematics and King Saud University.
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