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

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Abstract

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 A and the temperature dependent heat source parameter B) 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.

Introduction

The viscous flow over a stretching sheet has important industrial applications. For example, in metallurgical processes, such as drawing of continuous filaments through quiescent fluids, annealing and tinning of copper wires, glass blowing, manufacturing of plastic and rubber sheets, crystal growing, continuous cooling and fibers spinning, the sheets are stretched continuously. During the manufacture of these sheets, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. The final product with desired characteristics strictly depends upon the stretching rate, the rate of cooling in the process, and the process of stretching. In view of these applications, Sakiadis [1], [2] studied the boundary layer flow over a stretched surface. He employed a similarity transformation and obtained a numerical solution for the problem.

Later, Erickson et al. [3] extended the work of Sakiadis [2] to account for mass transfer at the stretched surface. The two dimensional boundary layer flow caused by a linear stretching sheet in an otherwise quiescent fluid was first discussed by Crane [4]. He obtained a closed form exponential solution. Singh [5] studied the effect of non-uniform heat source on hydromagnetic convective flow of a viscoelastic fluid. Grubka and Bobba [6] studied the heat transfer characteristics of a continuous stretching surface with variable temperature. Abel and Nandeppanavar [7], [8], [9] studied the effect of non-uniform heat source on viscoelastic boundary layer flows. Further, Abel et al. [10] investigated the effects of viscous dissipation and non-uniform heat source. Abel and Mahesha [11] studied the effects of non-uniform heat source with variable thermal conductivity. Ali [12] investigated the effects of power law index on heat transfer characteristics of a power law fluid flow. Tsai et al. [13] investigated the effects of non-uniform heat source on unsteady stretching sheet.

However, all these studies are restricted to linear stretching of the sheet. It is worth mentioning that the stretching is not necessarily linear. In view of this, Kumaran and Ramanaih [14] studied flow over a quadratic stretching sheet. Magyari and Keller [15], Elbashbeshy [16], Khan and Sanjayanand [17], Sanjayanand and Khan [18], Sajid and Hayat [19], Partha et al. [20] studied the heat transfer characteristics of viscous and viscoelastic fluid flows over an exponentially stretching sheet. Vajravelu [21], Vajravelu and Cannon [22], and Cortell [23], [24], [25] studied the effects of various parameters governing the flow of a viscous fluid over a nonlinearly stretching sheet. In all these studies with nonlinear stretching sheet, the authors ignored the effects of the heat source, which is very important in exothermic and endothermic processes.

The analysis of the temperature field as modified by the generation or absorption of heat in moving fluids is important in view of several physical problems, such as in a chemical reaction taking place and in problems concerned with dissociating fluids. The volumetric rate of heat generation has been assumed to be constant or a function of space variables whilst some other studies have considered directly the frictional heating and the expansion effect. Foraboschi and Federico [26] assumed volumetric rate of heat generation of the type Q = Q0(T  T0) when T  T0, and Q = 0 when T < T0 in their study of the steady state temperature profiles for linear, parabolic and piston-flow in circular pipes, The relations above, as explained by Foraboschi and Federico, are valid as an approximation of the state of some exothermic process increasing in temperature and having T0 as the onset temperature. When the inlet temperatures are not less than T0, they used Q = Q0(T  T0) and studied its effect on the heat transfer in laminar flow of non-Newtonian heat-generating fluids. Moalem [27] studied the effect of temperature-dependent heat sources of the form Q0  (a + bT)−1, such as the one occurring in electrical heating, on the steady-state heat transfer within a porous medium.

Hence in this paper we investigate the effects of non-uniform heat source as in Eq. (4) (which can bring out the effects of exothermic process and the effects of electrical heating) and the variable wall temperature on the heat transfer characteristics of a viscous fluid over a nonlinearly stretching sheet.

Section snippets

Mathematical formulation of the problem

Consider the two dimensional flow of an incompressible viscous fluid over a stretching surface. The x-axis is taken along the stretching surface in the direction of the motion and the y-axis is perpendicular to it; see for details [21] and Fig. 1a, Fig. 1b. It may be noted that the Navier–Stokes equations are elliptic, but when we use the boundary layer approximation they become parabolic. Hence, under the usual boundary layer approximations, the flow and heat transfer problems with non-uniform

Numerical solution

Analytical solution for the flow problem with n  1 does not exist so consequently, one has to use a numerical technique. The nonlinear differential Eqs. (7), (9) with boundary conditions (8), (10) are solved numerically by the shooting technique with a fourth-order Runge–Kutta method [28], [29]. The nonlinear differential equations are first decomposed into to a system of first order differential equationsdf0dη=f1,df1dη=f2,df2dη=(2nn+1)(f1)2-f0f2dθ0dη=θ1,dθ1dη=(2n+1){λPrf1-B}θ0-Prfθ1-Af1,with

Analytical solution (a special case)

In this special case, we investigate the solution of Eqs. (7), (9) with the boundary conditions (8), (10), when n = 1 and λ=2 as follows:

Results and discussion

Heat transfer characteristics of the viscous boundary layer flow over a nonlinearly stretching sheet with non-uniform heat source are investigated. The shooting technique with a fourth-order Runge–Kutta scheme is employed to obtain the solution for the one-way coupled nonlinear boundary value problem. Also, as a special case, we obtained an analytical solution (when n = 1 and λ=2) to the case of Newtonian fluid. The results for the Newtonian case are used to validate the numerical results for the

Acknowledgments

The authors are thankful to the reviewers for their insightful reviews, invaluable comments and suggestions, which have helped improvement of this article. Dr. Mahantesh M. Nandeppanavar would like to thank University Grants Commission, New-Delhi, India for supporting this work under Major Research Project [Grant No. 39-59/2010(SR)]. Dr. Chiu-On Ng would like to thank the support by the Research Grants Council of the Hong Kong Special Administrative Region, China, through Project No. HKU

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