1 Introduction

The study of flow and heat transfer of a Newtonian fluid over a stretching surface issuing from slit has gained considerable attention of many researchers due to its importance in many industrial applications,such as extraction of polymer sheet,wire drawing,paper production, glass-fiber production,hot rolling,solidification of liquid crystals,petroleum production,continuous cooling and fibers spinning,and exotic lubricants and suspension solutions. Much work on the boundary-layer Newtonian fluids has been carried out both experimentally and theoretically. Crane ^[1] was the first one who studied the stretching problem taking into account the fluid flow over a linearly stretched surface. There has been a great deal of the work done on Newtonian fluid flow and heat transfer over a stretchingsurface,but only a few recent studies are cited here. Gupta and Gupta ^[2] analyzed the stretching problemwith a constant surface temperature,while Soundalgekar and Ramana ^[3] investigated the constant surface velocity case with a power-law temperature variation. Grubka and Bobba ^[4] have analyzed the stretching problem for a surface moving with a linear velocity and with a variable surface temperature. Chen and Char ^[5] investigated the heat transfer characteristics over a continuous stretching sheet with variable surface temperature. Using the homotopy analysis method (HAM),series solutions wereobtainedbyHayatetal. ^[6] for the stretching sheet problem with mixed convection.

Despite the practical importance of the flow in a porous medium,all the above-mentioned works do not however consider the situations where the flow in fluid-saturated porous media arises. The study of the flow in fluid-saturated porous media due to a stretching sheet is important in engineering problems,such as the design of building components for energy consideration,soil science,mechanical engineering,control of pollutant spread in groundwater, thermal insulation systems,compact heat exchangers,solar power collectors,and food industries. Because of such important practical applications,many investigators have modeled the behavior of a boundary layer flow embedded in a porous medium. Then,extensive studies were conducted by many researchers ^[7,8,9,10] . In all the previous investigations,the effects of radiation on the flow and heat transfer have not been provided. Radiative heat transfer flow is very important in manufacturing industries for the design of reliable equipments,nuclear plants, gas turbines,and various propulsion devices for aircraft,missiles,satellites,and space vehicles. Also,the effects of thermal radiation on the forced and free convection flows are important in the context of space technology and processes involving high temperature. Based on these applications,Hossain et al. ^[11,12] and Elbashbeshy and Demain ^[13] studied the thermal radiation of a gray fluid which is emitting and absorbing radiation in non-scattering medium. Abel and Mahesha^[14] studied the effect of radiation in different situations. Recently,Battaler ^[15] studied the effect of thermal radiation on the laminar boundary layer about a flat plate. Historically, the study on boundary layer flows over a stretching sheet with variable thickness was studied by Fang et al. ^[16] . However,so far no attention has been given to the effects of the non-flatness on the stretching sheet problems considering a variable sheet thickness. The purpose of the present paper is to investigate the numerical solution for the variable thermal conductivity effect on the flow and heat transfer of a Newtonian fluid-saturated porous medium over a stretching sheet with variable thickness in the presence of thermal radiation.

Most non-linear differential equations do not have exact solutions,so approximate and numerical techniques^{[17,18,19,20,21,22]} must be used. The differential transformation method (DTM) is a semi-numerical-analytic-technique that formalizes the Taylor series in a totally different manner. It was first introduced by Zhou ina study about electrical circuits ^[23] . Borhanifar and Abazari^[24,25] used it for solving of the linear and non-linear problems. In this paper,we extended DTM for applying to the search for the numerical solutions of the introduced problem. The DTM plays an important rule in recent researches in this field. It has been shown that this procedure is a powerful tool for solving various kinds of problems ^{[26,27,28,29,30]} . This technique reduces the problem to a system of algebraic equations. In this work,we will use the Newton iteration method to solve the resulting system of algebraic equations. 2 Formulation of problem

Consider a steady,two-dimensional boundary layer flow of an incompressible Newtonian fluid over a continuously impermeable stretching sheet embedded in a porous medium. The origin is located at a slit,through which the sheet (see Fig. 1) is drawn through the fluid medium. The x-axis is chosen along the sheet andy-axis is taken normal to it. The stretching surface has the velocity ,where U₀ is the reference velocity. We assume that the sheet is not flat in which it is specified as ,where A is a very small constant so that the sheet is sufficiently thin andmis the velocity power index. We must observe that our problem is valid only for m ≠ 1,because for m= 1,the problem reduces to a flat sheet. Likewise, the fluid properties are assumed to be constant except for thermal conductivity variations in the temperature. Making the usual boundary layer approximations for the Newtonian fluid, the steady two-dimensional boundary-layer equations taking into account the thermal radiation effect in the energy equation can be written as

whereuandvare the velocity components inx-andy-directions,respectively. ρandκare the fluid density and the thermal conductivity,respectively.Tis the temperature of the fluid,νis the fluid kinematic viscosity,cpis the specific heat at constant pressure,μis the fluid viscosity, kis the permeability of the porous medium,and qr is the radiative heat flux. The radiative heat fluxqr is employed according to Rosseland approximation ^[31] such that where σ^* is the Stefan-Boltzmann constant,and k^* is the mean absorption coefficient. Following Raptis^[23] ,we assume that the temperature differences within the flow are sufficiently small such that T⁴ may be expressed as a linear function of the temperature. Expanding T⁴ in a Taylor series about T_∞ and neglecting higher-order terms,we have The physical and mathematical advantage of the Rosseland formula (5) consists of the fact that it can be combined with Fourier’s second law of conduction to an effective conduction-radiation flux q_eff in the form where is the effective thermal conductivity. Thus,the steady energy balance equation including the net contribution of the radiation emitted from the hot wall and absorbed in the colder fluid,takes the form To obtain the similarity solutions,it is assumed that the permeability of the porous medium k(x)isoftheform where k₀ is the permeability parameter. The boundary conditions can be written as The mathematical analysis of the problem is simplified by introducing the following dimensionless coordinates: where ζ is the similarity variable,ψ(x,y) is the stream function which is defined in the classical form as ,and Θ(ζ) is the dimensionless temperature.

In this study,the equation for the dimensionless thermal conductivity κ is generalized for the temperature dependence as follows ^[33,34] :

where κ_∞ is the ambient thermal conductivity,and is the thermal conductivity parameter.

Upon using these variables,the boundary layer governing equations (1)−(3) can be written in the following non-dimensional form:

where is the porous parameter, is the Prandtl number,and R = is the radiation parameter. The transformed boundary conditions are where is a parameter related to the thickness of the wall,and η = indicates the plate surface. In order to facilitate the computation,we defineF(ζ)= F(η−α)=f(η)andΘ(ζ)=Θ(η−α)=θ(η). The similarity equations and the associated boundary conditions become where the prime denotes differentiation with respect toη. Based on the variable transformation, the solution domain will be fixed from 0 to∞. The physical quantities of primary interest are the local skin-friction coefficientCf and the local Nusselt numberNu,which are defined as where is the local Reynolds number,and X=x+b. 3 Basic definitions of DTM

In the DTM,the given differential equation and related initial conditions are transformed into a recurrence equation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. This method is useful for obtaining exact and approximate solutions of linear and nonlinear differential equations. There is no need for linearization or perturbations,and large computational work and round-off errors are avoided. It has been used to solve a large class of linear and nonlinear differential equations with approximations effectively,easily,and accurately.

The basic definitions of DTM are introduced asfollows. With reference to the articles ^[26,28], we introduce in this section the basic definitions of the one-dimensional DTM. Definition 1 If f(η) is an analytic function in the domain Ω=[0,T],then it will be differentiated continuously with respect to time η,

For any point η=η_i in [0,T],we have φ(η,k)=φ(ηi,k),where k belongs to the set of nonnegative integersN0. Therefore,(22) can be written as where is called the spectrum of f(η)atη=ηi in the domain N0.Also,F(k)is called the differential transform of the functionf(η).

Definition 2 Iff(η) can be expressed by Taylor’s series about the fixed point η_i ,then f(η) can be represented as

If f_n(η) isthen-partial sums of Taylor’s series (24),then where f_n(η) is called thenth Taylor polynomial for f(η) about η_i,and R_n+1(η) is the remainder term. Now,using (23),(24) reduces to and the n-partial sum of Taylor’s series (25) reduces to For simplicity,we assume the particular case η_i = 0 (Maclaurin). Then,(26) reduces to It is clear that the concept of differential transformation is based upon the Taylor series expansion. The values of function F(k) at values of argumentkare referred to as discrete,i.e., F(0) is known as the zero discrete,F(1) as the first discrete,etc. The more discrete available, the more precise it is possible to restore the unknown function. The function f(η) consists of the transformed function F(k),and its value is given by the sum of the transformed function with η^k as its coefficient. In real applications,at the larger values of argumentk,the discrete of spectrum reduces rapidly.

From the above definitions,it can be found that the concept of the one-dimensional differential transform is derived from the Taylor series expansion. With (23) and (26),the fundamental mathematical operations performed by one-dimensional differential transform can readily be obtained and listed in Table 1.

4 Solution procedure using DTM

Our aim in this paper is to use the DTM to solve numerically (17)-(18) at the bounded domain (0,η_∞) with the boundary conditions (19)−(20). Using the DTM on (17)−(18),and from Table 1,weget

where F(k) and Θ(k) are the differential transforms of f(η) and θ(η),respectively. We choose suitable initial conditions for arbitrary constants c₀ and c₁. From (29)−(30),for k=0,1,···,and using (31),we get In the same manner,the rest of components can be obtained using the MATHEMATICA package. Substituted the quantities listed on (32) in (28),whenη0 = 0,the approximate solution in a series form of the proposed problems (17)−(18) is given by Now,we find the constants c₀ and c₁ using the boundary conditions (19)−(20),where we take the values α=0.2,m=0.5,D=0.5,R=0.5,P_r =1,and =0.2. These values are c₀=0.143 818 606 and c₁=−0.236 093 966. Having F(k),Θ(k),k=0,1,···,n,the solution are the same as (33). 5 Results and discussion

Tables 2 and 3 clearly reveal that present solution namely DTM shows excellent agreement with the existing solutions in the literature ^[16]. This analysis shows that DTM suits for the problems of boundary layer flow in fluid-saturated porous medium. This section provides the behavior of parameters involved in the expressions of heat transfer characteristics for the stretching sheet. Numerical evaluation for the solutions of this problem is performed and the results are illustrated graphically in Figs. 2−10. The study of flow in porous media is very important in approximating the shape of spherical particles or cylindrical fibers which better fit the model of permeability assumed for the analysis. Effects of the porous parameterDon velocity and temperature profiles are shown in Figs. 2 and 3,respectively. It is observed that the velocity decreases for increasing values ofporous parameter. Furthermore,the momentum boundary layer thickness decreases as porous parameterDincreases. Figure 3 elucidates that the fluid temperature enhances with anincrease in the porous parameter.

The effects of wall thickness parameter on the fluid flow and the temperature distribution have been analyzed and the results are presented in Figs. 4−5. From Fig. 4,it is clear that the velocity at any point near to the plate decreases as the wall thickness parameter increases for m<1 and the reverse is true for m>1. Also,it is obvious from these figures that the thickness of the boundary layer becomes thinner for a higher value ofαwhen m<1 and becomes thicker for a higher value of αwhen m>1.

Figure 5 displays that the wall thickness parameter decreases the thickness of the thermal boundary layer and enhances the rate of heat transfer for m<1 whereas reverse trend is observed as m>1. Physically,increasing the value ofαwhen m<1 will decrease the flow velocity,because under the variable wall thickness,not all the pulling force of the stretching sheet can be transmitted to the fluid causing a decrease for both friction between the fluid layers and temperature distribution for the fluid. However,when m>1 the velocity of the flow layers will increase causing an enhance for the friction force between this layers and thus increasing its temperature. Likewise,for a higher value of α,the thermal boundary layer becomes thinner when m<1 compared with the case of m>1.

Figure 6 shows that the velocity rises with a decrease in the values of the velocity power indexm. This implies the momentum boundary thickness becomes thinner asmincreases along the sheet,and the reverse is true away from it.Figure 7 displays the influence of the velocity power index parametermon the temperature profiles. It is clearly seen from this figure that increasing the value ofmproduces an increase in the temperature profiles. It further shows that the larger the value of m,the higher the magnitude of the thermal boundary thickness will be.In Fig. 8,we have varied the thermal conductivity parameter keeping the values of all other parameters fixed. Figure 8 reveals that the temperature profile as well as the thickness of the thermal boundary layer increases when increases.Figure 9 illustrates the effects of radiation parameterRon the temperature profiles when other parameters hold constant. It is depicted that the temperature field and the thermal boundary layer thickness increase with the increase in R.

It is observed from Fig. 10 that an increase inthe Prandtl number results in decreasing the heat transfer profiles. The reason is that increasing values of Prandtl number is equivalent to decreasing the thermal conductivities,and therefore heat is able to diffuse away from the heated sheet more rapidly. Hence,in the case of increasing Prandtl number,the boundary layer is thinner and the heat transfer is reduced.

Table 4 shows the influence of the porous parameterD,wall thickness parameter α,the velocity power index parameterm,the radiation parameter R,the Prandtl number Pr,and thermal conductivity parameters on the local skin friction coefficient and the local Nusselt number. It is noticed that increasing the wall thickness parameter leads to an increase in both the local skin-friction coefficient and the local Nusselt number. Likewise,the local Nusselt number is reduced but the skin-friction coefficient is increased with increasing for both values of porous parameter and velocity power index parameter. Also,an increase in the Prandtl number causes an increase in the local Nusselt number. This is because a fluid with larger Prandtl number possesses larger heat capacity,and henceintensifies the heat transfer. Moreover,it is observed that the values of the local Nusselt number decrease with increase in both the thermal conductivity parameter and the radiation parameter.

6 Conclusions

Here,we use the DTM to solve the resulting non-linear system of ordinary differential equations (ODEs) of the problem of flow and heat transfer in a quiescent Newtonian fluid flow caused solely by a stretching sheet,which embedded in a porous medium with variable thickness,variable thermal conductivity,and thermal radiation. The fluid thermal conductivity is assumed to vary as a linear function of temperature. Comparison with previously published work is performed and the results are found to be in excellent agreement. Asystematic study on the effects of the various parameters on flow and heat transfer characteristics is carried out. It is found that increasing values of the porous parameter,the velocity power index parameter, thermal conductivity parameter,and the radiation parameter reduces the local Nusselt number.On the other hand,it is observed that the local Nusselt number increases as the Prandtl number and the wall thickness parameter increase. Moreover,it is interesting to find that increasing the porous parameter,wall thickness parameter,and the velocity power index parameter in magnitude,causes the fluid to slow down past the stretching sheet and the increasing of skinfriction coefficient in magnitude. Summarizing these results,we can say that the DTM in its general form gives a reasonable calculation,easy to use,and can be applied for the differential equations in general form.