Department of Mathematics, Central College Campus, Bangalore University, Bangalore 560 001, India Received 12 July 2003; accepted 16 April 2004
Abstract
We study the **MHD** ﬂow and also **heat** **transfer** in a **viscoelastic** **liquid** **over** a **stretching** **sheet** in the presence of **radiation**. The **stretching** of the **sheet** is assumed to be proportional to the distance from the slit. Two different temperature conditions are studied, namely (i) the **sheet** with prescribed surface temperature (PST) and (ii) the **sheet** with prescribed wall **heat** ﬂux (PHF). 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. The resulting non-linear momentum differential equation is solved exactly. The energy equation in the presence of viscous dissipation (or frictional heating), internal **heat** generation or absorption, and **radiation** is a differential equation with variable coefﬁcients, which is transformed to a conﬂuent hypergeometric differential equation using a new variable and using the Rosseland approximation for the **radiation**. The governing differential equations are solved analytically and the **effects** of various parameters on velocity proﬁles, skin friction coefﬁcient, temperature proﬁle and wall **heat** **transfer** are presented graphically. The results have possible technological applications in **liquid**-based systems involving stretchable materials.

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The study of boundary layer **flow** passing a **stretching** **sheet** become an important and interesting challenge for research studies due to its practical utility in industry and engineering. It has potential applications in many areas such as cooling of metallic **sheet**, **stretching** of the plastic film, industrialized polymer **sheet**, metal spinning, crystal growing, electronic chips, filaments and wires, glass blowing, artificial fibers, paper production, metallurgical processes, rubber sheets, and polymer extrusion. The quality yet ultimate production formations among these techniques are dependent **over** the concerning cooling and **stretching**. The studies on boundary layer **flow** of nanofluid **over** a **stretching** **sheet** have attracted the attention of a large of a number of researchers. The boundary layer **flow** and **heat** **transfer** in a viscous fluid contacting metallic nanoparticles **over** a **stretching** **sheet** in the presence of thermal **radiation** have investigated by Hamad and Ferdows (2012). **Effects** of chemical reaction on the **MHD** **flow** of a visco-elastic fluid through porous medium have been presented by Nayak et al. (2014).

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The analysis of **radiation** effect has important applications in physics and engineering especially in space technology and high temperature pro- cesses. **Radiation** has a big impact on the boundary layer **flow**. However, very little is known about the **effects** of **radiation** on the boundary layer **flow**. Thermal **radiation** effect on the boundary layer may play important role in controlling **heat** **transfer** in polymer processing industry where the quality of the final product depends on the **heat** controlling factors to some degree. Radiative effect often observed in many engineering areas such as in electrical power generation, astrophysical flows, solar power technology, space vehicle re-entry, nuclear engineering applications and other industrial areas. Accordingly, researchers (Siddheshwar and Ma- habaleswar (2005), Mahmoud (2009), Shateyi and Motsa (2010), Abel and Mahesha (2008), Prasad et al. (2010)) have examined the **effects** of thermal **radiation** on **heat** **transfer** **over** **stretching** **sheet** by considering dif- ferent properties of a fluid. Moreover, when radiative **heat** **transfer** takes place, the fluid involved can be electrically conducting since it is ionized due to the high operating temperature. Therefore, it is important to ex- amine the effect of the magnetic field on the **flow**. Many researches have examined the **effects** of thermal **radiation** on fluid **flow** and **heat** trans- fer in the presence of magnetic field. Accordingly, Pal (2011) studied the **effects** of thermal **radiation** and non-uniform **heat** **source**/sink on **heat**

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Another effect which bears great importance on **heat** **transfer** is the viscous dissipation. The determination of the temperature distribution when the internal friction is not negligible is of utmost significance in different industrial fields, such as chemical and food processing, oil exploitation and bio-engineering. In view of this, **viscoelastic** **flow** and **heat** **transfer** **over** a flat plate with constant suction, thermal **radiation** and without viscous dissipation were studied by Salem [10] used a shooting technique to study numerically the **effects** of variable viscosity and thermal conductivity on the **MHD** **flow** and **heat** **transfer** of a **viscoelastic** fluid **over** a **stretching** **sheet** with variable surface temperature. The **flow** is induced due to an infinite elastic **sheet** which is stretched back and forth in its own plane. Temperature field and wall temperature gradient are obtained. The combined **effects** of Joule heating and viscous dissipation on the momentum and thermal transport have been examined by Chen [11] **Effects** of free convection, thermal **radiation**, and surface suction/blowing on the **flow** and **heat** **transfer** characteristics are also examined. Uddin et al. [12] investigated the **effects** of mass **transfer** on **MHD** mixed convective **flow** along inclined porous plate. R. Ravidran et al. [13] studied the effect of non-uniform single and double slot suction/injection into an unsteady mixed convection **flow** of an electrically conducting and **heat** generating/ absorbing fluid **over** a vertical cone in the presence of magnetic field and a first order chemical reaction. Yahaya et al. [14] presented a unified approach to solving the **MHD** **flow** due to influence of buoyancy and thermal **radiation** **over** a **stretching** porous **sheet** using homotopy analysis method. N. Sandeep et al. [15] investigated the influence of non-uniform **heat** **source**/sink, mass **transfer** and chemical reaction on an unsteady mixed convection boundary layer **flow** of a **MHD** micropolar fluid past a **stretching** **sheet** in presence of viscous dissipation and suction/injection, most recently.

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oscillatory **stretching** surface in the presence of **heat** generation or absorption. Hence we feel appropriate to consider the study of such flows, as such type of flows may find a number of industrial applications. The theory of couple stress fluid was developed by Stokes [34] as a simple generalization of classical viscous theory that sustains couple stresses and body couples. The main **effects** of couple stress fluid are to introduce a size dependent effect which is not present in classical viscous theo- ries. Blood, lubricants containing small amount of additives, electro-rheological and synthetic fluids are examples of couple stress fluid [35] . Couple stress theory has been successfully applied to many problems in biomechanics and lubrications area by Srivastava [36] , El-Shehawey and Mekheimer [37] , Pal et al. [38] , Chiang et al. [39] , Naduvinamani et al. [40] , Jian and Chen et al. [41] and Lu and Lin [42] .

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Abbas and Hayat [8] studied the **radiation** **effects** on the magnetohydrodynamic (**MHD**) **flow** of an incom- pressible viscous fluid in a porous space. In their study, they extended the analysis of Cortell [7] by considering a **MHD** **flow**, analyzed the **flow** in a porous medium, included the **radiation** **effects** and provided analytic solu- tion namely homotopy analysis method (HAM) instead of numerical technique applied in [7]. Hayat et al. [9] investigated the magnetohydrodynamic (**MHD**) boundary layer **flow** by employing the modified Adomian de- composition method and the Padé approximation and developed the series solution of the governing non-linear problem.

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two types of **viscoelastic** fluids **over** a **stretching** surface have been investigated by Turkyilmazoglu [17] . Moreover, Elbash- beshy and Ibrahim [18] investigated the effect of steady free convection **flow** with variable viscosity and thermal diffusivity along a vertical plate. Kafoussias and Williams [19] studied the thermal-diffusion and diffusion-thermo **effects** on the mixed free-forced convective and mass **transfer** steady laminar boundary layer **flow** **over** a vertical plate, with temperature- dependent viscosity. Sajid and Hayat [20] investigated the radi- ation **effects** on the mixed convection **flow** **over** an exponen- tially **stretching** **sheet** and solved the problem analytically using homotopy analysis method. The numerical solution for the same problem was then given by Bidin and Nazar [21] . Recently, Poornima and Bhaskar Reddy [22] presented an analysis of the **radiation** **effects** on **MHD** free convective boundary layer **flow** of nanofluids **over** a nonlinear **stretching** **sheet**. However, the interaction of **radiation** with mass **transfer** due to a **stretching** **sheet** has received little attention. Abol- bashari et al. [23] studied entropy analysis for an unsteady **MHD** **flow** past a **stretching** permeable surface in nanofluid. Rashidi and Erfani [24] applied an analytical method for solv- ing steady **MHD** convective and slip **flow** due to a rotating disk with viscous dissipation and Ohmic heating. Mixed con- vective **heat** **transfer** for **MHD** **viscoelastic** fluid **flow** **over** a porous wedge with thermal **radiation** is studied by Rashidi et al. [25] . Further, Rashidi et al. [26] studied an analytic approximate solution for **MHD** boundary layer **viscoelastic** fluid **flow** **over** continuously moving **stretching** surface by HAM with two auxiliary parameters.

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The **radiation** **effects** are neglected in the above studies In the processes involving high temperatures such as glass production, polymer processes and furnace design and space technology applications such as gas cooled nuclear reactors, gas turbines, propulsion system, rocket combustion chamber and plasma physics, the **radiation** **effects** play an important role in such cases and can not be neglected. As a result, many studies have been carried out on the influences of thermal **radiation** on the **heat** **transfer** characteristics in different situations [22-26]. The aim of the present analysis is to study the **heat** **transfer** characteristic from a linearly **stretching** surface with power- law surface temperature in quiescent fluid in the presence of internal **heat** **source** , a uniform transverse magnetic field and slip conditions. Exact solution to the energy equation in terms of Kummer’s functions is then obtained.

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studied buoyancy **effects** on **MHD** **flow** of nanofluid (with Copper and Copper Oxide as nanoparticles) **over** a **stretching** **sheet** in the presence of thermal **radiation** by adopting fourth order RK based shooting technique. They concluded that Cu-water has more temperature than Cuo-water nanofluid while the buoyancy decrease the temperature of the nanofluid. Sheikholeslami et al. (2016a) studied the effect of MFD (Magnetic Field Dependent) viscosity on free convective nanofluid **flow** in an enclosure with bottom wall has constant **heat** flux element by using control volume based Finite Element Method (CVFEM) with linear triangular elements. Makinde et al. (2013) concluded that dual solution exists for shrinking case while studying stagnation point **flow** and **heat** **transfer** of a nanofluid past a convectively heated **stretching**/shrinking **sheet** with buoyancy **effects**. Anwar et al. (2012) studied the **effects** of buoyancy, solutal buoyancy and power law velocity parameters by adopting Keller-Box Method (KBM). Chamkha (2000) considered similarity equations governing the steady hydromagnetic boundary layer **flow** **over** an accelerating permeable surface ith buoyancy **effects** and then these equations are solved numerically with IFDM. Patrick and Paul (2010) for narrow vertical flat plate with uniform surface **heat** flux and plate edge conditions. Sheikholeslami et al. (2015) adopted CVFEM developed in FOTRAN with triangular elements to study the forced convection **heat** **transfer** in a lid driven semi annulus enclosure filled with Ferro nanofluid in the presence of non-uniform magnetic field and this extended to FHD (Ferrohydrodynamics) by the Sheikholeslami et al. (2016b), in this work they used CVFEM to study convection **heat** **transfer** in semi annulus under the influence of a variable magnetic field considering both FHD and **MHD**. They concluded that Kelvin force is more pronounced for high Reynolds numbers. Magyari and Chamkha (2013) reported an exact solution for the **effects** of buoyancy force and chemical reaction on micropolar fluid **flow** **over** a permeable stretched surface. Very recently Barletta et al. (2017) investigated unstable buoyant **flow** in a vertical porous layer considering convective boundary conditions and few related studies can be seen in Chemseddine et al. (2017) for molten PB-SN alloys.

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Unsteady problems due to a **stretching** surface received less attention. The unsteady aspects become interesting in certain practical problems, where the motion of the **stretching** surface may start impulsively from rest. Elbashbeshy and Bazid [12] presented similarity solutions for unsteady **flow** and **heat** **transfer** **over** a **stretching** surface. They examined **effects** of unsteady parameter (A) and Prandtl number (Pr) on the **flow** and **heat** **transfer** characteristics. They observed that the unsteady parameter and Prandtl number increase **heat** **transfer** rate at the surface. These results were supported by Azid [13] and Ishak [14] and they obtained the exact solution of unsteady mixed convection boundary layer **flow** and **heat** **transfer**. The results show that the buoyancy parameter increases the **heat** **transfer** rate at the surface. Bachok et al. [15] investigated the effect of material parameter of the unsteady laminar **flow** of an incompressible micro polar fluid. They found that the skin friction coefficient decreases as the material parameter increases and the

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On the other hand, another physical phenomenon is the case in which the **sheet** stretched in a nonlinear fashion. On this domain, Mahdy and Elshehabey [15] studied the **flow** and **heat** **transfer** in a viscous fluid **over** a nonlinear **stretching** **sheet** utilizing nanofluid where, **effects** of vis- cous dissipation and **radiation** on the thermal boundary layer **over** a nonlinearly **stretching** **sheet** were studied by Cortell [16]. Vajravelu [17] studied viscous **flow** **over** a nonlinearly **stretching** **sheet**, where viscous **flow** and **heat** **transfer** **over** a nonlinearly **stretching** **sheet** were obtained by Cortell [18] then, series solution of **flow** **over** non- linearly **stretching** **sheet** with chemical reaction and mag- netic field was investigated by employing the Adomian decomposition method by Kechil and Hashim [19] where, Ziabakhsh et al. [20] used homotopy analysis method to present **flow** and diffusion of chemically reactive species **over** a nonlinearly **stretching** **sheet** immersed in a porous medium. Muhaimin et al. [21] studied the effect of che- mical reaction, **heat** and mass **transfer** on nonlinear boun- dary layer past a porous shrinking **sheet** in the presence of suction and, Robert [22] discussed high-order nonlin- ear boundary value problems admitting multiple exact solutions with application to the fluid **flow** **over** a **sheet**. Cortell [23] studied **heat** and fluid **flow** due to non-line- arly **stretching** surfaces where, existence and uniqueness results for a nonlinear differential equation arising in vis- cous **flow** **over** a nonlinearly **stretching** **sheet** were ob- tained by Robert et al. [24]. Finally, Vajravelu et al. [25] studied the diffusion of a chemically reactive species of a power-law fluid past a **stretching** surface.

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Recently, most of researchers have fascinated tremendous attention towards the theory of micropolar fluid and interest of extensive research has given towards the study of **MHD** micropolar fluid **over** a **stretching** **sheet** due to its industrial applications.Chaudaryet.al 1 investigated the **heat** and mass **transfer** processes on an unsteady **flow** of a micropolar fluid past through a porous medium bounded by a semi-infinite vertical plate.FaizAwadet.al 2 studied Dufour and Soret **effects** of a micropolar fluid in a horizontal channel. In the above article a couple of partial differential equations were solved analytically by Homotopy Analysis Method and numerically by bvp4c MATLAB. Mohamed Abd-El-aziz.et.al 3 discussed the effect of variable viscosity and variable thermal conductivity of an unsteady forced convective **flow** and **heat** **transfer** characteristics of a **viscoelastic** liquids film past on a horizontal **stretching** **sheet** in presence of viscous dissipation. Dakshinamoorthy.et.al 4 reported the steady, two-dimensional, boundary layer **flow** of an electrically conducting viscous incompressible fluid past in a continuously moving surface in presence of uniform transverse magnetic field. Kishoreet.al 5 presented **MHD** viscous incompressible fluid past in an oscillating vertical plate fixed in a porous medium in presence of variable **heat** and mass diffusion, **radiation** and viscous dissipation.

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generation/absorption. Bataller [7] investigated the effect of thermal **radiation** on **heat** **transfer** in a boundary layer viscoe- lastic second order ﬂ uid **over** a **stretching** **sheet** with internal **heat** **source**/sink. Recently, Hayat et al. [8] studied the **effects** of chemical reaction of unsteady three dimensional ﬂ ow of couple stress ﬂ uid **over** a **stretching** surface. Gireesha et al. [9] have studied the boundary-layer ﬂ ow and **heat** **transfer** of a dusty ﬂ uid ﬂ ow **over** a **stretching** **sheet** in presence of non-uniform **heat** **source**/sink and **radiation**. Parsa et al. [10] investigated the **MHD** boundary-layer ﬂ ow **over** a **stretching** surface with internal **heat** generation or absorption. Gupta and Gupta [11] have investigated **heat** and mass **transfer** in hydrodynamic ﬂ uid ﬂ ow **over** an isothermal **stretching** **sheet** with suction/blowing **effects**. Vajravelu and Rollins [12] have studied the ﬂ ow and **heat** **transfer** introducing the temperature dependent **heat** **source** and sink. But these studies are con ﬁ ned to hydrodynamic ﬂ ow and **heat** **transfer** in Newtonian ﬂ uids. However, most of the practical situations demand for ﬂ uids those are non-Newtonian in nature which are extensively used in many industrial and engineering applications.

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Another class of models is the rare-type fluidmodels, such as Oldroyd model, which has been modified by Walters.This modified model is referred to as the Walters’ **liquid** B. The steady two- dimensional boundary layer equations for Walters’ liquidB were derived by Beard and Walters [10] to first-order in elasticity(i.e., for short memory fluids with short relaxation times).Walters’ **liquid** B considered by Sidappa and Abel [13] exhibit normal stress-differences in simple shear flows. Rajagopal et al. [14]analyzed the **effects** of viscoelasticity on the **flow** of a second-order fluid with gradually fading memory and arrived to the boundarylayer equations as that in Ref. [13]. H.I.Andersson[15] considered **MHD** **flow** of a **viscoelastic** fluid past a **stretching** sheet.An exact analytical solution of the governing nonlinear boundary layer equation was obtained illustrating, that the effect of magnetic field is same as that of viscoelasticity, on **flow** and **heat** **transfer** charecteristics.

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Ishak [22] studied the **MHD** ﬂow and **heat** **transfer** charac- teristics **over** an unsteady **stretching** surface. Yusof et al. [23] extended Ishak [22] work by introducing the effect of **radiation** for the **MHD** ﬂow and **heat** **transfer** **over** an unsteady stretch- ing surface. **Radiation** is energy that comes from a **source** and travels through some material or through space. Light, **heat** and sound are types of **radiation**. **Radiation** is considered in his study due to the fact that thermal **radiation** effect might play a signiﬁcant role in controlling **heat** **transfer** process in polymer processing industry. Many new engineering processes such as fossil fuel combustion energy processes, solar power technology, astrophysical ﬂows, gas turbines and the various propulsion devises for aircraft, missiles, satellites, and space vehicle re-entry occur at high temperature. So knowledge of **radiation** plays a very important role and hence, its effect can- not be neglected. Also thermal **radiation** is of major impor- tance in many processes in engineering areas which occur at a high temperature for the design of many advance energy con- version systems and pertinent equipment. The Rosseland approximation is used to describe the radiative **heat** ﬂux in the energy equation.

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57 | P a g e with suction. Bidin and Nazar [11] presented the numerical solutions for the problem of boundary layer **flow** **over** an exponentially **stretching** **sheet** in the presence of **radiation**. The **flow** due to a heated surface immersed in a stable stratified viscous fluid has been investigated experimentally and analytically by Yang et al. [12]. Recently, Mukhopadhyay [13] analysed the **MHD** boundary layer **flow** and **heat** **transfer** towards an exponentially **stretching** **sheet** embedded in a thermally stratified medium by taking suction into account. Hence, the aim of the present work is to study the characteristics of **MHD** boundary layer **flow** and **heat** **transfer** of a **viscoelastic** fluid **over** an exponentially **stretching** **sheet** embedded in a thermally stratified medium in the presence of **radiation** using HAM [14].

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[1999] have investigated the effect of a chemical reaction on the **flow** along a semi infinite horizontal plate in the presence of **heat** **transfer**. Anjalidevi and Kandaswamy [2000] have studied the effect of a chemical reaction on the **flow** in the presence of **heat** **transfer** and magnetic field. Muthukumaraswamy and Ganesan [2000] have analyzed the effect of a chemical reaction on the unsteady **flow** past on impulsively started semi-infinite vertical plate, which is subject to uniform **heat** flux. McLeod and Rajagopal [1987] have investigated the uniqueness of the **flow** of a Navier Stoke’s fluid due to a linear **stretching** boundary. Raptis et. al. [2006], have studied the viscous **flow** **over** a non-linearly stretched **sheet** in the presence of a chemical reaction and magnetic field. Several authors (Sakiadis [1961], Erickson et.al. [1966], Fox et.al. [1968], Chen and Char [1988], Ali [1995], Magyari et.al. [2001], Crane[1970]) have discussed fluid **flow** past a **stretching** **sheet** under different conditions.

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