Department of Mathematics, Central College Campus, Bangalore University, Bangalore 560 001, India Received 12 July 2003; accepted 16 April 2004
We study the MHD ﬂow and also heattransfer in a viscoelasticliquidover a stretchingsheet 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 heattransfer, 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 heattransfer are presented graphically. The results have possible technological applications in liquid-based systems involving stretchable materials.
The study of boundary layer flow passing a stretchingsheet 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 stretchingsheet have attracted the attention of a large of a number of researchers. The boundary layer flow and heattransfer in a viscous fluid contacting metallic nanoparticles over a stretchingsheet in the presence of thermal radiation have investigated by Hamad and Ferdows (2012). Effects of chemical reaction on the MHDflow of a visco-elastic fluid through porous medium have been presented by Nayak et al. (2014).
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 heattransfer 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 heattransferoverstretchingsheet by considering dif- ferent properties of a fluid. Moreover, when radiative heattransfer 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 heatsource/sink on heat
Heatsource or sink has been assumed to be constant, space dependent. When technology processes take place at high temperatures thermal radiationheattransfer become very important and it effects cannot be neglected. Recent developments in hypersonic flights, missile re-entry rocket combustion chambers and gas cooled nuclear reactors, have focussed attention of researchers on thermal radiation as a mode of energy transfer and emphasize the need for inclusion of radiative transfer in these processes. For these many studies have appeared concerning the interaction of radiative flux with thermal convection flows. In certain porous media applications, working fluid heat generation (source) or absorption (sink) effects are important. Representative studies dealing with these effects have been reported by authors such that as Gupta and Sridhar , Abel and Veen  and Sharma . In recent years, the flow analysis of Nanofluids has been topic of extensive research due to characteristic in increasing thermal conductivity heat trans fer process.
Another effect which bears great importance on heattransfer 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, viscoelasticflow and heattransferover a flat plate with constant suction, thermal radiation and without viscous dissipation were studied by Salem  used a shooting technique to study numerically the effects of variable viscosity and thermal conductivity on the MHDflow and heattransfer of a viscoelastic fluid over a stretchingsheet 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  Effects of free convection, thermal radiation, and surface suction/blowing on the flow and heattransfer characteristics are also examined. Uddin et al.  investigated the effects of mass transfer on MHD mixed convective flow along inclined porous plate. R. Ravidran et al.  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.  presented a unified approach to solving the MHDflow due to influence of buoyancy and thermal radiationover a stretching porous sheet using homotopy analysis method. N. Sandeep et al.  investigated the influence of non-uniform heatsource/sink, mass transfer and chemical reaction on an unsteady mixed convection boundary layer flow of a MHD micropolar fluid past a stretchingsheet in presence of viscous dissipation and suction/injection, most recently.
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  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  . Couple stress theory has been successfully applied to many problems in biomechanics and lubrications area by Srivastava  , El-Shehawey and Mekheimer  , Pal et al.  , Chiang et al.  , Naduvinamani et al.  , Jian and Chen et al.  and Lu and Lin  .
Abbas and Hayat  studied the radiationeffects on the magnetohydrodynamic (MHD) flow of an incom- pressible viscous fluid in a porous space. In their study, they extended the analysis of Cortell  by considering a MHDflow, analyzed the flow in a porous medium, included the radiationeffects and provided analytic solu- tion namely homotopy analysis method (HAM) instead of numerical technique applied in . Hayat et al.  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.
two types of viscoelastic fluids over a stretching surface have been investigated by Turkyilmazoglu  . Moreover, Elbash- beshy and Ibrahim  investigated the effect of steady free convection flow with variable viscosity and thermal diffusivity along a vertical plate. Kafoussias and Williams  studied the thermal-diffusion and diffusion-thermo effects on the mixed free-forced convective and mass transfer steady laminar boundary layer flowover a vertical plate, with temperature- dependent viscosity. Sajid and Hayat  investigated the radi- ation effects on the mixed convection flowover an exponen- tially stretchingsheet and solved the problem analytically using homotopy analysis method. The numerical solution for the same problem was then given by Bidin and Nazar  . Recently, Poornima and Bhaskar Reddy  presented an analysis of the radiationeffects on MHD free convective boundary layer flow of nanofluids over a nonlinear stretchingsheet. However, the interaction of radiation with mass transfer due to a stretchingsheet has received little attention. Abol- bashari et al.  studied entropy analysis for an unsteady MHDflow past a stretching permeable surface in nanofluid. Rashidi and Erfani  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 heattransfer for MHDviscoelastic fluid flowover a porous wedge with thermal radiation is studied by Rashidi et al.  . Further, Rashidi et al.  studied an analytic approximate solution for MHD boundary layer viscoelastic fluid flowover continuously moving stretching surface by HAM with two auxiliary parameters.
The radiationeffects 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 radiationeffects 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 heattransfer characteristics in different situations [22-26]. The aim of the present analysis is to study the heattransfer characteristic from a linearly stretching surface with power- law surface temperature in quiescent fluid in the presence of internal heatsource , a uniform transverse magnetic field and slip conditions. Exact solution to the energy equation in terms of Kummer’s functions is then obtained.
studied buoyancy effects on MHDflow of nanofluid (with Copper and Copper Oxide as nanoparticles) over a stretchingsheet 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 heattransfer 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 flowover 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 heattransfer 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 heattransfer 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 flowover 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.
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  presented similarity solutions for unsteady flow and heattransferover a stretching surface. They examined effects of unsteady parameter (A) and Prandtl number (Pr) on the flow and heattransfer characteristics. They observed that the unsteady parameter and Prandtl number increase heattransfer rate at the surface. These results were supported by Azid  and Ishak  and they obtained the exact solution of unsteady mixed convection boundary layer flow and heattransfer. The results show that the buoyancy parameter increases the heattransfer rate at the surface. Bachok et al.  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
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  studied the flow and heattransfer in a viscous fluid over a nonlinear stretchingsheet utilizing nanofluid where, effects of vis- cous dissipation and radiation on the thermal boundary layer over a nonlinearly stretchingsheet were studied by Cortell . Vajravelu  studied viscous flowover a nonlinearly stretchingsheet, where viscous flow and heattransferover a nonlinearly stretchingsheet were obtained by Cortell  then, series solution of flowover non- linearly stretchingsheet with chemical reaction and mag- netic field was investigated by employing the Adomian decomposition method by Kechil and Hashim  where, Ziabakhsh et al.  used homotopy analysis method to present flow and diffusion of chemically reactive species over a nonlinearly stretchingsheet immersed in a porous medium. Muhaimin et al.  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  discussed high-order nonlin- ear boundary value problems admitting multiple exact solutions with application to the fluid flowover a sheet. Cortell  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 flowover a nonlinearly stretchingsheet were ob- tained by Robert et al. . Finally, Vajravelu et al.  studied the diffusion of a chemically reactive species of a power-law fluid past a stretching surface.
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 stretchingsheet 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 heattransfer characteristics of a viscoelastic liquids film past on a horizontal stretchingsheet 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.
generation/absorption. Bataller  investigated the effect of thermal radiation on heattransfer in a boundary layer viscoe- lastic second order ﬂ uid over a stretchingsheet with internal heatsource/sink. Recently, Hayat et al.  studied the effects of chemical reaction of unsteady three dimensional ﬂ ow of couple stress ﬂ uid over a stretching surface. Gireesha et al.  have studied the boundary-layer ﬂ ow and heattransfer of a dusty ﬂ uid ﬂ ow over a stretchingsheet in presence of non-uniform heatsource/sink and radiation. Parsa et al.  investigated the MHD boundary-layer ﬂ ow over a stretching surface with internal heat generation or absorption. Gupta and Gupta  have investigated heat and mass transfer in hydrodynamic ﬂ uid ﬂ ow over an isothermal stretchingsheet with suction/blowing effects. Vajravelu and Rollins  have studied the ﬂ ow and heattransfer introducing the temperature dependent heatsource and sink. But these studies are con ﬁ ned to hydrodynamic ﬂ ow and heattransfer 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.
flowover a moving continuous solid surfaces. Crane  found a closed form exact solution for Sakiadis problem. The effects of magneto hydrodynamics and thermal radiation on convective heattransfer play vital role in the phenomena of electrically conducting fluid past a heated surface and thermal processes involving high temperatures such as power generators, nuclear power plants etc. Swati  analysed these effects on boundary layer flowover an exponentially stretchingsheet.
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  to first-order in elasticity(i.e., for short memory fluids with short relaxation times).Walters’ liquid B considered by Sidappa and Abel  exhibit normal stress-differences in simple shear flows. Rajagopal et al. 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. . H.I.Andersson considered MHDflow 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 heattransfer charecteristics.
MHD boundary layer flow due to an exponentially stretchingsheet with radiation effect was presented by Ishak (2011). Yao et al. (2011) studied heattransfer on a generalized stretching/shrinking wall with convective boundary condition. Heattransfer in a fluid through a porous medium over a permeable stretching surface with thermal radiation and variable thermal conductivity was analyzed by Cortell (2012). Hayat (2012) considered three-dimensional flow of a Jeffery fluid over a linearly stretchingsheet. Hydromagnetic boundary layer flowoverstretching surface with thermal radiation has been discussed by Soid et al. (2012). Mandal and Mukhopadhyay (2013) presented heattransfer analysis for fluid flowover an exponentially stretching porous sheet with surface heat flux in porous medium. Slip effects on MHD boundary layer flowover an exponentially stretchingsheet with suction/blowing and thermal radiation shown by Mukhopadhyay (2013). Norhafizah et al. (2013) studied numerical solution of flow and heattransferover a stretchingsheet with Newtonian heating using the Keller Box Method. Singh and Makinde (2015) presented a similarity solution for the combined effects of velocity slip and temperature jump on boundary layer flowover a moving surface. The MHD slip flow of a conducting Casson nanofluid over a convectively heated stretchingsheet was numerically studied by Ibrahim and Makinde (2016a). Other relevant papers with respect to MHDflowover a stretchingsheet include Ibrahim and Makinde (2016b); Khan et al. (2016).
Ishak  studied the MHD ﬂow and heattransfer charac- teristics over an unsteady stretching surface. Yusof et al.  extended Ishak  work by introducing the effect of radiation for the MHD ﬂow and heattransferover 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 heattransfer 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.
57 | P a g e with suction. Bidin and Nazar  presented the numerical solutions for the problem of boundary layer flowover an exponentially stretchingsheet 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. . Recently, Mukhopadhyay  analysed the MHD boundary layer flow and heattransfer towards an exponentially stretchingsheet 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 heattransfer of a viscoelastic fluid over an exponentially stretchingsheet embedded in a thermally stratified medium in the presence of radiation using HAM .
 have investigated the effect of a chemical reaction on the flow along a semi infinite horizontal plate in the presence of heattransfer. Anjalidevi and Kandaswamy  have studied the effect of a chemical reaction on the flow in the presence of heattransfer and magnetic field. Muthukumaraswamy and Ganesan  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  have investigated the uniqueness of the flow of a Navier Stoke’s fluid due to a linear stretching boundary. Raptis et. al. , have studied the viscous flowover a non-linearly stretched sheet in the presence of a chemical reaction and magnetic field. Several authors (Sakiadis , Erickson et.al. , Fox et.al. , Chen and Char , Ali , Magyari et.al. , Crane) have discussed fluid flow past a stretchingsheet under different conditions.