Abstract —The steady, laminar incompressible MHD stagnation-point flows andheattransferwithvariableconductivity of a Non-Newtonian Fluidover a stretching sheets are analyzed for three cases of heating conditions, namely, (i)the sheetwith the con- stant surface temperature; (ii) the sheetwith the pre- scribed surface temperature; (iii) surface temperature with the prescribed surface heat flux. The governing system of partial differential equations is first trans- formed into a system of dimensionless ordinary dif- ferential equations. The numerical solutions are pre- sented to illustrate the influence of the various values of the ratio of free stream velocity andstretching ve- locity, the magnetic field parameter, Prandtl number, the wall temperature exponent and the power-lawin- dex. These effects of the different parameters on the velocity and temperature as well as the skin friction and wall heattransfer are presented in tables and graphically. The results are found to be in good agre- ment with those of earlier investigations reported in existing scientific literatures.
The combined influence of thermal and magnetic field gradients on the saturated ferrofluid flowing along a flat plate was investigated by NeuringerJ.L. (1966). The flow of a viscous fluid past a linearly stretching surface was considered by Crane L.J. (1970) for a Newtonian fluid. Andersson and Valnes (1998) extended Crane’s problem by studying the influence of the magnetic field, due to a magnetic dipole, on a shear driven motion (flowover a stretchingsheet) of a viscous non-conducting ferrofluid. It was concluded that the primary effect of the magnetic field was to decelerate the fluid motion as compared to the hydrodynamic case. At the present time there are enumerable papers on the stretchingsheet problem using different continua and considering various effects such as non-Newtonian characteristics, radiation, and magnetic field and so on. The above discussions can be found in Abel et al. 2008, 2009a, 2009b, 2009c, 2009d, 2011; Andersson 1998, 1992, 2006; Cortell 2010, 2008, 2007a, 2007b, 2006; Dandapat 2011, 2010, 2007; Dulal Pal 2010a, 2010b; Siddheshwar and Mahabaleshwar 2005; Hayat et al. 2010a, 2010b; Abbas et al.2010; Wang C.Y. 2007; Hamad 2007; Arnold et al. 2010; Seddeek 2007; Prasad et al. 2010; Magyari and Keller 2006; Van Gorder and Vajravelu 2010; Vajravelu and Cannon 2006; Abdoul and Ghotbi 2009; Tzirtzilakis and Kafoussias 2003 and the references there in. In many of the physical situation the sheet may be stretched vertically, rather than horizontally, into the ambient liquid. In this case the liquid flowand the heattransfer characteristics are determined by the motion of the stretchingsheetand the buoyant force. There are no studies in literature concerning the flowandheattransferin a ferrofluid due to a vertical stretchingsheetin the presence of external magnetic field. This paper aims at studying the same using two different types of boundary heating, namely, prescribed surface temperature (PST) and prescribed surface heat flux (PHF). Shooting method based on Runge-Kutta-Fehlberg and Newton Raphson schemes is used in arriving at the numerical solution of the proposed problem.
In recent years several industries deal with the Non-Newtonian fluids under the influence of magnetic field. In view of this, some researchers [Sarpakaya (1961), Saponkov (1967), Martinson and Pavlov (1971), Samokhen (1987), Andersson et. al (1992), Cortell (2005a), Liao (2005)] have presented works on MHD flowandheattransferin an electrically conducting powerlawfluidover a stretchingsheet. Motivated by this, we produce similarity analysis via deductive group method based on general group transformation is, probably first time, to derive symmetry group and similarity solutions for boundary layer flow of an electrically conducting power-lawfluidover non-linear surface. The aim of the present work is twofold: first to derive systematically the similarity transformation using general group theoretic method under similarity requirement for the governing equations, secondly to incorporate the effects of applied magnetic field for an electrically conducting fluidand to carry out the heattransfer analysis.
A new dimension is added to the study of flowandheattransferin a viscous fluidover a stretching surface in the presence of thermal radiation. The radiative effects have important applications in physics and engineering particularly in space technology and high temperature processes. Thermal radiation effect might play a significant role in controlling heattransfer process in polymer processing industry. Bakier and Gorla (1996) investigated the effect of thermal radiation on mixed convection from horizontal surfaces in saturated porous media.The quality of the final product depends to a great extent on the heat controlling factors and the knowledge of radiative heattransferin the system can perhaps lead to a desired product with a sought characteristic. Pal and Malashetty (2008) have presented similarity solutions of the boundary layer equations to analyze the effects of thermal radiation on stagnation point flowover a stretchingsheetwith internal heat
486 transfer. Afify (2009) discussed the MHD free convective heatand mass transferflowover a stretchingsheetin the presence of suction/injection with thermal diffusion and diffusion thermo effects. The influence of dust particles on the flow of a viscous fluid has several important applications. The dust particles tend to retard the flowand to decrease the fluid temperature. Such flows are encountered in a wide variety of engineering problem such as nuclear reactor cooling, rain erosion, paint spraying, transport, waste water treatment, combustion, etc. The presence of solid particles such as ash or soot in combustion energy generators and their effect on performance of such devices led to studies of particulate suspension in electrically conducting fluidin the presence of magnetic field. Saffman (1962) initiated the study of dusty fluids and discussed the stability of the laminar flow of a dusty gas in which the dust particles are uniformly distributed Chamkha (2000b) investigated the unsteady laminar hydromagnetic fluid particle flowandheattransferin channels and circular pipes considering two phase continuum models. The effects of Hall current on the Couette flowwithheattransfer of a dusty conducting fluidin the presence of uniform suction/injection was studied by Attia (2005). Ghosh and Ghosh (2008) considered the problem of hydromagnetic rotating flow of a dusty fluid near a pulsating plate when the flow is generated in the fluid particle system due to velocity tooth pulses subjected on the plate in the presence of a transverse magnetic field. Makinde and Chinyoka (2010) investigated the unsteady fluidflowandheattransfer of a dusty fluid between two parallel plates withvariable viscosity and thermal conductivity
Cortell  studied heattransferin a moving fluidover a moving surface numerically by means of a fourth-order Rung-Kutta method. Ephraim and Abraham  investigated the streamwise variation of the temperature of a moving sheetin the presence of moving fluid. They applied an iterative method for solving boundary layer equations. Their solution does not de- pend on the properties of sheetandfluid. Ming et al.  studied conjugate heattransfer from a continuous, moving flat plate numerically by employing the cubic spline collocation. They in- vestigated effects of Prandtl number, the convection-conduction parameters and the Peclet num- ber on the heattransfer from a continuous, moving plate. The investigation of mixed convection heattransfer along a continuously moving heated vertical plate with suction and blowing was carried out by Al-Sanea . He applied the finite volume method to solve boundary layer equa- tions. He used the published results available under special condition to validate numerical data, and the comparison indicated an excellent agreement. The buoyancy force and thermal radiation effects in magnetohydodynamics (MHD) boundary layer visco-elastic fluidflowover continu- ously moving surface were performed by Abel et al. . Lee and Tsai  studied cooling of a continuous moving sheet of finite thickness. The effect of the buoyancy force is also taken into account. They obtained the temperature distribution along the solid-fluid interface by solving numerically a conjugate heattransfer problem that consists of heat conduction inside the sheetand induced mixed convection adjacent to the sheet surface. Other conjugate convection-con- duction researches have been presented by Choudhry and Jaluria , and Mendez and Trevino , among others. The heattransfer of a moving material in a non-Newtonian fluid was first studied by Fox et al. . They applied an exact solution for boundary layer equations. Howell et al.  studied heattransfer on a continuous moving plate in non-Newtonian powerlawfluid. They applied Merk-Chao series expansion to generate ordinary differential equation from the partial differential momentum andheattransfer equations in order to obtain universal velocity and temperature functions. Torabi et al.  investigated convective-radiative non-Fourier heat conduction withvariable coefficients by employing homotopy perturbation method (HPM) Some of the other studies that investigated the heattransfer of a continuous moving material inpowerlawfluid have been reported by Sahu et al.  and, Zheng and Zhang .
blowing, continuous casting of metals, and spinning of fibers also involve the flowover a stretching surface. During the manufacturing process of these sheets, the mixture issued from a slot is stretched to reach the desired thickness. At last, in view of acquiring the top-grade final product, this sheet solidifies as it passes through the air/water-cooled systems. In water cooling systems, inclusion of nanoparticles can enhance the cooling process efficiency and can also reduce the transient time. There is a vast literature on the boundary layer flowover a stretchingsheet, but we only refer to few recent studies Ziabakhsh et al. (2010), Hassani et al. (2011), Hayat et al. (2011), Postelnicu and Pop (2011), Malvandi et al. (2013), Malvandi et al. (2013), Reddy (2013). All these investigations employ no-slip condition at the boundary. However, the non-adherence of the fluid to a solid boundary at the presence of nanoparticles, known as slip velocity condition, has been reported by numerous researchers Abbas et al. (2008), Hayat et al. (2008), Hamad et al. (2012), Noghrehabadi et al. (2012), Malvandi et al. (2014), Sharma et al. (2014). Recently, impacts of the convective boundary condition of nanofluid over a stretchingsheetwith no slip condition have been studied by Makinde and Aziz (2011). As stated earlier, slip condition occurs in the
Heatand mass transfer past over a stretchingsheet is an excellent study in industrial applications such as glass fiber production, hot rolling and wire drawing, the aerodynamic extrusion of plastic sheets, glass blowing, metal spinning and drawing plastic films under different heating processes. The quality of the final product depends on the rate of heattransfer at the stretching surface. A comprehensive study on boundary layer flow caused by the stretching of an elastic flat sheet was studied by McCormack and Crane (1973). Crane (1970) investigated the flow caused by a stretching plate. Many authors such as Gupta and Gupta (1977); Chen and Char (1988); Dutta et al. (1985) extended the study of Crane (1970) by including the effects of heatand mass transfer under various situations.
The convective flowoverstretching surfaces immersed in porous media has paramount importance because of its potential applications in industrial purposes like soil physics, filtration of solids from liquids, chemical engineering and biological systems. In addition with the recent improvements in modern technology many researchers are concentrating on the study of heatand mass transferinfluid flows due to its broad applications in geothermal engineering as well as other geophysical and astrophysical studies. Radiative heatand mass transfer play an important role in manufacturing industries for the design of reliable equipment. Nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles, etc.
The phenomena of momentum andheattransferin boundary layer flowover a flat heated sur- face are experienced widely in industrial engineering applications. The momentum andheattransfer due to a heated stretching surface have gained considerable attention because of their practical importance in diverse engineering disciplines. Plastic and rubber sheets are manufac- tured by this process, where it is often necessary to blow a gaseous medium through the not yet solidified material. Further example that belongs to the above class of problems is the cooling of a large metallic plate in a bath, which may be an electrolytic . The quality of finished product is strongly dependent upon the final cooling of the product. Various aspects of such problems, including unidirectional and bidirectional stretching surface, have been the focal point of many theoretical researchers. Some previous works regarding bidirectional stretching surface was carried out by Wang , who presented exact similar solutions for a three- dimensional flow due to stretching of sheetin two lateral directions. Later on, Ariel  addressed this problem by finding the approximate analytical solutions using the homotopy perturbation method. Liu and Andersson  also explored numerically the heattransfer char- acteristics of fluid, when the sheet is stretched in two lateral directions withvariable thermal
Now a days, the energy efficiency is an extremely important topic in view of thermal conductivity enhancement amongst the researchers. For this purpose the researchers considered the involvement of nanoparticles in the base fluid. Originally Masuda et al. (1996) reported the liquid dispersions of submicron particles or nanoparticles. After that, first time nanofluid term is used by Choi (1995). In comparison to the base fluids, thermal conductivity of nanofluid is too high that's why these have been used in many energetic systems such as cooling of nuclear systems, radiators, natural convection in enclosures etc. The model proposed by Buongiorno (2006) studies the Brownian motion and the thermophoresis on the heattransfer characteristics. Recently, the analytical solutions for the laminar axisymmetric mixed convection boundary layer nanofluid flow past a vertical cylinder is obtained by Rashidi et al. (2012a). Stagnation point flow of nanofluid near a permeable stretched surface with thermal convective condition is provided by Alseadi et al. (2012). Mustafa et al. (2013) discussed the boundary layer flow of nanofluid over an exponentially stretchingsheetwith convective boundary conditions. Rashidi et al. (2014b) presented the analytical solutions of transport phenomena in nanofluid adjacent to a nonlinearly porous stretchingsheet. Sheikholeslami and Ganji (2013a) studied the heattransfer of Cu-water nanofluid flow between the parallel plates. Turkyilmazoglu (2013) studied the unsteady mixed convection flow of nanofluids over a moving vertical flat plate withheattransfer. Sheikholeslami et al. (2013b) determined free convection flow of nanofluid. Hayat et al. (2014) presented the mixed convection peristaltic flow of magnetohydrodynamic (MHD) nanofluid in presence of Brownian motion and thermophoresis. Casson fluid model is one of the base fluids which exhibits yield stress. However such fluid behaves like a solid when shear stress less than the yield stress is applied and it moves if applied shear stress is greater than yield stress. Examples of Casson fluid include jelly, soup, honey, tomato sauce, concentrated fruit juices, blood and many others. In fact several substances like protein, fibrinogen and globin in an aqueous base plasma, human red cells form a chain like structure, known as aggregates or rouleaux. If the rouleaux behaves like a plastic solid then there exists a field stress that can be identified with the constant yield stress in Casson fluid by Dash et al. (1996). Recently, Mukhopadhyay (2013a) provided the boundary layer flow of Casson fluidover a non-linearly stretchingsheet. Some of the recent studies about flow of Casson fluid are [Shehzad et al. (2013), Mukhopadhyay and Vajravelu (2013b), Hayat et al. (2012a)].
considered the boundary layer flow of a Williamson fluidover a stretchingsheet. Stretchingsheet flows are of great importance in many engineering applica- tions like extrusion of a polymer sheet from the die, the boundary layer in liquid film condensation processes, emulsion coating on photographic films, etc. Sakiadis (1961) initiated the study of boundary layer flows over a continuous surface and formulated the two dimensional boundary layer equations. Tsou et al. (1967) extended the work of Sakiadis and considered the heattransferin the boundary layer flowover a continuous surface and experimentally verified Sakiadis’ results. Erickson et al. (1966) included the heatand mass transfer on a stretching surface with suction or injection. Many researchers later investigated boundary layer flowover a stretching surface, such as Gupta and Gupta (1977), Ishak (2008), and Nadeem (2010).
There are many fluids which are important from the industrial point of view, and display non-Newtonian behavior. Due to the complexity of such fluids, several models have been proposed but the micropolar model has been found to be the most appropriate one. It has been experimentally predicted that the fluids which could not be characterized by Newtonian relationships, indicated significant reduction in shear stress near a rigid body. The micropolar model has been successful in explaining such behaviors of the non-Newtonian fluids. Since its introduction, the micropolar fluid has been a hot area of research, and therefore many investigators have studied the related flowandheattransfer problems in different geometries. For example, natural convection heattransfer between two differentially heated concentric isothermal spheres utilizing micropolar fluid has been numerically investigated by Khoshab and Dehghan, (2011). Govardhan and Kishan (2011) studied the MHD effects on the unsteady boundary layer flow of an incompressible micropolar fluidover a stretchingsheet when the sheet was stretched in its own plane. Ashmawy (2014) considered the problem of fully developed natural convective micropolar fluidflowin a vertical channel, under the slip boundary conditions for fluid velocity. The effect of the presence of a thin perfectly conductive baffle on the fully developed laminar mixed convection in a vertical channel containing
d to alternating ed the thermal ments of wa et al. (2011) radiation on M ching sheetwith rected a valuabl vity of nanoflu eat transfer inte y Makinde layer flow of a a convective bou and Gupta (2002 -point flow tow ntioned studies e best of our kn oblem of non-al hylene Glycol b cles. In this st andheattransfer
The technological application of the hydromagnetic flowwith slip flow effects has become the centre of attraction of many scientists, engineers and researchers. Beaver and Joseph  proposed a slip flow condition at the boundary. Of late, there has been a revival of interest in the flow problems with partial slip. Martin et al.  presented the Blasius boundary layer solution with slip flow conditions. Wang  undertook the study of the flow of a Newtonian fluid past a stretchingsheetwith partial slip and purportedly gave an exact solution. Slip flow past a stretching surface was analysed by Andersson . Martin et al.  analysed the momentum andheattransferin a laminar boundary with slip flow. Wang  carried out the stagnation slip flowandheattransfer on a moving plate. Matthews et al.  gave a note on the boundary layer equations with linear slip boundary conditions. Abbas et al.  analysed the slip effects andheattransfer effects of a viscous fluidover an oscillatory stretching surface. Fang et al.  gave an exact solution of the slip MHD viscous flowover a stretchingsheet. Wang  carried out an analysis of viscous flow due to a stretchingsheetwith surface slip and suction. Recently, the effects of slip conditions on stretchingflowwith ohmic dissipation and thermal radiation was given by Qasim .
It is interesting to note that the Brownian motion of nanoparticles at molecular and nanoscale levels are a key nanoscale mechanism governing their thermal behaviors. In nanofluid systems, due to the size of the nanoparticles, the Brownian motion takes place, which can affect the heattransfer properties. As the particle size scale approaches to the nanometer scale, the particle Brownian motion and its effect on the surrounding liquids play an important role in the heattransfer. In view of these applications, Nield and Kuznetsov ([22, 23]) analyzed the free convective boundary layer flows in a porous medium saturated by nanofluid by taking Brownian motion and thermophoresis effects into consideration. In the first article, the authors have assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology and hence they have concluded that this prevents particles from agglomeration and deposition on the porous matrix. Chamkha et al.  carried out a boundary layer analysis for the natural convection past an isothermal sphere in a Darcy porous medium saturated with a nanofluid. Nield and Kuznetsov  investigated the cross-diffusion in nanofluids, with the aim of making a detailed comparison with regular cross diffusion effects and the cross- diffusion effects peculiar to nanofluids, and at the same time investigating the interaction between these effects when the base fluid of the nanofluid is itself a binary fluid such as salty water. Recently, a boundary layer analysis for the natural convection past a horizontal plate in a porous medium saturated with a nanofluid is analyzed by Gorla and Chamkha , N. Kishan et.al , studied the unsteady MHD flow of heatand mass transfer of Cu-water and TiO 2 -water nanofluids overstretchingsheetwith a non-uniform heat/source/sink
flow past a stretchingsheet. Vajravelu and Nayfeh  analyzed hydromagnetic flow of a dusty fluidover a stretchingsheet. Ishak et al.  studied the effect of a uniform transverse magnetic field on the stagnation point flow toward a vertical stretchingsheet. Cortell  examined the flowandheattransfer of an electrically conducting second grade fluidin the presence of transverse magnetic field past a semi-infinite stretchingsheet. Sweet et al.  obtained the analytical solution of the MHD flow of a viscous fluid between two moving parallel plates via the homotopy analysis method. Further, Robert and Vajravelu  obtained explicit exact solutions for fourth-order nonlinear differential equations arising in the hydromagnetic flow of a second grade fluidover a stretching or shrinking sheet. Recently, Abbasbandy et al.  obtained both numerical and analytical solutions for Falkner-Skan flow of MHD Maxwell fluidand emphasized the variations of viscoelastic and magnetic parameters. Many researchers have made the valuable contribution to the literature of hydromagnetic flowandheattransferover a stretchingsheet by considering different geometry and obtained exact solutions [10-15].
Polymeric suspensions such as waterborne coatings are identi- fied to be non-Newtonian in nature and are proven to follow the Sisko fluid model . The Sisko fluid model was originally proposed for high shear rate measurements on lubricating greases . Khan et al.  examined the steady flowandheattransfer of a Sisko fluidin annular pipe. Then, Khan and Shahzad [17,18] developed the boundary layer equations for Sisko fluidover planer and radially stretching sheets and found the analytical solutions for only integral values of the power-law index. The utmost studies relating to the heattransfer of Sisko fluid involve only one dimensional flows and literature survey indicates that no work has so far been communicated with regards to heattransferin a boundary layer flow for Sisko fluidover a nonlinear stretchingsheetwithvariable surface temperature andvariableheat flux.
fluidover a permeable stretching/shrinking sheet. Mukhopadhyay (2013) studied the Casson fluidflowandheattransferover a nonlinear stretchingsheet. At the current investigation, we refer some latest studies on stretched flows. Rashidi and Mohimanian Pour (2010) obtained the analytic solutions for unsteady boundary-layer flowandheattransfer due to a stretchingsheet by means of homotopy analysis method. Further Rashidi and Keimanesh (2010) use the differential transform method and Padé approximant for solving MHD flowin a laminar liquid film from a horizontal stretching surface. Rana and Bhargava (2012) studied the flowandheattransfer of a nanofluid over a nonlinearly stretchingsheet. Radiation effect on a steady two-dimensional boundary layer flow of a dusty fluidover a stretchingsheet is analyzed by Ramesh and Gireesha (2013). Flow near stagnation-point is very interesting influid dynamics. Actually, the stagnation flow takes place whenever the flow impinges to any solid object and the local velocity of the fluid at the stagnation-point is zero. It is an important bearing on several industrial and technical applications such as cooling of electronic devices by fans, cooling of nuclear reactors during emergency shutdown, heat
specific heat, k the thermal conductivity of the fluid, j~ n=c ð Þ is microinertia per unit mass, c ~(mzk=2)j and k are the spin gradient viscosity and vortex viscosity, respectively. Here k~0 corresponds to situation of viscous fluidand the boundary parameter n varies in the range 0ƒnƒ1: Here n~0 corresponds to the situation when microelements at the stretchingsheet are unable to rotate and denotes weak concentrations of the microelements at sheet. The case n~1=2 corresponds to the vanishing of anti-symmetric part of the stress tensor and it shows weak concentration of microelements and the case n~1 is for turbulent boundary layer flows.