Abstract —The steady, laminar incompressible MHD stagnation-point flows **and** **heat** **transfer** **with** **variable** **conductivity** of a Non-Newtonian **Fluid** **over** a **stretching** sheets are analyzed for three cases of heating conditions, namely, (i)the **sheet** **with** the con- stant surface temperature; (ii) the **sheet** **with** 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 **and** **stretching** ve- locity, the magnetic field parameter, Prandtl number, the wall temperature exponent **and** the **power**-**law** **in**- dex. These effects of the different parameters on the velocity **and** temperature as well as the skin friction **and** wall **heat** **transfer** 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.

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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 (**flow** **over** a **stretching** **sheet**) 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 **stretching** **sheet** 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 **flow** **and** the **heat** **transfer** characteristics are determined by the motion of the **stretching** **sheet** **and** the buoyant force. There are no studies **in** literature concerning the **flow** **and** **heat** **transfer** **in** a ferrofluid due to a vertical **stretching** **sheet** **in** 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.

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A new dimension is added to the study of **flow** **and** **heat** **transfer** **in** a viscous **fluid** **over** 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 **heat** **transfer** 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 **heat** **transfer** **in** 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 **flow** **over** a **stretching** **sheet** **with** internal **heat**

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486 **transfer**. Afify (2009) discussed the MHD free convective **heat** **and** mass **transfer** **flow** **over** a **stretching** **sheet** **in** 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 **flow** **and** 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 **fluid** **in** 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 **flow** **and** **heat** **transfer** **in** channels **and** circular pipes considering two phase continuum models. The effects of Hall current on the Couette **flow** **with** **heat** **transfer** of a dusty conducting **fluid** **in** 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 **fluid** **flow** **and** **heat** **transfer** of a dusty **fluid** between two parallel plates **with** **variable** viscosity **and** thermal **conductivity**

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Cortell [3] studied **heat** **transfer** **in** a moving **fluid** **over** a moving surface numerically by means of a fourth-order Rung-Kutta method. Ephraim **and** Abraham [4] investigated the streamwise variation of the temperature of a moving **sheet** **in** 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 **sheet** **and** **fluid**. Ming et al. [5] studied conjugate **heat** **transfer** 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 **heat** **transfer** from a continuous, moving plate. The investigation of mixed convection **heat** **transfer** along a continuously moving heated vertical plate **with** suction **and** blowing was carried out by Al-Sanea [6]. 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 **fluid** **flow** **over** continu- ously moving surface were performed by Abel et al. [7]. Lee **and** Tsai [8] 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 **heat** **transfer** problem that consists of **heat** conduction inside the **sheet** **and** induced mixed convection adjacent to the **sheet** surface. Other conjugate convection-con- duction researches have been presented by Choudhry **and** Jaluria [9], **and** Mendez **and** Trevino [10], among others. The **heat** **transfer** of a moving material **in** a non-Newtonian **fluid** was first studied by Fox et al. [11]. They applied an exact solution for boundary layer equations. Howell et al. [12] studied **heat** **transfer** on a continuous moving plate **in** non-Newtonian **power** **law** **fluid**. They applied Merk-Chao series expansion to generate ordinary differential equation from the partial differential momentum **and** **heat** **transfer** equations **in** order to obtain universal velocity **and** temperature functions. Torabi et al. [13] investigated convective-radiative non-Fourier **heat** conduction **with** **variable** coefficients by employing homotopy perturbation method (HPM) Some of the other studies that investigated the **heat** **transfer** of a continuous moving material **in** **power** **law** **fluid** have been reported by Sahu et al. [14] **and**, Zheng **and** Zhang [15].

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blowing, continuous casting of metals, **and** spinning of fibers also involve the **flow** **over** 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 **flow** **over** a **stretching** **sheet**, 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 **stretching** **sheet** **with** no slip condition have been studied by Makinde **and** Aziz (2011). As stated earlier, slip condition occurs **in** the

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The convective **flow** **over** **stretching** 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 **heat** **and** mass **transfer** **in** **fluid** flows due to its broad applications **in** geothermal engineering as well as other geophysical **and** astrophysical studies. Radiative **heat** **and** 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.

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The phenomena of momentum **and** **heat** **transfer** **in** boundary layer **flow** **over** a flat heated sur- face are experienced widely **in** industrial engineering applications. The momentum **and** **heat** **transfer** 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 [1]. 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 [2], who presented exact similar solutions for a three- dimensional **flow** due to **stretching** of **sheet** **in** two lateral directions. Later on, Ariel [3] addressed this problem by finding the approximate analytical solutions using the homotopy perturbation method. Liu **and** Andersson [4] also explored numerically the **heat** **transfer** char- acteristics of **fluid**, when the **sheet** is stretched **in** two lateral directions **with** **variable** thermal

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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 **heat** **transfer** 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 **stretching** **sheet** **with** convective boundary conditions. Rashidi et al. (2014b) presented the analytical solutions of transport phenomena **in** nanofluid adjacent to a nonlinearly porous **stretching** **sheet**. Sheikholeslami **and** Ganji (2013a) studied the **heat** **transfer** 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 **with** **heat** **transfer**. 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 **fluid** **over** a non-linearly **stretching** **sheet**. Some of the recent studies about **flow** of Casson **fluid** are [Shehzad et al. (2013), Mukhopadhyay **and** Vajravelu (2013b), Hayat et al. (2012a)].

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considered the boundary layer **flow** of a Williamson **fluid** **over** a **stretching** **sheet**. **Stretching** **sheet** 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 **heat** **transfer** **in** the boundary layer **flow** **over** a continuous surface **and** experimentally verified Sakiadis’ results. Erickson et al. (1966) included the **heat** **and** mass **transfer** on a **stretching** surface **with** suction or injection. Many researchers later investigated boundary layer **flow** **over** a **stretching** surface, such as Gupta **and** Gupta (1977), Ishak (2008), **and** Nadeem (2010).

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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 **flow** **and** **heat** **transfer** problems **in** different geometries. For example, natural convection **heat** **transfer** 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 **fluid** **over** a **stretching** **sheet** when the **sheet** was stretched **in** its own plane. Ashmawy (2014) considered the problem of fully developed natural convective micropolar **fluid** **flow** **in** 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

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d to alternating ed the thermal ments of wa et al. (2011) radiation on M ching **sheet** **with** 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 **and** **heat** **transfer**

The technological application of the hydromagnetic **flow** **with** slip **flow** effects has become the centre of attraction of many scientists, engineers **and** researchers. Beaver **and** Joseph [17] 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. [18] presented the Blasius boundary layer solution **with** slip **flow** conditions. Wang [19] undertook the study of the **flow** of a Newtonian **fluid** past a **stretching** **sheet** **with** partial slip **and** purportedly gave an exact solution. Slip **flow** past a **stretching** surface was analysed by Andersson [20]. Martin et al. [21] analysed the momentum **and** **heat** **transfer** **in** a laminar boundary **with** slip **flow**. Wang [22] carried out the stagnation slip **flow** **and** **heat** **transfer** on a moving plate. Matthews et al. [23] gave a note on the boundary layer equations **with** linear slip boundary conditions. Abbas et al. [24] analysed the slip effects **and** **heat** **transfer** effects of a viscous **fluid** **over** an oscillatory **stretching** surface. Fang et al. [25] gave an exact solution of the slip MHD viscous **flow** **over** a **stretching** **sheet**. Wang [26] carried out an analysis of viscous **flow** due to a **stretching** **sheet** **with** surface slip **and** suction. Recently, the effects of slip conditions on **stretching** **flow** **with** ohmic dissipation **and** thermal radiation was given by Qasim [27].

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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 **heat** **transfer** 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 **heat** **transfer**. **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. [24] 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 [25] 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 [26], N. Kishan et.al [27], studied the unsteady MHD **flow** of **heat** **and** mass **transfer** of Cu-water **and** TiO 2 -water nanofluids **over** **stretching** **sheet** **with** a non-uniform **heat**/source/sink

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Polymeric suspensions such as waterborne coatings are identi- fied to be non-Newtonian **in** nature **and** are proven to follow the Sisko **fluid** model [14]. The Sisko **fluid** model was originally proposed for high shear rate measurements on lubricating greases [15]. Khan et al. [16] examined the steady **flow** **and** **heat** **transfer** of a Sisko **fluid** **in** annular pipe. Then, Khan **and** Shahzad [17,18] developed the boundary layer equations for Sisko **fluid** **over** 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 **heat** **transfer** of Sisko **fluid** involve only one dimensional flows **and** literature survey indicates that no work has so far been communicated **with** regards to **heat** **transfer** **in** a boundary layer **flow** for Sisko **fluid** **over** a nonlinear **stretching** **sheet** **with** **variable** surface temperature **and** **variable** **heat** flux.

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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 **fluid** **and** the boundary parameter n varies **in** the range 0ƒnƒ1: Here n~0 corresponds to the situation when microelements at the **stretching** **sheet** 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.

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