The boundary layer **flow** past a **stretching** **surface** in the presence **of** a magnetic field has much practical relevance in polymer processing and in other several industrial processes. Along **with** this, a new dimension is added to the study **of** **flow** and heat transfer **effects** **over** a **stretching** **surface** by considering the effect **of** **thermal** **radiation**. **Thermal** **radiation** **effects** may play an important role in controlling heat transfer in industry where the quality **of** the final product depends to a great extend **on** the heat controlling factors and the knowledge **of** radiative heat transfer in the system can perhaps lead to a desired product **with** sought qualities. High temperature plasmas, cooling **of** nuclear reactors and liquid metal fluids are some important applications **of** radiative heat transfer. The radiative **flow** **of** an electrically conducting fluid **with** high temperature in the presence **of** a magnetic field are encountered in electrical power generation, astrophysical flows, solar power technology, space vehicle re-entry, nuclear engineering applications and in other industrial areas.

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design **of** heat exchangers, induction pumps, and nuclear reactors, in oil exploration and in space vehicle propulsion. **Thermal** **radiation** in fluid dynamics has become a significant branch **of** the engineering sciences and is an essential aspect **of** various scenarios in mechanical, aerospace, chemical, environmental, solar power and hazards engineering. Bhaskara Reddy and Bathaiah [18, 19] analyze the Magnetohydrodynamic free convection laminar **flow** **of** an incompressible Viscoelastic fluid. Later, he was studied the MHD combined free and forced convection **flow** through two parallel porous walls. Elabashbeshy [20] studied heat and mass transfer along a vertical plate in the presence **of** magnetic field. Samad, Karim and Mohammad [21] calculated numerically the effect **of** **thermal** **radiation** **on** steady MHD free convectoin **flow** taking into account the Rosseland diffusion approximaion. Loganathan and Arasu [22] analyzed the **effects** **of** thermophoresis particle deposition **on** non-Darcy MHD mixed convective heat and mass transfer past a porous wedge in the presence **of** suction or injection. Ghara, Maji, Das, Jana and Ghosh [23] analyzed the **unsteady** MHD Couette **flow** **of** a viscous fluid between two infinite non-conducting horizontal porous plates **with** the consideration **of** both Hall currents and ion-**slip**. The **radiation** effect **on** steady free convection **flow** near isothermal **stretching** sheet in the presence **of** magnetic field is investigated by Ghaly et al. [24]. Also, Ghaly [25] analyzed the effect **of** the **radiation** **on** heat and mass transfer **on** **flow** and **thermal** field in the presence **of** magnetic field for horizontal and inclined plates.

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Rajagopal (1983) studied the heat transfer in the forced convection **flow** **of** a visco-elastic fluid **of** Walter’s model. Rahmann and Sarkar (2004) investigated the **unsteady** MHD **flow** **of** a viscoelastic Oldroyd fluid under time varying body forces through a rectangular channel. Singh and Singh (1996) analyzed MHD **flow** **of** a dusty visco-elastic (Oldroyd B-liquid) through a porous medium between two parallel plates inclined to the horizon. Ibrahim et. al. (2004) discussed the **flow** **of** a viscoelastic fluid between coaxial rotating porous disks **with** uniform suction or injection .Oscillatory motion **of** an electrically conducting visco-elastic fluid **over** a **stretching** sheet in a saturated porous medium was studied by Rajagopal (2006). Prasuna et al. (2010) examined an **unsteady** **flow** **of** a visco-elastic fluid through a porous media between two impermeable parallel plates. When the strength **of** the magnetic field is strong one cannot neglect the **effects** **of** the Hall currents. It is **of** considerable importance and interest to study how the results **of** the hydro dynamical problems are modified by the **effects** **of** the Hall currents. Singh and Kumar (2010) examined the exact solution **of** an oscillatory MHD **flow** through a porous medium bounded by rotating porous channel in the presence **of** Hall current. Chaudhary and Jain (2006) investigated the **effects** **of** Hall current and **radiation** **on** MHD mixed convection **flow** **of** a visco- elastic fluid past an infinite vertical plate. Biswal and Sahoo (1999) also studied Hall current **effects** **on** free convective **hydromagnetic** **flow** **of** visco-elastic fluid past an infinite vertical plate.

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w w w . a j e r . o r g Page 69 The boundary layer **flow** **over** a shrinking **surface** is encountered in several technological processes. Such situations occur in polymer processing, manufacturing **of** glass sheets, paper production, in textile industries and many others. Crane [4] initiated a study **on** the boundary layer **flow** **of** a viscous fluid towards a linear **stretching** sheet. An exact similarity solution for the dimensionless differential system was obtained. Carragher and Carane [5] discussed heat transfer **on** a continuous **stretching** sheet. Afterwards, many investigations were made to examine **flow** **over** a **stretching**/shrinking sheet under different aspects **of** MHD, suction/injection, heat and mass transfer etc. [6 –13]. In these attempts, the boundary layer **flow**, due to **stretching**/shrinking has been analyzed. Magyari and Keller [14] provided both analytical and **numerical** solutions for boundary layer **flow** **over** an exponentially **stretching** **surface** **with** an exponential temperature distribution. The combined **effects** **of** viscous dissipation and mixed convection **on** the **flow** **of** a viscous fluid **over** an exponentially **stretching** sheet were analyzed by Partha et al. [15], Elbashbeshy [16] numerically studied **flow** and heat transfer **over** an exponentially **stretching** **surface** **with** wall mass suction. Madhu.M and Naikoti Kishan[17] studied the Two-dimensional MHD mixed convection boundary layer **flow** **of** heat and mass transfer stagnation-point **flow** **of** a non-Newtonian power-law nanofluid towards a **stretching** **surface** in the presence **of** **thermal** **radiation** and heat source/sink.

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It is well known that the characteristics **of** heat transfer are dependent **on** the **thermal** boundary conditions. Here a conjugate convective type **flow** or Newtonian heating is considered. Newtonian heating is a kind **of** wall-to-ambient heating process where the rate **of** heat transfer from the bounding **surface** **with** a finite heat capacity is proportional to the local **surface** temperature. This type **of** situation occurs in many important engineering devices such as in heat exchangers, gas turbines and also in seasonal **thermal** energy storage systems. Therefore, the interaction **of** conduction-convection coupled **effects** is **of** much significance from practical point **of** view and it must be considered when evaluating the conjugate heat transfer processes in many engineering applications. Merkin (1994) initiated the study **of** free convection boundary layer **flow** **over** a vertical **surface** **with** Newtonian heating while Lesnic et al. (1999, 2000) analyzed free convection boundary layer **flow** past vertical and horizontal surfaces in a porous medium generated by Newtonian heating. Chaudhary and Jain (2006) investigated **unsteady** free convection **flow** past an impulsively started vertical plate **with** Newtonian heating. Salleh et al. (2009) discussed forced convection boundary layer **flow** at a forward stagnation point **with** Newtonian heating. Narahari and Ishak (2011) investigated the **effects** **of** **thermal** **radiation** **on** **unsteady** free convection **flow** **of** an optically thick fluid past a moving vertical plate **with** Newtonian heating. They considered three cases **of** interest, namely, (i) impulsive movement **of** the plate; (ii) uniformly accelerated movement **of** the plate and (iii) exponentially accelerated

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Heat transfer in a radiating fluid **with** slug **flow** in a parallel plate channel was in- vestigated by Viskanta [18] who formulated the problem in terms **of** integro-differential equa- tions and solved by an approximate method. Helliwell [19] discussed the stability **of** thermally radiative magnetofluiddynamic channel **flow**. Elsayed et al. [20] provided **numerical** solution for simultaneous forced convection and **radiation** in parallel plate channel and presented anal- ysis for the case **of** non-emitting “blackened” fluid. Helliwell et al. [21] discussed the radia- tive heat transfer in horizontal magnetohydrodynamic channel **flow** considering the buoyancy **effects** and an axial temperature gradient. Elbashbeshy et al. [22] studied heat transfer **over** an **unsteady** **stretching** **surface** embedded in a porous medium in the presence **of** **thermal** radia- tion and heat source or sink. The viscous heating aspects in fluids were investigated for its practical interest in polymer industry and the problem was invoked to explain some rheologi- cal behavior **of** silicate melts. The importance **of** viscous heating has been demonstrated by Gebhart [23], Gebhart et al. [24], Magyari et al. [25], and Rees et al. [26].

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The study **of** boundary layer **flow** **over** porous **surface** moving **with** constant velocity in an ambient fluid was initiated by Sakiadis [1]. Erickson et al. [2] extended Sakiadis [1] problem to include blowing or suction at the moving porous **surface**. Subsequently Tsou et al. [3] presented a combined analytical and experimental study **of** the **flow** and temperature fields in the boundary layer **on** a continuous moving **surface**. R. Ellahi et al. [4] investigated **numerical** analysis **of** **unsteady** flows **with** viscous dissipation and nonlinear **slip** **effects**. Excellent reviews **on** this topic are provided in the literature by Nield and Bejan [5], Vafai [6], Ingham and Pop [7] and Vadasz [8]. Recently, Cheng and Lin [9] examined the melting effect **on** mixed convective heat transfer from a permeable **over** a continuous **Surface** embedded in a liquid saturated porous medium **with** aiding and opposing external flows. The **unsteady** boundary layer **flow** **over** a **stretching** sheet has been studied by Devi et al. [10], Elbashbeshy and Bazid [11], Tsai et al. [12] and Ishak [13].

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Heat generation/absorption **effects** **on** the free convective boundary layer **flow** **of** a viscous incompressible, electrically conducting fluid under the influence **of** a magnetic field are encountered in several industrial applications, such as underground disposal **of** radioactive waste materials, exothermic and/or endothermic chemical reactions, heat removal from nuclear fuel debris and dissociating flu-ids in packed-bed reactors etc. Due to this fact several researchers studied the **effects** **of** heat absorption by the fluid **on** the **flow** and heat transfer **of** a viscous, incompressible and electrically conducting fluid in the presence **of** a magnetic field. Chamkha (2000a) investigated magneto- hdyrdynamic (MHD) boundary layer **flow** **over** an accelerating permeable **surface** in the presence **of** **thermal** **radiation**, **thermal** buoyancy force and heat generation or absorption. It was found that heat absorption coefficient reduces fluid temperature which resulted in decrease in the fluid velocity. The rate **of** heat transfer decreases as the heat absorption coefficient increases. Also heat absorption coefficient has tendency to reduce the rate **of** heat transfer. Ibrahim et al. (2008) discussed the **effects**

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Some specific industrial applications such as in polymer processing technology that involves cooling **of** continuous strip or filaments. During the process, strips are sometimes stretched. The properties **of** the final product depend **on** the rate **of** cooling. The rate **of** cooling can be controlled by the use **of** electrically conducting fluid **with** the application **of** the magnetic field. Numerous attempts have been made to analyze the **effects** **of** transverse magnetic field **on** the boundary layer **flow**, heat and mass transfer characteristics **of** electrically conducting fluid. Vajravelu and Rollins (1992) studied heat transfer in an electrically conducting fluid **over** a **stretching** **surface** by taking into account **of** magnetic field. Mostafa et al. (2012) investigated the MHD **flow** and heat transfer **of** a micropolar fluid **over** a **stretching** **surface** **with** heat generation/absorption and **slip** velocity. It is well known that the heat transfer due to concentration gradient is called the Dufour effect (or diffusion-thermo) whereas the mass transfer caused by temperature gradient is called Soret effect (or **thermal**-diffusion). In other words, Soret effect is referred to the species differentiation developed in an initial homogeneous mixture submitted to a **thermal** gradient whereas Dufour effect is referred to heat flux produced by the concentration gradient. Alam et al. (2006) studied the Soret and Dufour **effects** **on** a steady MHD combined free-forced convective and mass transfer **flow** past a semi- infinite vertical plate. Postelnicu (2004) discussed the influence **of** a magnetic field **on** heat and mass transfer by natural convection from a vertical **surface** in porous media in the presence **of** Soret and Dufour **effects**. Pal and Chatterjee (2011) analyzed mixed convection magnetohydrodynamic heat and mass transfer past a **stretching** **surface** in a micropolar fluid--saturated porous medium in the presence **of** Ohmic heating, Soret and Dufour **effects**. Reddy and Rao (2012) analyzed thermo- diffusion and diffusion –thermo **effects** **on** convective heat and mass transfer through a porous medium in a circular cylindrical annulus **with** quadratic density temperature variation. Recently, Gangadha (2013) studied Soret and Dufour **effects** **on** hydro magnetic heat and mass transfer **over** a vertical plate **with** a convective **surface** boundary condition and chemical reaction.

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This study presents a **numerical** analysis **on** the **effects** **of** Soret, variable **thermal** conductivity, viscous-Ohmic dissipation, non-uniform heat sources, **on** steady two-dimensional **hydromagnetic** mixed convective heat and mass transfer **flow** **of** a micropolar fluid **over** a **stretching** sheet embedded in a non-Darcian porous medium **with** **thermal** **radiation** and chemical reaction. The governing differential equations are transformed into a set **of** non-linear coupled ordinary differential equations which are then solved numerically by using the fifth- order Runge-Kutta-Fehlberg method **with** shooting technique. **Numerical** solutions are obtained for the velocity, angular velocity, temperature and concentration profiles for various parametric values, and then results are presented graphically as well as skin-friction coefficient, and also local Nusselt number and local Sherwood number for different physical parameters are shown graphically and in tabular form. A critical analysis **with** earlier published papers was done, and the results were found to be in accordance **with** each other.

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Copyright: © 2015 Ahmad Khan et al. This is an open access article distributed under the terms **of** the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Based on the present predictions it seems clear that two-dimensional flow around the cylinder undergoes a bifurcation from a steady two-dimensional regime to an unsteady periodic twodime[r]

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Problems involving free-**surface** inviscid **flow** **over** an obstacle are studied by many re- searchers to model various situations arising in oceanography and atmospheric sciences. The study **of** such **flow** problems is important to an- alyze the qualitative insight **of** the mechanism **of** wave generation by submerged bodies. Various mathematical techniques have been employed to study free-**surface** flows **over** different kinds **of** obstacles situated at the bottom **of** a channel. For example, Lamb (1932) studied the **flow** **over** a cylindrical obstruction lying **on** the bottom and calculated the drag force **on** the obstruc- tion. Forbes and Schwartz (1982) considered the **flow** **over** a semi-circular obstruction and calculated the wave resistance offered by the semicircle. Vanden-Broeck (1987) solved nu- merically the problem **of** Forbes and Schwartz (1982), and discussed the existence **of** the su- percritical solutions. Forbes (1988) presented a **numerical** solution for critical free-**surface** **flow** **over** a semicircular obstruction attached to the bottom **of** a running stream. Dias and Vanden- Broeck (1989) studied the problem involving free-**surface** **flow** past a submerged triangular obstacle at the bottom **of** a channel and solved

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Let us consider an **unsteady** electrically conducting viscous incompressible, laminar fluid **flow** through a vertical porous plate. The fluid is assumed to be in the x-direction which is taken along the porous plate in upward direction and y-axis is normal to it. Let the **unsteady** fluid **flow** starts at t=0 afterward the whole frame is allowed to rotate about y-axis **with** t>0,the plate started to move in its own plate **with** constant velocity U and temperature **of** the plate is raised to T w to T ∞ A strong uniform

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T HE fluid in a hydrodynamic torque converter (H.T.C.) is responsable for the torque conversion and power transfer from the engine to the transmission and influences the propulsion efficiency. H.T.C. are commonly used in vehicle power transmission systems such as cars, buses and locomotives. Its typical configuration consists **of** a pump, driven by the engine that transmits the generated angu- lar momentum, a turbine that transmits the torque to the transmission and a stator, which makes possible the torque conversion through the redirection **of** the **flow** to the pump. Some advantages are the capacity to provide torque am- plification during the start-up conditions, a soft start from standstill and the capacity for damping transmission through the absorbtion **of** torsional vibrations introduced from the engine. One disadvantage is the higher fuel consumption and lower efficiency compared **with** the gear transmission. So it is necessary to optimize its operating work through understanding the **flow** behaviour. The internal **flow** within H.T.C. is three-dimensional, turbulent, viscous, complex, highly **unsteady** and difficult to analize because **of** the operating conditions that consist **of** three elements rotating at different velocities.

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Here we choose the Cartesian coordinate system in such a manner that the x − axis is along the stretched sheet and the y − axis is normal to it. We consider the MHD mixed convection **flow** **of** a vis- cous fluid **with** convective boundary conditions **over** a porous **stretching** sheet. All the physical properties **of** the fluid except the **thermal** conductivity are taken to be constant. The **thermal** conductivity varies linearly **with** the temperature. Neglecting the radia- tive and viscous dissipation **effects**, the boundary layer equations are:

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Since the 1970s, computational modelling **of** fluid **flow** through porous media has increased rapidly [1, 2]. Porous media are diverse and include different scales. The flows through macro and micro-scales are not the same. As flows approach microscopic scales, increasing deviations from the well-established continuum laws are reported [3, 4]. The K nudsen number, deﬁned as the ratio **of** the molecular mean free path to the characteristic length **of** pores, allows establishing four regimes [5]: 0 < Kn 0.001 (no-**slip**), 0.001 < Kn 0.1 (**slip**), 0.1 < Kn 10 (transition), and Kn > 10 (free molecular, ballistic). Navier-Stokes equation is only adequate for no-**slip** regime and can be extended into the **slip**-**flow** regime provided the velocity **slip** and temperature jump boundary conditions [4, 6]. In this regard, the limit **of** validity **of** the continuum **flow** description is Kn 0.1. Discrete Boltzmann models are based **on** a kinetic representation **of** the fluid dynamics, and avoid the drawbacks associated to the conventional Navier-Stokes description. Lattice-Boltzmann equation method (LBM) is appropriate for complexes geometries and covers all these four regimes (i. e., is also valid for transition and free molecular regime) [7, 8]. Within the continuum regime, LBM has been shown to be equivalent to a finite difference approximation **of** the incompressible Navier-Stokes equation [8].

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The basic equations for the LES model are the spatially fil- tered continuity equation, Navier-Stokes equation and the transport equation for concentration. The subgrid-scale (SGS) Reynolds stress is parameterized by using the standard Smagorinsky model (Smagorinsky, 1963) **with** a Van Driest damping function (Van Driest, 1956), where the Smagorin- sky constant is set to 0.12 (Iizuka and Kondo, 2004) for es- timating the eddy viscosity. The subgrid-scale scalar flux is also parameterized by an eddy viscosity model and the tur- bulent Schmidt number is set to 0.5. Various SGS models for LES have been proposed besides the standard Smagorin- sky model. For example, dynamic Smagorinsky models have been proposed by Germano et al. (1991), Lilly (1992), and Meneveau et al. (1996). However, Iizuka and Kondo (2004) examined the influence **of** various SGS models **on** the predic- tion accuracy **of** LESs **of** turbulent **flow** **over** a hilly terrain and showed that the prediction accuracy **of** LESs **with** the standard Smagorinsky model is better than that **of** the LESs **with** the dynamic Smagorinsky type models. This indicates that the dynamic Smagorinsky type models are not always effective for determining model constant. As a static type SGS model, Nicoud and Ducros (1999) proposed the wall- adapting local eddy-viscosity (WALE) model. This model can capture the **effects** **of** both the strain and the rotation rate **of** the small-scale turbulent motions without a damping func- tion from the wall. According to Temmerman et al. (2003), the WALE model shows better performance when compared **with** the dynamic/standard Smagorinsky model. However, the conventional Smagorinsky model that has the advantage **of** simplicity and low computational costs is adopted in our LES model because the focus **of** our research is not the small order **effects** **of** turbulent **flow**.

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Focus in this section is **on** the airfoil wake in a pitch sinusoidal motion. Figure 17 shows the instantaneous voltages **of** three sensors in α(t) = 3 + 3 sin (2πt/T – π/2) pitch oscillation motion at frequency **of** 3 Hz. Comparing consecutive cycles reveals their repeatability **with** an appropriate precision; if a complete cycle is considered as a pitch up-pitch down sequence, then every sensor will cover the same pattern through all cycles. Deviations may appear only in very small fluctuations, which are likely related to noise or any other unsteadiness that may instantaneously and irregularly affect the sensors voltages, but not to the **flow** physics.

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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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