The boundary layer flow past a stretchingsurface 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 offlow and heat transfer effectsover a stretchingsurface by considering the effect ofthermalradiation. Thermalradiationeffects 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 flowof 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.
design of heat exchangers, induction pumps, and nuclear reactors, in oil exploration and in space vehicle propulsion. Thermalradiation 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 flowof an incompressible Viscoelastic fluid. Later, he was studied the MHD combined free and forced convection flow through two parallel porous walls. Elabashbeshy  studied heat and mass transfer along a vertical plate in the presence of magnetic field. Samad, Karim and Mohammad  calculated numerically the effect ofthermalradiationon steady MHD free convectoin flow taking into account the Rosseland diffusion approximaion. Loganathan and Arasu  analyzed the effectsof 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  analyzed the unsteady MHD Couette flowof 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. . Also, Ghaly  analyzed the effect of the radiationon heat and mass transfer onflow and thermal field in the presence of magnetic field for horizontal and inclined plates.
Rajagopal (1983) studied the heat transfer in the forced convection flowof a visco-elastic fluid of Walter’s model. Rahmann and Sarkar (2004) investigated the unsteady MHD flowof a viscoelastic Oldroyd fluid under time varying body forces through a rectangular channel. Singh and Singh (1996) analyzed MHD flowof 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 flowof 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 unsteadyflowof 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 effectsof the Hall currents. It is of considerable importance and interest to study how the results of the hydro dynamical problems are modified by the effectsof 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 effectsof Hall current and radiationon MHD mixed convection flowof a visco- elastic fluid past an infinite vertical plate. Biswal and Sahoo (1999) also studied Hall current effectson free convective hydromagneticflowof visco-elastic fluid past an infinite vertical plate.
w w w . a j e r . o r g Page 69 The boundary layer flowover 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  initiated a study on the boundary layer flowof a viscous fluid towards a linear stretching sheet. An exact similarity solution for the dimensionless differential system was obtained. Carragher and Carane  discussed heat transfer on a continuous stretching sheet. Afterwards, many investigations were made to examine flowover 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  provided both analytical and numerical solutions for boundary layer flowover an exponentially stretchingsurfacewith an exponential temperature distribution. The combined effectsof viscous dissipation and mixed convection on the flowof a viscous fluid over an exponentially stretching sheet were analyzed by Partha et al. , Elbashbeshy  numerically studied flow and heat transfer over an exponentially stretchingsurfacewith wall mass suction. Madhu.M and Naikoti Kishan studied the Two-dimensional MHD mixed convection boundary layer flowof heat and mass transfer stagnation-point flowof a non-Newtonian power-law nanofluid towards a stretchingsurface in the presence ofthermalradiation and heat source/sink.
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 surfacewith 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 flowover a vertical surfacewith 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 effectsofthermalradiationonunsteady free convection flowof 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
Heat transfer in a radiating fluid with slug flow in a parallel plate channel was in- vestigated by Viskanta  who formulated the problem in terms of integro-differential equa- tions and solved by an approximate method. Helliwell  discussed the stability of thermally radiative magnetofluiddynamic channel flow. Elsayed et al.  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.  discussed the radia- tive heat transfer in horizontal magnetohydrodynamic channel flow considering the buoyancy effects and an axial temperature gradient. Elbashbeshy et al.  studied heat transfer over an unsteadystretchingsurface embedded in a porous medium in the presence ofthermal 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 , Gebhart et al. , Magyari et al. , and Rees et al. .
The study of boundary layer flowover porous surface moving with constant velocity in an ambient fluid was initiated by Sakiadis . Erickson et al.  extended Sakiadis  problem to include blowing or suction at the moving porous surface. Subsequently Tsou et al.  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.  investigated numerical analysis ofunsteady flows with viscous dissipation and nonlinear slipeffects. Excellent reviews on this topic are provided in the literature by Nield and Bejan , Vafai , Ingham and Pop  and Vadasz . Recently, Cheng and Lin  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 flowover a stretching sheet has been studied by Devi et al. , Elbashbeshy and Bazid , Tsai et al.  and Ishak .
Heat generation/absorption effectson the free convective boundary layer flowof 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 effectsof 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 flowover an accelerating permeable surface in the presence ofthermalradiation, 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
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 effectsof 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 stretchingsurface by taking into account of magnetic field. Mostafa et al. (2012) investigated the MHD flow and heat transfer of a micropolar fluid over a stretchingsurfacewith 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 effectson 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 stretchingsurface 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 effectson 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 effectson hydro magnetic heat and mass transfer over a vertical plate with a convective surface boundary condition and chemical reaction.
This study presents a numerical analysis on the effectsof Soret, variable thermal conductivity, viscous-Ohmic dissipation, non-uniform heat sources, on steady two-dimensional hydromagnetic mixed convective heat and mass transfer flowof a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium withthermalradiation 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.
Problems involving free-surface inviscid flowover 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 flowover a cylindrical obstruction lying on the bottom and calculated the drag force on the obstruc- tion. Forbes and Schwartz (1982) considered the flowover 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-surfaceflowover a semicircular obstruction attached to the bottom of a running stream. Dias and Vanden- Broeck (1989) studied the problem involving free-surfaceflow past a submerged triangular obstacle at the bottom of a channel and solved
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
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.
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 flowof 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:
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 : 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 .
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 flowover 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 effectsof 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 effectsof turbulent flow.
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.
considered the boundary layer flowof 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 flowover a continuous surface and experimentally verified Sakiadis’ results. Erickson et al. (1966) included the heat and mass transfer on a stretchingsurfacewith suction or injection. Many researchers later investigated boundary layer flowover a stretchingsurface, such as Gupta and Gupta (1977), Ishak (2008), and Nadeem (2010).