Top PDF MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity

MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity

MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity

In the above mentioned studies, the effect of slip condition has not been taken into consideration. Navier [21] proposed a slip boundary condition where the slip velocity depends linearly on the shear stress. Since then the effects of slip velocity on the boundary layer flow of Newtonian and non-Newtonian fluids have been studied by several authors. Ariel [22] studied the flow of an elastico-viscous fluid past a stretching sheet with partial slip. Hayat et al. [23] analytically studied the solutions of the equations of motion and energy of a second grade fluid for the developed flow over a stretching sheet with slip condition. Hayat et al. [24] studied the effect of the slip condition on flows of an Oldroyd 6-constant fluid. Roux [25] discussed the solvability of the equations of motion for an incompressible fluid of grade two subject to nonlinear partial slip boundary conditions in a bounded simply-connected domain. Liakos [26] studied the steady-state, isothermal flow of a viscoelastic fluid obeying an Oldroyd-type constitutive law with slip bound- ary condition. Asghar et al. [27] analytically studied the rotating flow of a third grade fluid past a porous plate with partial slip effects. Khan [28] presented the exact analytical solutions for three basic fluid flow problems in a porous medium when the no-slip condition is no longer valid. Asghar et al. [29] obtained the exact analytical solutions for general periodic flows of a second grade fluid in the presence of partial slip and a porous medium. Mahmoud [30] studied the effects of slip velocity on flow and heat transfer of a non-Newtonian power-law fluid on a stretching surface with thermal radiation. Hayat et al. [31] studied the flow and heat transfer of a Newtonian fluid over an unsteady permeable stretching sheet with slip conditions. Mahmoud and Waheed [32] investigated the effect of slip veloc- ity on mixed convection flow of a micropolar fluid over a heated stretching surface in the presence of a uniform magnetic field and heat generation or absorption. The aim of the present work is to investigate the flow and heat transfer of an electrically conducting micropolar fluid over a permeable stretching surface with variable heat flux in the presence of slip velocity at the surface, heat generation (absorption) and a uniform transverse magnetic field.
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EFFECTS OF HOMOGENEOUS-HETEROGENEOUS CHEMICAL REACTION AND SLIP VELOCITY ON MHD STAGNATION FLOW OF A MICROPOLAR FLUID OVER A PERMEABLE STRETCHING/SHRINKING SURFACE EMBEDDED IN A POROUS MEDIUM

EFFECTS OF HOMOGENEOUS-HETEROGENEOUS CHEMICAL REACTION AND SLIP VELOCITY ON MHD STAGNATION FLOW OF A MICROPOLAR FLUID OVER A PERMEABLE STRETCHING/SHRINKING SURFACE EMBEDDED IN A POROUS MEDIUM

boundary layer flow of a viscous fluid flow past a flat plate was studied by Merkin (1996). Effects of homogeneous and heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness was reported by Hayat et al., (2016) and observed that the homogenous and heterogeneous parameters have opposite behaviors for concentration profile. Ziabakhsh et al. (2010) studied the problem of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos (1956) studied an isothermal chemical reaction on a catalytic in a laminar boundary layer flow. They found the actual surface concentration without introducing unnecessary assumptions related to the reaction mechanism. The effect of flow near the two- dimensional stagnation point flow on an infinite permeable wall with a homogeneous-heterogeneous reaction was studied by Khan and Pop (2010). They solved the governing nonlinear equations using the implicit finite difference method. It was observed that the mass transfer parameter considerably affects the flow characteristics. Melting and homogeneous/heterogeneous reactions effects in nanofluid flow by a cylinder are addressed by Hayat et al., (2016). It is found that maximum heat transfer and minimum thermal resistance for base fluid suspended multi-wall carbon nanotubes (MWCNTs) when compared with other nanofluids. The behavior of homogeneous parameter K on concentration profile is sketched for water and kerosene oil base fluids by Hayat et al., (2016). It is analyzed that the concentration field is decreasing function of homogeneous parameter K for base fluids water and kerosene oil. In fact higher values of homogeneous reaction parameter correspond to larger chemical reaction which consequently reduces the concentration distribution. Hayat et al., (2016) developed numerical analysis for homogeneous-heterogeneous reactions and Newtonian heating in magnetohydrodynamic (MHD) flow of Powell- Eyring fluid by a stretching cylinder and noticed that the flow accelerates for large values of Powell-Eyring fluid parameter. Hayat et al., (2016) disclose the effects of homogeneous–heterogeneous reactions and melting heat phenomenon in the Magnetohydrodynamic second grade fluid flow. Heat transfer is tackled with heat generation/absorption. Khan and Pop (2012) studied the effects of homogeneous-heterogeneous reactions on the viscoelastic fluid toward a stretching sheet. They observed that the concentration at the surface decreased with an increase in the viscoelastic parameter.
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Heat Transfer in MHD Flow over A Stretching Sheet with Velocity and Thermal Slip Condition

Heat Transfer in MHD Flow over A Stretching Sheet with Velocity and Thermal Slip Condition

Keywords: MHD; Heat transfer; Slip conditions, Kumer’s function, Similarity transformation 1. Introduction The problem of flow and heat transfer in the boundary layer induced by a stretching surface in an otherwise ambient fluid is important in many industrial applications, such as the extrusion of plastic sheets, electronic chips, glass blowing, continuous casting and spinning of fibers.Crane [1] is the first to investigate analytically the problem of boundary layer flow of an incompressible viscous fluid over a linearly stretching surface. This problem was then extended by many authors. Gupta and Gupta[2] studied the effect of suction/blowing on heat and mass transfer over a stretching surface. Grubka and Bobba[3] analyzed heat transfer characteristics of a continuous stretching surface with variable temperature. Dutta et al.[4] studied the temperature field in the flow over a stretching sheet with uniform heat flux. Cortell[5] studied the effects of heat generation/absorption and suction/blowing on the flow and heat transfer of a fluid through a porous medium over a stretching surface.
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Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet in a micropolar fluid

Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet in a micropolar fluid

lyzed the magnetohydrodynamics (MHD) convective flow of a micropolar fluid past a continu- ously moving vertical porous plate in the presence of heat generation/absorption. Eldabe and Ouaf [37] studied the heat and mass transfer past a stretching horizontal surface immersed in an electrically conduction fluid which are solved numerically using Chebyshev finite difference method. Chen [38] considered the problem of MHD flow and heat transfer of an electrically conducting of a non-Newtonian power-law fluid past a stretching sheet in the presence of a transverse magnetic field. Recently, Ishak et al. [39] studied the steady 2-D MHD stagnation point flow towards a stretching sheet with variable surface temperature. Motivated by the above-mentioned investigations, the present paper considers the problem of hydromagnetic flow and heat transfer past a stretching vertical sheet immersed in an incompressible micropolar fluid.
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MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium

MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium

or absorption. The stretching sheet problem for viscoelastic Oldroyd-B model was extended by Rajagopal and Bhatnagar [8] . Cortell [9] presented a study of flow and heat transfer of hydromagnetic second-order fluid past a stretching sheet. In another paper [10] , Cortell included magnetic and suction effects while discussing the flow and heat transfer of second grade fluid due to stretching sheet. The study of stagnation- point flow of micropolar fluid toward a stretching sheet was initiated by Nazar et al. [11] . Combined forced and nat- ural convection flow of micropolar fluid over a stretching sheet in the vicinity of stagnation point was discussed by Ishak et al. [12] . Ariel [13] put forward a study of flow of sec- ond grade fluid past a radially stretching sheet. Ariel work was extended by Hayat and Sajid [14] to include the heat transfer characteristics. Some other contribution by Hayat and his co-workers regarding heat transfer analysis over a stretching surface can be found in references [15–18] . The investigation of non-similar flow of third order fluid over a stretching sheet was due to Sajid and Hayat [19] . Hayat et al. [20] analyzed combined heat and mass transfer charac- teristics in boundary layer flow of viscoelastic fluid over lin- early stretching vertical surface by using homotopy analysis method. Later on, Turkyilmazoglu [21] computed exact solu- tion of hydromagnetic flow of viscoelastic fluid in the pres- ence of Soret and Dufour effects. Turkyilmazoglu [22] also developed closed-form solutions for three dimensional flow and heat transfer of a electrically conducting viscoelastic fluid over a porous stretching/shrinking sheet. The multiplicity of the solutions in boundary layer flows over stretching/shrink- ing sheet is also addressed by Turkyilmazoglu. In fact, it was shown by him that multiple solutions exist for two- dimensional flow over stretching sheet and in the case of shrinking sheet triple solutions can be detected [23–25] . In another paper, Turkyilmazoglu [26] computed exponential series type solution in the hydromagnetic and heat transfer in two–three dimensional flow over stretching or shrinking surface. According to him exponential series type solution exists with finite and infinite terms in the expansion depend- ing upon prevailing parameters in the governing equations.
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MHD FLOW OVER A STRETCHING CONVECTIVE SURFACE WITH DUFOUR EFFECT, SLIP AND RADIATIVE HEAT

MHD FLOW OVER A STRETCHING CONVECTIVE SURFACE WITH DUFOUR EFFECT, SLIP AND RADIATIVE HEAT

stagnation point flow of a nanofluid over an exponentially stretching sheet. Mahantha et al. [6] discussed MHD stagnation point flow of a nanofluid with velocity slip, Non-linear radiation and Newtonian heating. Mabood et al. [7] explained MHD stagnation point flow heat and mass transfer of nanofluids in porous medium with radiation, viscous dissipation and chemical reaction. Mohammad et al. [8] investigated MHD convective stagnation point flow of nanofluid over a shrinking surface with thermal radiation, heat radiation and chemical reaction. Stagnation point flow towards a stretching surface studied by Mahapatra and Gupta [9].Sreenivasulu and Reddy [10] have analyzed the effects of thermal radiation and chemical reaction on MHD stagnation point flow of a nanofluid over a porous stretching sheet embedded on a porous medium with heat absorption/ generation.
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MHD mixed convective Micropolar Fluid flow over an exponentially stretching sheet with heat source and Mass transfer

MHD mixed convective Micropolar Fluid flow over an exponentially stretching sheet with heat source and Mass transfer

Numerical analysis of a steady two-dimensional hydromagnetic stagnation point flow of an electrically conducting nanofluid past in a stretching surface with induced magnetic field was studied by Gireeshaet.al 20 . KhairyZaimiet.al 21 discussed the influence of partial slip on stagnation point flow and heat transfer processes towards a stretching vertical sheet and the problem was solved numerically by using the shooting method. Sheri Siva Reddyet.al 22 analyzed boundary layer analysis on an unsteady MHD free convective micropolar fluid flow past in a semi-infinite vertical porous plate with the presence of diffusion Thermo, heat absorption, and homogeneous chemical reaction. Shahirah Abu Bakeret.al 23 investigated a mathematical analysis of forced convective boundary layer stagnation-point slip flow in Darcy-Forchheimer porous medium through a shrinking sheet.Sahin.Ahmad.et.al 24 analyzed perturbation analysis of combined heat and mass transfer processes in MHD steady mixed convective flow of an incompressible, viscous, Newtonian, electrically conducting and chemical reacting fluid past over an infinite vertical porous plate in presence of the homogeneous chemical reaction of first order.
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MHD Convection Fluid Flow and Heat Transfer in an Inclined Microchannel with Heat Generation

MHD Convection Fluid Flow and Heat Transfer in an Inclined Microchannel with Heat Generation

electrically conducting and heat generating fluids through channel or pipe flow attracted many scholars due to its used in many applications such as Magnetohydrodynamic (MHD) generators, pumps, accelerators and flow meter. The MHD channel flow was first investigated by Hartmann [17]. Osterle and Young [19] later investigated the natural convection between heated vertical plates with magnetic field. Seigal [18], Perlmutter and Siegel [20], Romig [21], Affiliated with Department of Mathematics, Indian Institute of Technology Chamkha [11], Umavathi [22] and Naseem Ahmad et al [23]. have presented detailed analysis of free and forced convection heat transfer to in the presence of magnetic field for the vertical channel. Recently Chamkha et al. [12, 13] studied the MHD free convection flow of a nanofluid past a vertical Plate in the presence of heat generation or absorption effects. B. K. Jha et al [10], investigated the MHD natural convection in a vertical parallel plate microchannel considering velocity slip and temperature jump conditions at the fluid- wall interface. Hasan Nihal Zaidi and Naseem Ahmad had been discussed MHD convection flow of two immiscible fluids in an inclined channel with heat generation or absorption [15]. Chi Chuan Wang et al [16] experimentally investigated the effect of inclination on the convective boiling performance of a microchannel heat sink. They found that the heat transfer coefficient for 45 upward considerably exceeds other configurations.
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Slip Flow of a Casson Fluid Flow over an Exponentially Stretching Surface and Heat and Mass Transfer

Slip Flow of a Casson Fluid Flow over an Exponentially Stretching Surface and Heat and Mass Transfer

occurrence in many branches of science and engineering. Chambre and Young [14] have presented a first order chemical reaction in the neighborhood of a horizontal plate. Akyildiz et al. [15] analyzed the transport and diffusion of chemically reactive species over a stretching sheet in a non-Newtonian fluid (fluid of differential type, second grade). Muthcumaraswamy and Ganesan [16] studied effect of the chemical reaction and injection on the flow characteristics in an unsteady upward motion of an isothermal plate. Chamkha [17] studied the MHD flow of a numerical of uniform stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction. Seddeek et al.[18] analyzed the effects of chemical reaction, radiation and variable viscosity on hydromagnetic mixed convection heat and mass transfer Hiemenz flow through porous media. Gangadhar [19] studied the heat and mass transfer over a vertical plate with a convective surface boundary condition and chemical reaction and also conclude that the spices boundary layer thickness decreases with increasing the chemical reaction parameter. Gangadhar and Bhaskar Reddy [20] studied the MHD boundary layer flow of heat and mass transfer over a moving vertical plate in a porous medium with suction. Gangadhar et al. [21] investigated the hydro magnetic flow of heat and mass transfer over a vertical plate with a convective boundary condition and chemical reaction and also conclude that concentration of the fluid decreases with an increasing the chemical reaction parameter.
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Heat and Mass Transfer Effects on MHD Boundary Layer Flow of Casson Fluid over an Exponentially Stretching Surface

Heat and Mass Transfer Effects on MHD Boundary Layer Flow of Casson Fluid over an Exponentially Stretching Surface

The study of boundary layer flow by stretching surface is related in industrial and engineering applications such as drawing of copper wires, condensation process, die forging and extrusion of polymer in melt spinning, metal extrusion, paper production and fiber production. Prabhakar et al. [1] studied effects of inclined magnetic field and chemical reaction on flow of a Casson Nanofluid with second order velocity slip and thermal slip over an exponentially stretching sheet. Hayat et al. [2] reported on diffusion of chemically reactive species in third grade fluid flow over an exponentially stretching sheet considering magnetic field effects. Gireesha et al. [3]investigate MHD three dimensional double diffusive flow of Casson nanofluid with buoyancy forces and nonlinear thermal radiation over a stretching surface. Sharma et al. [4] presented on MHD slip flow and heat transfer over an exponentially stretching permeable sheet embedded in a porous medium with heat source. Nadeem et al. [5] discussed non aligned stagnation point flow of radiating Casson fluid over a stretching surface. Tasawar Hayat et al. [6-8] developed radiative three dimensional flow with soret and dufour effects. MHD stagnation point flow accounting variable thickness and slip conditions and three dimensional flow of CNTs nanofluids with heat generation/absorption effect: A numerical study.
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Heat transfer in a MHD Viscoelastic Fluid over a Stretching Sheet with Internal heat generation,viscous dissipation and Work due to Deformation

Heat transfer in a MHD Viscoelastic Fluid over a Stretching Sheet with Internal heat generation,viscous dissipation and Work due to Deformation

however, later, Abel et al. [14] studied the effect of heat transfer on MHD viscoelastic fluid over a stretching surface and an important finding was that the effect of visco-elasticity is to decrease the dimensionless surface temperature profiles in that flow. Furthermore, Char [15] studied MHD flow of a viscoelastic fluid over a stretching sheet; however, only the thermal diffusion is considered in the energy equation. Vajravelu and Rollins [16] obtained analytical solution for heat transfer characteristics in viscoelastic second order fluid over a stretching sheet with frictional heating and internal heat generation. Later, Sarma and Rao [17] extended the work of
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Heat and Mass Transfer in MHD Micropolar Fluid Flow over a Stretching Sheet with Velocity and Thermal Slip Conditions

Heat and Mass Transfer in MHD Micropolar Fluid Flow over a Stretching Sheet with Velocity and Thermal Slip Conditions

This study presents the problem of a steady, two-dimensional, heat and mass transfer of an incompressible, electrically conducting micropolar fluid flow past a stretching surface with velocity and thermal slip conditions. Also, the influences of temperature dependent viscosity, thermal radiation and non-uniform heat generation/absorption and chemical reaction of a general order are examined on the fluid flow. The governing system of partial diffe- rential equations of the fluid flow are transformed into non-linear ordinary differential equations by an appropriate similarity variables and the resulting equations are solved by shooting method coupled with fourth order Runge-Kutta integration scheme. The effects of the controlling parameters on the velocity, temperature, microrotation and concentration profiles as well as on the skin friction, Nusselt number, Sherwood and wall couple stress are in- vestigated through tables and graphs. Comparison of the present results with the existing results in the literature in some limiting cases shows an excellent agreement.
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Effect of MHD on Nanofluid flow, Heat and Mass Transfer over a Stretching Surface Embedded in a Porous Medium

Effect of MHD on Nanofluid flow, Heat and Mass Transfer over a Stretching Surface Embedded in a Porous Medium

The combination of heat and mass transfer in the presence of a chemical reaction is a growing need in the chemical industries such as combustion systems, solar collectors. Patil et al. [30] studied the effects of chemical reaction on mixed convection flow of a polar fluid through a porous medium in the presence of internal heat generation. The effects of chemical reactions on unsteady MHD free convection and mass transfer for flow past a hot vertical porous plate with heat generation absorption through porous medium was studied by Singh and Kumar [31]. It noted that temperature, velocity skin-friction coefficient increase and Nusselt number decreases for generative chemical reactions (ˠ <0). However, destructive chemical reactions (ˠ >0) have opposite effect on temperature, velocity, skin-friction coefficient and Nusselt number.
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Heat generation and radiation effects on steady MHD free convection flow of micropolar fluid past a moving surface

Heat generation and radiation effects on steady MHD free convection flow of micropolar fluid past a moving surface

The study of heat generation or absorption in moving fluids is important in the problems dealing with chemical reactions and those concerned with dissociating fluids. Possible heat generation effects may alter temperature distribution; hence, the particle deposition rate in nuclear reactors, electronic chips and semi- conductor wafers. Recently, Gnaneswara Reddy [17] analyzed heat generation and thermal radiation effects over a stretching sheet in a micropolar fluid.
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Unsteady MHD Flow and Heat Transfer over a Stretching Permeable Surface with Suction or Injection

Unsteady MHD Flow and Heat Transfer over a Stretching Permeable Surface with Suction or Injection

A theoretical investigation is presented to describe a two-dimensional unsteady flow of a viscous incompressible electrically conducting fluid over a stretching permeable surface taking into account a transverse magnetic field of constant strength. The governing boundary layer equations for the problem are formulated and converted to nonlinear ordinary differential equations and then solved numerically using fourth order Runge-Kutta method with shooting technique. Influences of many parameters, like, suction or injection parameter, unsteadiness parameter, magnetic parameter, Prandtl number and Eckert number on velocity and temperature distributions are plotted on graphs while skin-friction coefficient and Nusselt number are shown numerically. Computational results are compared with previously presented results for non-magnetic case.
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MHD SLIP FLOW AND HEAT TRANSFER OVER AN EXPONENTIALLY STRETCHING PERMEABLE SHEET EMBEDDED IN A POROUS MEDIUM WITH HEAT SOURCE

MHD SLIP FLOW AND HEAT TRANSFER OVER AN EXPONENTIALLY STRETCHING PERMEABLE SHEET EMBEDDED IN A POROUS MEDIUM WITH HEAT SOURCE

Steady two dimensional laminar boundary layer flow and heat transfer of a viscous incompressible and electrically conducting fluid past over a flat exponentially non-conducting stretching porous sheet in the presence of non uniform transverse magnetic field and non uniform heat source are analyzed numerically. Effect of different physical parameters on fluid velocity, fluid temperature, skin friction coefficient and Nusselt number are investigated and the following observations are made:

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Influence of Thermal Stratification on Mhd Heat Transfer Flow Over an Exponentially Stretching Surface

Influence of Thermal Stratification on Mhd Heat Transfer Flow Over an Exponentially Stretching Surface

Flow of an incompressible viscous fluid and heat transfer phenomena over a stretching sheet have received great attention during the last decades owing to the abundance of practical applications in chemical and manufacturing processes, such as polymer extrusion, drawing of copper wires, continuous casting of metals, wire drawing and glass blowing. Since the pioneering work of Sakiadis[1], various aspects of the problem have been investigated by many authors. Crane[2] studied a steady flow past a stretching sheet and presented a closed form solution to it. Gupta and Gupta [3] stressed that realistically, stretching surface is not necessarily continuous. Most of the available literature deals with the study of boundary layer flow over a stretching surface where the velocity of the stretching surface is assumed linearly proportional to the distance from the fixed origin. However, it is often argued that (Gupta and Gupta [3]) realistically, stretching of plastic sheet may not necessarily be linear. This situation was beautifully dealt by Kumaran and Ramanaiah [4] in their work on boundary layer fluid flow where,
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Melting Heat Transfer in a Steady Stagnation Point Flow of a Jeffrey Fluid over a Stretching Sheet with Radiation and Velocity Slip

Melting Heat Transfer in a Steady Stagnation Point Flow of a Jeffrey Fluid over a Stretching Sheet with Radiation and Velocity Slip

We Consider the two-dimensional stagnation-point flow of a steady incompressible Jeffrey fluid towards a horizontal linearly stretching sheet at = 0, which is melting at a steady rate. The x-coordinate is measured along the plate and the y-coordinate normal to it. The flow being confined to ≥ 0. The flow is generated, due to stretching of the sheet, caused by the simultaneous application of two equal and opposite forces along the x-axis. The sheet is stretched, keeping the origin fixed, with a velocity ( ) = where ‘a’ is a positive constant with dimensions of (time) . It is assumed that the velocity of the external flow is ( ) = where ‘b’ is a positive constant with dimensions of (time) . It is also assumed that the temperature of the melting surface is , while the temperature in the free stream condition is where > . In addition, the temperature of the solid medium far from the interface is constant and is denoted by where < . The species concentration is assumed to be constant at the stretching surface. When y tends to infinity, the ambient value of the concentration is denoted by .
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MHD Three Dimensional Flow of a Williamson Fluid over a Stretching Sheet with Velocity Slip

MHD Three Dimensional Flow of a Williamson Fluid over a Stretching Sheet with Velocity Slip

Fluids used in engineering and industrial processes are modeled as non-Newtonian fluids. For example, mercury amalgams, lubrications with heavy oils and greases, paints molten plastics and slurries. The rheological properties of such fluids cannot be explained by the Navier Stokes equations alone. It is also pertinent that one single model is not sufficient to describe these non-Newtonian fluids. To overcome this deficiency rheological models have been proposed. Several rheological models have been introduced such as Ellis model Carreau model, Cross model, power law model and Williamson fluid model. Williamson fluid is non-Newtonian fluid model with shear thinning property. Williamson [1] while studying the flow of psuedoplastic materials, proposed a model equation to describe the flow of psuedoplatic fluids and validated with the experimental results. Dapra and Scarpi [2] examined approximate analytical expression for the velocity field for a pulsatile flow of a non-Newtonian Williamson fluid in a rock fracture. Nadeem and Akram [3] analyzed the peristaltic flow of a Williamson fluid in an asymmetric channel. Nadeem et al. [4] investigated the two dimensional flow of a Williamson fluid over stretching sheet. In later study, Nadeem and Hussain [5] examined the effects of nano-sized particle of Williamson fluid over a stretching sheet. Krishnamurthy et al. [6] obtained the effects of radiation and chemical reaction on the steady boundary layer flow of MHD Williamson fluid through porous medium toward a horizontal linearly stretching in the presence of nanoparticles. Hayat et al. [7] discussed the two- dimensional boundary layer flow of an incompressible Williamson fluid over an unsteady permeable stretching surface with thermal radiation. Sinivas Reddy e al. [8] investigated the MHD flow and heat transfer characteristics of Williamson nanofluid over a stretching sheet with variable thickness and variable thermal conductivity. Ellahi et al. [9] discussed the three dimensional peristaltic flow of a Williamson fluid in a rectangular duct. Very recently, Malik et al. [10] explored the three dimensional flow of a Williamson fluid over a linearly stretching surface under the influence of a magnetic field.
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Magnetohydrodynamic three dimensional flow and heat transfer over a stretching surface in a viscoelastic fluid

Magnetohydrodynamic three dimensional flow and heat transfer over a stretching surface in a viscoelastic fluid

Problems involving flows over stretching sheet are widely applied in various industries. Some of the applications can be found in the extrusion of a polymer in a melt-spinning, cooling of an infinite metallic plate in a cooling bath, hot rolling, wire drawing, glass–fiber production, manufacturing of plastic and rubber sheet and paper production. Pioneering work done by Crane (1970) to get the analytical solution for the steady two-dimensional boundary layer flow caused by the stretching surface with velocity varying linearly with distance from a fixed point has initiated many researchers to study different aspects of this problem either by integrating the problem with heat and mass transfer, MHD, chemical reaction, suction/injection, non-Newtonian fluids or other various situations. Some of the huge collections of research papers existing in literature can be found in Chakrabarti and Gupta (1979), Chen (1998), Fan et al. (1999), Sajid (2007), Xu and Liao (2009), Jat and Chaudary (2009), Ishak et al. (2009, 2010), Salleh et al. (2010) and Ali et al. (2011), to name just a few.
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