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It is noticed that when the density of **an** electrically conducting **fluid** is low and/or applied magnetic field is strong, Hall current plays a vital role **in** determining **flow**-features of the **fluid** **flow** problems because it induces secondary **flow** **in** the **flow**-field (Sutton and Sherman (1965). Taking into account of this fact, Aboeldahab and Elbarbary (2001) and Seth et al. (2012) investigated the effects of Hall current on hydromagnetic free **convection** **boundary** **layer** **flow** past a **flat** **plate** considering different aspects of the problem. It is noteworthy that Hall current induces secondary **flow** **in** the **flow**-field which is also the characteristics of Coriolis force. Therefore, it is essential to compare and contrast the effects of these two agencies and also to study their combined effects on such **fluid** **flow** problems. Narayana et al. (2013) studied the effects of Hall current and radiation- absorption on MHD natural **convection** heat and mass transfer **flow** of a micropolar **fluid** **in** a rotating frame of reference. Recently, Seth et al. (2013a) investigated the effects of Hall current and rotation on unsteady hydromagnetic natural **convection** **flow** of a **viscous**, incompressible, electrically conducting and heat absorbing **fluid** past **an** impulsively **moving** **vertical** **plate** **with** ramped temperature **in** a porous medium taking into account the effects of thermal diffusion. Aim of the present investigation is to study unsteady hydromagnetic natural **convection** heat and mass transfer **flow** **with** Hall current of a **viscous**, incompressible, electrically conducting, temperature dependent heat absorbing and optically thin heat radiating **fluid** past **an** accelerated **moving** **vertical** **plate** through **fluid** saturated porous medium **in** a rotating environment when temperature of the **plate** has a temporarily ramped profile. This problem has not yet received any attention from the researchers although natural **convection** heat and mass transfer **flow** of a heat absorbing and radiating **fluid** resulting from such ramped temperature profile of a **plate** **moving** **with** time dependent velocity may have strong bearings on numerous problems of practical interest where initial temperature profiles are of much significance **in** designing of so many hydromagnetic devices and **in** several industrial processes occurring at high temperatures where the effects of thermal radiation and heat absorption play a vital role **in** the **fluid** **flow** characteristics.

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tion to the energy equation has been neglected. However, **in** recent years it has been noted that **in** **mixed** **convection** and vigorous natural con- vection flows **in** porous media, **viscous** dissipa- tion may become more significant. Also grav-ity has been assumed to be constant **in** most of the experimental and theoretical studies. But this assumption may not give accurate result while considering large scale flows, e.g. flows **in** ocean, atmosphere or earth’s mantle, be-cause the gravity field is varying **with** height from earth’s surface. **In** this cases consider-ing gravity as variable will help one to pro-duce more accurate results. The **convection** of a **fluid** through a **flat** **layer** bounded above and below by perfectly conducting media **with** **vertical** temperature gradient is considered by Horton and Rogers (1945). The instability of a horizontal **fluid** **layer** where the gravitational field is varying **with** height is investigated by Pradhan and Samal (1987). Later Straughan (1989) done the linear instabilty and nonlin-ear energy stability analysis for **convection** **in** a horizontal porous **layer** **with** variable gravity **effect**. Alex et al. (2001) investigated the ef-fect of variable gravity on the onset of convec-tion **in** **an** isotropic porous medium **with** inter-nal heat source and inclined temperature gradi-ent. The **effect** of variable gravity field on the onset of thermosolutal **convection** **in** a **fluid** sat-urated isotropic porous **layer** is studied by Alex and Patil (2001). Barletta et al. (2009) con-sidered a horizontal porous **layer** **with** **an** adia-batic lower **boundary** and **an** isothermal upper **boundary** and discussed the **effect** of **viscous** **dissipation** on parallel Darcy **flow** by means of linear stability analysis. The **effect** of **viscous** **dissipation**, on the stability of **flow** **in**

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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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unsteady MHD free convective heat transfer **flow** along a **vertical** porous **flat** **plate** **with** internal heat generation. **In** many chemical engineering processes, there does occur the chemical reaction between a foreign mass and the **fluid** **in** which the **plate** is **moving**. These processes take place **in** numerous industrial applications viz., Polymer production, manufacturing of ceramics or glassware and food procession. Cramer K. R. and Pai, S. I.et al.[11] taken transverse applied magnetic field and magnetic Reynolds number are assumed to be very small, so that the induced magnetic field is negligible. Muthucumaraswamy et al.[12] have studied the **effect** of homogeneous chemical reaction of first order and free **convection** on the oscillating infinite **vertical** **plate** **with** variable temperature and mass diffusion. Das et al.[13] have studied the effects of mass transfer on **flow** past **an** impulsively started infinite **vertical** **plate** **with** constant heat flux and chemical reaction. K.Sudhakar and R. Srinivasa Raju et al.[14] have studied chemical reaction **effect** on **an** unsteady MHD free **convection** **flow** past **an** infinite **vertical** accelerated **plate** **with** constant heat flux, thermal diffusion and diffusion thermo. S. Shivaiah and J. Anand Rao et al.[15] studied chemical reaction **effect** on **an** unsteady MHD free **convection** **flow** past a **vertical** porous **plate** **in** the presence of suction or injection. Chaudhary and Jha [16] studied the effects of chemical reactions on MHD micropolar **fluid** **flow** past a **vertical** **plate** **in** slip-**flow** regime. Anjalidevi et al.[17] have examined the **effect** of chemical reaction on the **flow** **in** the presence of heat transfer and magnetic field. Moreover, Al-Odat and Al-Azab [18] studied the influence of magnetic field on unsteady free convective heat and mass transfer **flow** along **an** impulsively started semi-infinite **vertical** **plate** taking into account a homogeneous chemical reaction of first order. The chemical reaction, heat and mass transfer on MHD **flow** **over** a **vertical** stretching surface **with** heat source and thermal stratification have been presented by Kandasamy et al.[19]. Ahmed Sahin.et al.[20] have studied influence of chemical reaction on transient MHD free convective **flow** **over** a **vertical** **plate** **in** slip-**flow** regime.

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when the **fluid** is driven by a constant pressure gradient and subjected to a uniform **external** magnetic field applied perpendicular to the plates **with** Navier slip **boundary** condition. **In** all the above investigations a solution for the **flow** and heat transfer is obtained assuming the temperature at the interface of the **plate** as constant. However, there exist several problems of physical interest which may require non-uniform conditions. Gireesha et al. (2011) obtained the solution for the **boundary** **layer** **flow** and heat transfer of a dusty **fluid** **over** a stretching sheet **with** non-uniform heat source/sink. They considered two types of heating processes namely (i) prescribed surface temperature and, (ii) prescribed surface heat flux. Ramesh et al. (2012) analyzed the steady two-dimensional MHD **flow** of a dusty **fluid** near the stagnation point **over** a permeable stretching sheet **with** the **effect** of non- uniform source/sink. Recently, the effects of ramped surface temperature on the **flow** and heat transfer of a **viscous**, incompressible, and electrically conducting dusty **fluid** **in** the presence of a transverse magnetic field are studied by (Nandkeolyar et al. 2013 ; Nandkeolyar and Das 2013). They assumed that the surface temperature increases up to a specific time and then it becomes constant. They also compared the **flow** of dusty fluids through a wall having ramped temperature **with** that of a **flow** past **an** isothermal wall.

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The study of **flow**, heat and mass transfer about natural **convection** of non-Newtonian fluids **in** porous media has gained much attention from the researchers because of its engineering and industrial applications. These applications include design of chemical processing equipment, formation and dispersion of fog, distributions of temperature and moisture **over** agricultural fields and groves of fruit trees and damage of crops due to freezing and pollution of the environment, etc. Several investigators have extended the **convection** of heat and mass transfer problems to fluids exhibiting non-Newtonian rheology. Different models have been proposed to explain the behavior of non-Newtonian fluids. Among these, the power law model gained importance. Although this model is merely **an** empirical relationship between the stress and velocity gradients, it has been successfully applied to non-Newtonian fluids experimentally. Free **convection** from a horizontal line heat source **in** a power-law **fluid**-saturated porous medium was studied by Nakayama (1993). The study of free **convection** **in** **boundary** **layer** flows of power law fluids past a **vertical** **flat** **plate** **with** suction/injection was done by Sahu and Mathur (1996). They observed that the suction/injection has significant **effect** on the velocity and temperature fields. Free **convection** heat and mass transfer of non-Newtonian power law fluids **with** yield stress from a **vertical** **flat** **plate** **in** a saturated porous media was studied by Rami and Arun (2000). They concluded that the velocity, temperature, and concentration profiles as well as the local heat and mass transfer rates are significantly affected by the **fluid** rheology **in** addition to the buoyancy ratio and the Lewis number of the **fluid**. The **flow** of natural **convection** heat and mass transfer

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When the temperature distribution of the solid body and that of the **fluid** **flow** are coupled, the resulting problem is known conjugate heat transfer. This problem, for the first time, was studied by Perelman [1] and Luikov et al. [2]. Perelman [1] used the method of the asymptotic solution to solve the integral equation occurring **in** the conjugate heat- transfer problem. A generalized Fourier sine transform was presented by Luikov et al. [2] for the semi-infinite **plate**. Trevino and Linan [3] modeled the **external** heating of a **flat** **plate** cooled under a convective laminar **flow** **with** and accounted for the axial heat condu c- tion **in** the **plate** by solving the integro-diffrential equation using perturbation. Determina- tion of **plate** temperature **in** case of combined conduction, **convection** and radiation heat exchange is carried out by Sohal and Howel [4]. Later, Karvinen [5] presented **an** approxi- mate method is presented by for calculating heat transfer from a **flat** **plate** **in** forced **flow** and compared the results **with** experimental data and previous results obtained **in** [4] for the case of combined convective heat exchange **with** the environment, conduction **in** the **plate** and internal heat sources. Forced **convection** conjugate heat transfer **in** a laminar plane wall jet was considered by Kanna and Das [6]. A problem of conduction-**convection** **in** fins [7, 8] and **in** cavities [9, 10] and the combined **effect** of conduction and radition **in** a T-Y shaped fin [11] are carried out **in** recent years.

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Because of its application for MHD natural **convection** **flow** **in** the nuclear engineering where **convection** aids the cooling of reactors, the natural **convection** **boundary** **layer** **flow** of **an** electrically conducting **fluid** up a hot **vertical** wall **in** the presence of strong magnetic field has been studied by several authors, such as Sparrow and Cess [7], Reley [8] and Kuiken [9]. Simultaneous occurrence of buoyancy and magnetic field forces **in** the **flow** of **an** electrically conducting **fluid** up a hot **vertical** **flat** **plate** **in** the presence of a strong cross magnetic field was studied by Sing and Cowling [10] who had shown that regardless of strength of applied magnetic field there will always be a region **in** the neighborhood of the leading edge of the **plate** where electromagnetic forces are unimportant. Creamer and Pai [11] presented a similarity solution for the above problem **with** uniform heat flux by formulating it **in** terms of both a regular and inverse series expansions of characterizing coordinate that provided a link between the similarity states closed to and far from the leading edge. Hossain and Ahmed [13] studied the combined **effect** of the free and forced **convection** **with** uniform heat flux **in** the presence of strong magnetic field. Hossain et al [14] also investigated

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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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generalized 3-D MHD **flow** **over** a porous stretching sheet [17], the wire coating analysis using MHD Oldroyd 8-constant **fluid** [18], the **viscous** **flow** **over** a non-linearly stretching sheet [19], the off-centered stagnation **flow** towards a rotating disc [20], the nano **boundary** **layer** flows [21], the **boundary**-**layer** **flow** about a heated and rotating down-pointing **vertical** cone [22], the 2-D **viscous** **flow** **in** a rectangular domain bounded by two **moving** porous walls [23], the unsteady laminar MHD **flow** near forward stagnation point of **an** impulsively rotating and translating sphere **in** presence of buoyancy forces [24], the non-simi- larity **boundary**-**layer** flows **over** a porous wedge [25], the steady **flow** and heat transfer of a Sisko **fluid** **in** annular pipe [26], the steady **flow** of **an** Oldroyd 8-constant **fluid** due to a suddenly moved **plate** [27], the MHD **flow** of non-Newtonian nanofluid and heat transfer **in** coaxial po- rous cylinder [28], the non-Newtonian nanofluids **with** Reynolds' model and Vogel's model [29], and the **flow** of non-Newtonian nanofluid **in** a pipe [30]. These new solutions have never been reported by all other previous analytic methods. This shows the great potential of the HAM for strongly non-linear problems **in** science and engineering.

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growth. Until recently this study has been largely concerned **with** **flow** and heat transfer characteristics **in** various physical situations. Radiative magnetohydrodynamic flows arise **in** many areas of technology and applied physics including oxide melt materials processing (Shu et al. (2004)), astrophysical **fluid** dynamics (Stone and Norman (1992); Vishwakarma et al. (1987)), plasma **flow** switch performance (Bowers et al. (1990)), MHD energy pumps operating at very high temperatures (Biberman et al.(1979)) and hypersonic aerodynamics (Ram and Pandey (1980)). Takhar et al. (1996) investigated the effects of radiation on the MHD free **convection** **flow** of radiating gas past a semi-infinite **vertical** **plate**. Raptis and Masslas (1998) studied unsteady magnetohydrodynamics **convection** **in** a gray, absorbing- emitting but non-scattering **fluid** regime using the Rosseland radiation model. A similar study was communicated by Raptis and Perdikis(2000). Azzam (2002) considered thermal radiation flux influence on hydromagnetic **mixed** convective steady optically-thick laminar **boundary** **layer** **flow** also using Rosseland approximation. Helliwell and Mosa (1979) reported on thermal radiation effects **in** buoyancy-driven hydromagnetic **flow** **in** a horizontal channel **flow** **with** **an** axial temperature gradient **in** the presence of Joule and **Viscous** heating. Yasar and Moses (1992) developed a

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MHD free **convection** **over** **an** inclined **plate** **in** a thermally stratified high porous medium **in** the presence of a magnetic field has been studied. The dimensionless momentum and temperature equations have been solved numerically by explicit finite difference technique **with** the help of a computer programming language Compaq Visual Fortran 6.6a. The obtained results of these studies have been discussed for the different values of well known parameters **with** different time steps. Also, the stability conditions and convergence criteria of the explicit finite difference scheme has been analyzed for finding the restriction of the values of various parameters to get more accuracy. The effects of various governing parameters on the **fluid** velocity, temperature, local and average shear stress and Nusselt number has been investigated and presented graphically.

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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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Since the boundaries **in** the x direction are of infinite dimensions, without loss of generality, we assume that the physical quantities depend on y only. The **fluid** properties are assumed to be constant except for density variations **in** the buoyancy force term. **In** addition, the thermo diffusion **with** thermal radiation effects considered. The **flow** is a **mixed** **convection** **flow** taking place under thermal buoyancy and uniform pressure gradient **in** the **flow** direction. The **flow** configuration and the coordinates system are shown **in** Figure 1. The **fluid** velocity u is assumed to be parallel to the x-axis, so that only the x-component u of the velocity vector does not vanish but the transpiration cross-**flow** velocity v 0 remains

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To make sure that the results are independ- ent of the generated grid, four different grids **with** 60 × 40 × 80, 75 × 50 × 120, 90 × 60 × × 160, and 105 × 70 × 200 elements along the x-, y-, and z-axes have been generated. Differ- ent parameters such as dimensionless tempera- ture (θ) along the x and y, dimensionless axial velocity along the centerline, and the local Nusselt number have been calculated for these four grids and their values have been compared **with** one another **in** fig. 2. As this figure shows, all the four generated grids successfully pass the grid independency test for the temperature along the x-axis and axial velocity, but the 60 × 40 × 80 grid fails the grid independency test for the other two parameters. Therefore, the 75 × 50 × 120 grid will be used **in** all the simula- tions. Figure 3 shows the structure non-uniform generated grid for the **flat** tube of present paper. As shown **in** this figure, the grids are finer near the tubes entrance and near the wall where the velocity and temperature gradients are high. Validations of the numerical simulations

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Abo-Eldahab and Elbarbary [8] have studied the Hall current effects on MHD free-**convection** **flow** past a semi-infinite **vertical** **plate** **with** mass transfer. The **effect** of Hall current on the steady magneto hydrodynamics **flow** of **an** electrically conducting, incompressible Burger’s **fluid** between two parallel electrically insulating infinite planes have been studied by M. A. Rana and A. M. Siddiqui [9].

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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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