Top PDF MHD Natural Convection Flow of an incompressible electrically conducting viscous fluid through porous medium from a vertical flat plate

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

Magnetohydrodynamic (MHD) free convection of a viscous incompressible fluid along a vertical wall in porous medium must be studied if we are to understand the behavior of fluid motion in many applications as for exam- ple, in MHD electrical power generation, geophysics, astrophysics, etc. The problem of free convection flows of viscous incompressible fluids past a semi-infinite vertical wall has received a great deal of attention in recent years because of its many practical applications, such as in electronic components, chemical processing equipment, etc. Magneto hydrodynamic free convection flow of an electrically conducting fluid in different porous geometries is of considerable interest to the technical field due to its frequent occurrence in industrial, technological and geother- mal applications. As an example, the geothermal region gases are electrically conducting and undergo the influence of magnetic field. Also, it has applications in nuclear engineering in connection with reactors cooling. The inter- est in this field is due to the wide range of applications in engineering and geophysics, such as the optimization of the solidification processes of metals and metal alloys, the study of geothermal sources, the treatment of nuclear fuel debris, the control of underground spreading of chemical wastes and pollutants and the design of MHD power

Consider an unsteady MHD free convection flow past infinite vertical porous plate which is thermally stratified. Let us consider an unsteady free convective flow of an electrically conducting viscous fluid through a porous medium along a semi-infinite vertical porous plate y  0 in a rotating system under the influence of transversely applied magnetic field. The flow is assumed to be in the x -direction which is taken along the plate in the upward direction and y -axis is normal to it. Initially the fluid is at rest, after the whole system is allowed to rotate with a constant angular velocity  about the y-axis. Since the systems rotate about the y-axis, so it is assumed as   (0,  ,0) . The temperature of the plate raised from T w to T  ,

It is well known that Newtonian fluids such as air, water, ethanol, benzene and mineral oils form a basis for classical fluid mechanics. However, many important fluids, such as blood, polymers, paint, and foods show non-Newtonian behavior. Due to the diversity of non-Newtonian fluids in nature no unique relationship is available in the literature that can describe the rheology of all the non-Newtonian fluids. Of course, the mathematical systems for non-Newtonian fluids are of higher order and complicated in comparison to the Newtonian fluids. Therefore, a variety of constitutive equations have been suggested to predict the behavior of non-Newtonian fluids. Despite of all these difficulties, the recent researchers in the field have made valuable contributions in study of flows of non-Newtonian fluids [1–12]. Amongst the different categorizations of non-Newtonian fluids, there is one simplest model of differential type fluids known as second grade fluid [13,14]. Keeping the importance of non- Newtonian fluids in mind, for the present problem, we have chosen second grade fluid as a non-Newtonian fluid. Amongst the different studies on second grade fluids [15–25], Nazar et al. [26] provided some interesting results. They considered the second grade fluid over an oscillating plate and obtained exact solutions using the Laplace transform technique, expressed them as the sum of steady-state and transient solutions. Recently, Farhad et al. [27] extended the work of Nazar et al. [26] by considering the second grade fluid to be electrically conducting and passes through a porous medium. As a special case, it is observed that their results in the absence of MHD and porosity effects are reduced to those obtained by Nazar et al. [26].

order of magnitude than the effects described by Fourier’s and Fick’s laws. But these effects are considered as second order phenomena and may become significant in areas such as hydrology, petrology, geosciences, etc. The Soret effect, for instance, has been utilized for isotope separation and in mixture between gases of very light molecular weight and of medium molecular weight. The importance of these effects in convective transport in clear fluids has been reported in the book by Eckert and Drake (1972). Dursunkaya and Worek (1992) studied diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface. Kafoussias and Williams (1995) presented the same effects on mixed convective and mass transfer steady laminar boundary layer flow over a vertical flat plate with temperature dependent viscosity. They concluded that to predict more accurate results the variable viscosity effect and the thermal- diffusion and diffusion thermo effects have to be taken into consideration in the fluid, heat and mass transfer flow. Postelnicu (2004) has analyzed the simultaneous heat and mass transfer by natural convection from a vertical flat plate embedded in an electrically conducting fluid saturated porous medium using the Darcy Boussinesq model in the presence of Dufour and Soret effects and made a remark that as magnetic parameter increases, thickness of the hydrodynamic/thermal/concentration boundary layer increases. Both free and forced convection boundary layer flows with Soret and Dufour have been addressed by Abreu et al. (2006). The effect of Soret and Dufour parameters on free convection heat and mass transfers from a vertical surface in a doubly stratified Darcian porous medium has been reported by Lakshmi

An investigation of unsteady hydromagnetic natural convection heat and mass trans fer flow with Hall current of a viscous, incompressible, electrically conducting, heat absorbing and optically thin radiating fluid past an accelerated moving vertical plate through fluid saturated porous medium in a rotating environment is carried out when temperature of the plate has a temporarily ramped profile. The exact solutions of momentum, energy and concentration equations are obtained in closed form by Laplace transform technique. The expressions of skin friction, Nusselt number and Sherwood number are also derived. For both ramped temperature and isothermal plates, Hall current tends to accelerate primary and secondary fluid velocities whereas heat absorption and radiation have reverse effect on it. Rotation tends to retard primary fluid velocity whereas it has a reverse effect on secondary fluid velocity. Heat absorption and radiation have tendency to enhance rate of heat transfer at the plate.

However, not much attention has been paid to the effects of Hall currents. When the magnetic field is strong and the density of electrons is small, the Hall effect cannot be ignored as it has a significant effect on the flow pattern of an ionized gas. However, for ionized gases, the conventional MHD is not valid. In an ionized conducting fluid where the density is low and/or magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and a current is induced in a direction normal to both electric and magnetic fields. This phenomenon is called Hall effect. The effect of Hall current cannot be neglected when the medium is rarefied or in a strong magnetic field. The study of MHD flows with Hall current has important industrial applications in many geophysical and astrophysical situations and in several engineering problems such as Hall accelerators, Hall effect sensors, constrictions of turbines, centrifugal machines, and flight magneto-hydrodynamics. In view of these applications, it is essential to analyze the influence of Hall current on the fluid flow. Gupta [16] considered a strong magnetic field which permeated the fluid and investigated the effect of Hall current on the fluid past an infinite porous flat plate and Jana et al. [17] continued the work of Ref. [16] for different physical situations. Hossain and Rashid [18] extended the work of Gupta [16] to unsteady free convection flow. Further, Rana et al. [19] analyzed the effect of Hall current on Hartmann flow between two parallel electrically insulated infinite planes. Chaudhary and Jha [20] studied the effect of heat and mass transfer on the flow of elastico-viscous fluid past an impulsively started infinite vertical plate with mass transfer and Hall effect taken into account. Recently, Hayat et al

In all the investigations mentioned above, viscous mechanical dissipation is neglected. A number of authors have considered viscous heating effects on Newtonian flows. Mahajan et al. [25] reported the influence of viscous heating dissipation effects in natural convective flows, showing that the heat transfer rates are reduced by an increase in the dissipation parameter. Isreal- Cookey et al. [26] investigated the influence of viscous dissipation and radi- ation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. Zueco [27] used net- work simulation method (NSM) to study the effects of viscous dissipation and radiation on unsteady MHD free convection flow past a vertical porous plate. Suneetha et al. [28] have analyzed the thermal radiation effects on hydromagnetic free convection flow past an impulsively started vertical plate with variable surface temperature and concentration by taking into account of the heat due to viscous dissipation. Recently Suneetha et al. [29] stud- ied the effects of thermal radiation on the natural conductive heat and mass transfer of a viscous incompressible gray absorbing-emitting fluid flowing past an impulsively started moving vertical plate with viscous dissipation. Very recently Hiteesh [30] studied the boundary layer steady flow and heat trans- fer of a viscous incompressible fluid due to a stretching plate with viscous dissipation effect in the presence of a transverse magnetic field.

The study of an electrically conducting fluid in engineering applications is of considerable interest, especially in metallurgical and metal working processes or in the separation of molten metals from non-metallic inclusions by the application of a magnetic field. The phase change problem occurs in casting welding, purification of metals and in the formation of ice layers on the oceans as well as on aircraft surfaces. Because of its importance, accurate and robust methods of modelling phase change problems are of great interest. The velocity field in the liquid phase has a significant effect in determining the quality of the final product, and therefore, it is a great interest to study the fluid and solid phases. Recently, a significant number of studies have been conducted using computational fluid dynamics to enhance the physical and understand mathematical modelling capabilities in liquid – solid phase change as mentioned in introduction to convection in porous medium. High temperature plasmas, cooling of nuclear reactors, liquid metal fluids, magneto hydrodynamics (MHD) accelerators, and power generation systems are important applications for radiation heat transfer from a vertical wall to conductive grey fluids. Hossain and Takhar (1996)

In many practical situations such as condensation, evaporation and chemical reactions the heat transfer process is always accompanied by the mass transfer process. Perhaps, it is due to the fact that the study of combined heat and mass transfer is helpful in better understanding of a number of technical transfer processes. Besides, free convection flows with conjugate effects of heat and mass transfer past a vertical plate have been studied extensively in the literature due to its engineering and industrial applications in food processing and polymer production, fiber and granular insulation and geothermal systems [1–3]. Some recent attempts in this area of research are given in [4–9]. On the other hand, considerable interest has been developed in the study of interaction between magnetic field and the flow of electrically conducting fluids in a porous medium due to its applications in modern technology [10]. Toki et al. [11] have studied the unsteady free convection flows of incompressible viscous fluid near a porous infinite plate with arbitrary time dependent heating plate. The effects of chemical reaction in two dimensional steady free convection flow of an electrically conducting viscous fluid through a porous medium bounded by vertical surface with slip flow region has been studied by Senapati1 et al. [12]. Khan et al. [13] analyzed the effects of radiation and thermal diffusion on MHD free convection flow of an incompressible viscous fluid near an oscillating plate embedded in a porous medium.

We consider the unsteady flow of a viscous incompressible and electrically conducting viscoelastic fluid with oscillating temperature. The flow occurs over an infinite vertical porous plate. The ∗ -axis is assumed to be oriented vertically upwards along the plate and ∗ -axis is taken normal to the plane of the plate. It is assumed that the plate is electrically non-conducting and a uniform magnetic field of strength is applied normal to the plate. The induced magnetic field is assumed to be negligible so that , , . The plate is subjected to a constant suction velocity .

one dimensional adaptive-grid finite-differencing computer code for thermal radiation magnetohydrodynamic simulation of fusion plasmas. Excellent studies of thermal radiation-convection magnetohydrodynamics include Duwairi and Damseh (2004), Raptis et al. (2004) who considered axisymmetric flow and Duwairi and Duwairi (2005) who studied thermal radiation heat transfer effects on the hydrodynamic Rayleigh flow of a gray viscous fluid. Aboeldahab and Azzam(2005) have described the effects of magnetic field on hydromagnetic mixed free forced heat and mass convection of a gray, optically-thick, electrically conducting viscous fluid along a semi-infinite inclined plane for high temperature and concentration using the Rosseland approximation. Ghosh and Pop (2007) and Jana and Ghosh (2011) have studied thermal radiation of an optically-thick gray gas in the presence of indirect natural convection showing that the pressure rise region leads to slightly increase in the velocity with an increase of radiation parameter. Patra et.al (2012) considered transient approach to radiation heat transfer free convection flow with ramped wall temperature. Rajput and Kumar (2012) examined the radiation effects on MHD flow past an impulsively started vertical plate with variable heat and mass transfer. Ahmed and Kalita (2013) presented an analytical and numerical study for MHD radiating flow over an infinite vertical surface bounded by a porous medium in presence of chemical reaction. Al-Odat and Al-Azab (2007) have examined the influence of chemical reaction on transient MHD free convection over a moving vertical plate. Mbeledogu and Ogulu (2007) have presented the heat and mass transfer of an unsteady MHD natural convection flow of a rotating fluid past a vertical porous plate in the presence of radiative heat transfer. The MHD transient free convection and chemically reactive flow pasta porous vertical plate with radiation and temperature gradient dependent heat source in slip flow regime have been studied by Rao et al. (2013). Reddy et al. (2012) have investigated the heat and mass transfer effects on unsteady MHD free convection flow past a vertical permeable moving plate with radiation.

In astrophysical and geophysical studies, the MHD boundary layer flows of an electrically conducting fluid have also vast applications. Many researchers studied the laminar flow past a vertical porous plate for the application in the branch of science and technology such as in the field of mechanical engineering, plasma studies, petroleum industries Magneto hydrodynamics power generator cooling of clear reactors, boundary layer control in aerodynamics and chemical engineering. Many authors have studied the effects of magnetic field on mixed, natural and force convection heat and mass transfer problems which have many modern applications like missile technology used in army, nuclear power plant, parts of aircraft and ceramic tiles. Indeed, MHD laminar boundary layer behavior over a stretching surface is a significant type of flow having considerable practical applications in chemical engineering, electrochemistry and polymer processing. This problem has also an important bearing on metallurgy where magneto hydrodynamic (MHD) techniques have recently been used.

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.

temperature dependent heat absorption on steady stagnation point flow and heat transfer. Several physical problems exist for possible application in industry where heat generation and absorption take place, namely, fire and combustion modeling, fluids undergoing exothermic and/or endothermic chemical reaction, development of metal waste from spent nuclear fuel, nuclear thermal power generation etc. Keeping in view the importance of such study, Moalem (1976) considered steady state heat transfer in a porous medium with temperature- dependent heat generation. Ramadan and Chamkha (2000) investigated hydromagnetic natural convection of a particulate suspension from an inclined plate with heat absorption. Kamel (2001) considered unsteady hydromagnetic natural convection flow due to heat and mass transfer through a porous medium bounded by an infinite vertical porous plate with temperature-dependent heat sources/sinks. Chamkha (2004) analyzed unsteady hydromagnetic natural convection heat and mass transfer flow past a semi-infinite vertical moving plate with heat absorption. Singh and Makinde (2012) investigated steady hydromagnetic natural convection flow along an inclined plate with Newtonian heating in the presence of volumetric heat generation. Prasad et al. (2013) considered the effects of heat generation/absorption, thermal radiation, magnetic field, and temperature- dependent thermal conductivity on the flow and heat transfer characteristics of a non-Newtonian Maxwell fluid over a stretching sheet. Kumar (2013) studied MHD free convection flow over a stretching porous sheet in the presence of heat source and radiation. Das et al. (2014b) investigated unsteady hydromagnetic flow of a heat absorbing dusty fluid past a permeable vertical plate with ramped temperature.

An unsteady two-dimensional magnetohydrodynamic free convection flow of a viscous incompressible and electrically conducting fluid past a vertical porous plate in the presence of suction or injection is considered. The x- axis is taken along the plate in the upward direction and the y- axis is taken normal to the plate. A uniform magnetic field is applied in the direction perpendicular to the plate. The fluid is assumed to be slightly conducting, and hence the magnetic Reynolds number is much less than unity and the induced magnetic field is negligible in comparison with the applied magnetic field. It is assumed that the external electric field is zero and the electric field due to the polarization of charges is negligible. Initially, the plate and the fluid are at the same temperature T ∞ and the concentrationC ∞ . At time t > 0,

Raptis et al. [11] discussed the effect of thermal radiation on MHD Flow. Saha and Hossain [12] studied the natural convection flow with combined buoyancy effects due to thermal and mass diffusion in a thermally stratified media. Das et al. [13] estimated numerically the effect of mass transfer on unsteady flow past an accelerated vertical porous plate with suction. Mazumdar and Deka [14] analyzed the MHD flow past an impulsively started infinite vertical plate in presence of thermal radiation. Das and his co- workers [15] discussed the magnetohydrodynamic unsteady flow of a viscous stratified fluid through a porous medium past a porous flat moving plate in the slip flow regime with heat source. In a separate paper Das et al. [16] analyzed the mass transfer effects on MHD flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source. Recently, Das and his associates [17] reported the hydromagnetic convective flow past a vertical porous plate through a porous medium with suction and heat source. More recently, Das and Tripathy [18] estimated the effect of periodic suction on three dimensional flow and heat transfer past a vertical porous plate embedded in a porous medium.

The study of heat generation or absorption in moving fluids is important in problems dealing with chemical re- actions and those concerned with dissociating fluids. Possible heat generation effects may alter the temperature distribution; consequently, the particle deposition rate in nuclear reactors, electronic chips, and semiconductor wafers. In addition, natural convection with heat generation can be applied to combustion modeling. Heat gener- ation or absorption may be constant, space-dependent or temperature-dependent. Moalem [1] studied the effect of temperature dependent heat sources on the steady convective flow taking place in electrically conducting fluid within a porous medium. Vajravelu and Nayfeh [2] have reported results on the hydromagnetic convection at a cone and a wedge in the presence of temperature-dependent heat generation or absorption effects. Heat transfer in a vis- cous fluid over a stretching sheet with viscous dissipation and internal heat generation was studied by Vajravelu and Hadjinicolaou [3]. They have found that the wall temperature increases as the heat source/sink parameter increases. Molla et al. [4] studied the natural convection flow along a vertical wavy surface with uniform surface temperature in the presence of heat generation or absorption. The effect of Magnetohydrodynamic natural convection flow on

Taking into account that the general similarity method gives good results not only for simple boundary layer problems but also for very complicated cases (cf. [11,12,13]) we make an attempt here to evolve this method in Loicijanskij version [6] to the described problem.

The derivations for velocities, amplitude ratios and energy ratios involve a large number of parameters. Then, in order to study the dependence of amplitude and energy ratios on the direction of oblique incidence, we confine our numerical work to a particular model. Keeping in view the availability of numerical data, we consider the model consisting of a reservoir rock (sandstone) saturated with water and CO 2 is chosen for the numerical model of porous medium (Garg and Nayfeh; 1986) in welded contact with water-saturated sandstone which is assumed to be an impervious porous solid.