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

Mostrar mais
6 Ler mais

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

Mostrar mais
13 Ler mais

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 ,

Mostrar mais
9 Ler mais

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].

Mostrar mais
15 Ler mais

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

Mostrar mais
6 Ler mais

14 Ler mais

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

Mostrar mais
20 Ler mais

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.

Mostrar mais
27 Ler mais

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)

Mostrar mais
10 Ler mais

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**.

Mostrar mais
12 Ler mais

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 .

Mostrar mais
10 Ler mais

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.

Mostrar mais
9 Ler mais

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.

Mostrar mais
9 Ler mais

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.

Mostrar mais
10 Ler mais

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.

Mostrar mais
11 Ler mais

24 Ler mais

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**.

Mostrar mais
8 Ler mais

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

Mostrar mais
10 Ler mais

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.

13 Ler mais

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.

Mostrar mais
26 Ler mais