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Role of Induced Magnetic Field on MHD Natural Convection Flow In Alexandria Engineering Journal (2016) xxx, xxx–xxx HOSTED BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com ORIGINAL ARTICLE Role of induced magnetic field on MHD natural convection flow in vertical microchannel formed by two electrically non-conducting infinite vertical parallel plates Basant K. Jha, Babatunde Aina * Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria Received 2 March 2016; revised 25 May 2016; accepted 25 June 2016 KEYWORDS Abstract The present work consists of theoretical investigation of MHD natural convection flow Microchannel; in vertical microchannel formed by two electrically non-conducting infinite vertical parallel plates. Transverse magnetic field; The influence of induced magnetic field arising due to the motion of an electrically conducting fluid Induced magnetic field; is taken into consideration. The governing equations of the motion are a set of simultaneous ordi- Velocity slip and temperature nary differential equations and their exact solutions in dimensionless form have been obtained for jump the velocity field, the induced magnetic field and the temperature field. The expressions for the induced current density and skin friction have also been obtained. The effects of various non- dimensional parameters such as rarefaction, fluid wall interaction, Hartmann number and the mag- netic Prandtl number on the velocity, the induced magnetic field, the induced current density, and skin friction have been presented in graphical form. It is found that the effect of Hartmann number and magnetic Prandtl number on the induced current density is found to have a decreasing nature at the central region of the microchannel. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction such as, fire engineering, combustion modelling, geophysics, the cooling of nuclear reactors, operation of magnetohydrody- Study associated with natural convection flow of an electrically namic (MHD) generators, and plasma studies. Application of conducting fluid in the presence of an external magnetic field a magnetic field has been found to be effective in controlling has received considerable interest due to the enormous applica- the melt convection during crystal growth from melts under tions in various branches of industry, science and technology terrestrial conditions and has now been widely practises in the metals and semiconductor industries. Several studies have * Corresponding author. been reported on MHD convective flow under different phys- E-mail addresses: [email protected] (B.K. Jha), ainavicdydx@ ical situations. Record of such investigations can be found in gmail.com (B. Aina). the works of Cramer and Pai [1], Chawla [2], Das et al. [3], Peer review under responsibility of Faculty of Engineering, Alexandria Sheikholesslami and Gorgi-Bandpy [4], Sheikholesslami et al. University. http://dx.doi.org/10.1016/j.aej.2016.06.030 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: B.K. Jha, B. Aina, Role of induced magnetic field on MHD natural convection flow in vertical microchannel formed by two elec- trically non-conducting infinite vertical parallel plates, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.06.030 2 B.K. Jha, B. Aina Nomenclature b gap between the plates y0 dimensional coordinate perpendicular to the plates Cq specific heat of the fluid at constant pressure y dimensionless coordinate perpendicular to the g gravitational acceleration plates 0 H0 constant strength of applied magnetic field 0 Hx dimensional induced magnetic field Greek letters H dimensionless induced magnetic field b coefficient of thermal expansion b ; b ln fluid wall interaction parameter t v dimensionless variables J induced current density c ratio of specific heats ðCq=CvÞ bmKn Knudsen number h dimensionless temperature M Hartmann number q density l Pm magnetic Prandtl number e magnetic permeability Pr Prandtl number v fluid kinematic viscosity Qm dimensionless volume flow rate k molecular mean free path T0 temperature of the fluid k thermal conductivity T0 temperature of the fluid and plates in reference r electrical conductivity of the fluid state rt; rv thermal and tangential momentum accommoda- u0 dimensional velocity of the fluid tion coefficients, respectively U dimensionless velocity of the fluid [5,6], Chauhan and Rastogi [7], Ibrahim and Makinde [8], Far- and Singh [20] considered the effects of Hall and wall conduc- had et al. [9,10]. tance on mixed convection hydromagnetic flow in a rotating Although there are many studies on natural convection channel. Seth et al. [21] studied the unsteady hydromagnetic flow of an electrically conducting fluid in channels, there are natural convection flow of a heat absorbing fluid within a only a few studies regarding natural convection flow of an elec- rotating vertical channel in porous medium with Hall effect. trically conducting fluid in microchannel and annular In another work, Seth et al. [22] investigated the effect of Hall microchannel. In recent years, the present authors and their current on unsteady MHD convective Couette flow of heat collaborators have carried out a number of studies on MHD absorbing fluid due to accelerated movement of one of the natural convection covering several aspects. For instance, plates of the channel in a porous medium. Jha et al. [11] analytically studied the fully developed steady The above studies on MHD natural convective heat and natural convection flow of conducting fluid in a vertical paral- mass transfer in vertical microchannel and annular microchan- lel plate microchannel in the presence of transverse magnetic nel have been limited to the cases in which the induced mag- field. The effect of Hartmann number was reported to decrease netic field is neglected in order to facilitate the mathematical the volume flow rate. The combined influence of externally analysis of the problem as simple. However, the induced mag- applied transverse magnetic field and suction/injection on netic field also generates its own magnetic field in the fluid and steady natural convection flow of conducting fluid in a vertical as a result of which it modifies the applied magnetic field and microchannel was carried out by Jha et al. [12]. In another motion of the fluid. Therefore, it is known that in several phys- work, Jha et al. [13] examined the effect of wall surface curva- ical situations, it will be necessary to include the effect of ture on transient MHD free convective flow in vertical micro- induced magnetic field in the MHD equations when magnetic concentric-annuli. Jha et al. [14] studied exact solution of Reynolds number is large enough [1]. Singh et al. [23] pre- steady fully developed natural convection flow of viscous, sented numerical studies on the hydromagnetic free convective incompressible, and electrically conducting fluid in a vertical flow in the presence of induced magnetic field. Jha and Sani annular microchannel. Recently, Jha and Aina [15] presented [24] presented the MHD natural convection flow of an electri- the MHD natural convection flow in a vertical micro- cally conducting and viscous incompressible fluid in a vertical porous-annulus (MPA) in the presence of radial magnetic field. channel due to symmetric heating in the presence of induced Also, the MHD natural convection flow in vertical micro- magnetic field. A study on hydromagnetic free convective flow concentric-annuli (MCA) in the presence of radial magnetic in the presence of induced magnetic field has been carried out field has been analysed by Jha et al. [16]. by Ghosh et al. [25]. In another related work, Kumar and Some recent works related to the present investigation are Singh [26] studied the unsteady MHD free convective flow past found in the literature [17–22]. Seth and Ansari [17] presented a semi-infinite vertical wall by taking into account the induced a study on magnetohydrodynamics convective flow in a rota- magnetic field. Recently, Sarveshanand and Singh [27] analyt- tion channel with Hall effect. Combined free and forced con- ically studied the MHD free convective flow between vertical vection flow in a rotating channel with arbitrary conducting parallel porous plates in the presence of induced magnetic field walls was conducted by Seth et al. [18]. Also, Seth et al. [19] and found that the induced current density profile increases studied the unsteady MHD convective flow within a parallel with increase in the magnetic Prandtl number. plate rotating channel with thermal source/sink in a porous The induced magnetic field has many important applica- medium under slip boundary conditions. More recently, Seth tions in the experimental and theoretical studies of MHD flow Please cite this article in press as: B.K. Jha, B. Aina, Role of induced magnetic field on MHD natural convection flow in vertical microchannel formed by two elec- trically non-conducting infinite vertical parallel plates, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.06.030 Role of induced magnetic field on MHD natural convection flow 3 due to its use in many scientific and technological phenomena, pled with the corresponding wall boundary conditions are for example in MHD electrical power generation, geophysics, valid only if the fluid flow adjacent to the surface is in thermal purification of crude oil, and glass manufacturing. The role equilibrium. However, they are not valid for fluid flow at of induced magnetic field is important when the magnetic Rey- microscale. For this case, the fluid no longer reaches the veloc- nolds number is large enough [1]. ity or the temperature of the surface and therefore a slip con- The objective of this work is to present a comprehensive dition for the velocity and a jump condition for the theoretical study of steady hydromagnetic fully developed nat- temperature should be adopted.
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