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 magnetic 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/).

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

where Ts is the temperature of the fluid at the wall, Tw is the A steady laminar fully developed natural convection flow of an wall temperature, and rt is the thermal accommodation coeffi- electrically conducting, viscous incompressible fluid in a verti- cient, which depends on the gas and surface materials. How- cal microchannel formed by two electrically non-conducting ever, for air, it assumes typical values near unity [28]. For infinite vertical parallel plates is considered. The flow is the rest of the analysis, rv and rt will be assumed to be 1. assumed to be steady and fully developed, i.e., the transverse By taking into account the conducting fluid, transverse velocity is zero. The x0–axis is taken vertically upward along magnetic field and induced magnetic field, the governing equa- the plates and the y0–axis normal to it as presented in the tions of the system are given in dimensional form: Fig. 1. The distance between the plates is b. The plates are 2 0 l 0 0 d u eH0 dHx 0 heated asymmetrically with one plate maintained at a temper- t þ þ gbðT T0Þ¼0 ð3Þ dy02 q dy0 ature T1 while the other plate at a temperature T2 where T1 > T2. Due to temperature gradient between the plates, nat- 1 d2H0 du0 ural convection flow occurs in the microchannel. A magnetic x þ H0 ¼ 0 ð4Þ 0 rl 02 0 dy0 field of uniform strength H0 is assumed to be applied in the e dy direction perpendicular to the direction of flow. Both plates y0 = 0 and y0 = b are taken to be non-conducting. As stated d2T0 ¼ 0 ð5Þ earlier, our emphasis in this study is to investigate the effect dy02 of induced magnetic field on the flow formation inside the ver- with the boundary conditions for the velocity, and induced tical microchannel. It is shown that fluid flow and heat transfer magnetic field and temperature field are as follows: at microscale differ greatly from those at macroscale. At macroscale, classical conservation equations successfully cou- 2 r du0 0 0 v k ; 0 0 ; u ðy Þ¼ 0 Hx0 ðy Þ¼0 rv dy r c k 0 0 0 2 t 2 dT 0 T ðy Þ¼T2 þ 0 at y ¼ 0 ð6Þ rt c þ 1 Pr dy

2 r du0 0 0 v k ; 0 0 ; u ðy Þ¼ 0 Hx0 ðy Þ¼0 rv dy r c k 0 0 0 2 t 2 dT 0 T ðy Þ¼T1 0 at y ¼ b ð7Þ rt c þ 1 Pr dy In the above Eqs. (3)–(7), T0 is the temperature of the fluid, ! 0 ; 0 ; m H½¼ ðHx H0 0Þ the magnetic field, the kinematic viscosity, g the acceleration due to gravity, b the coefficient of thermal l q r expansion, e the magnetic permeability, the density, and is electrical conductivity of the fluid. Using the following non-dimensional quantities

0 m 0 0 y ; u ; h T T0 ; y ¼ U ¼ 2 ¼ b gbb ðT1 T0Þ T1 T0 Pm ¼ mrl ; ð8Þ Figure 1 Flow conﬁguration and coordinate system. e

Referring to the values of rv and rt given in Eckert and b 4. Results and discussion Drake [29] and Goniak and Duffa [30], the value of v is near b unity, and the value of t ranges from near 1 to more than 100 for actual wall surface conditions and is near 1.667 for many The present study on magnetohydrodynamic natural convection flow in vertical microchannel formed by two electrically engineering applications, corresponding to rv ¼ 1, rt ¼ 1, c : : b ; b : non-conducting infinite vertical parallel plates in the presence s ¼ 1 4 and Pr ¼ 0 71 ð v ¼ 1 t ¼ 1 667Þ. The physical quantities used in the above equations are of induced magnetic field is controlled by a number of physical defined in the nomenclature. parameters such as the wall-ambient temperature difference ratio ðnÞ, rarefaction ðbmKnÞ, fluid wall interaction (ln), Hart- mann number (M), and the magnetic Prandtl number (Pm). 3. Method of solution The effects of these parameters on the velocity profile, induced magnetic field profile, the induced current density profile and Eqs. (9)–(11) are coupled system of ordinary differential equa- the skin friction are shown using the line graphs. The present tions with constant coefficients. This system of linear ordinary parametric study has been carried out over reasonable ranges differential equations has been solved in closed form by the of 0 6 bmKn 6 0:1, 0 6 ln 6 10, 0 6 M 6 5, and 0 6 Pm 6 1 theory of simultaneous ordinary differential equations. The and the selected reference values of bmKn; ln, M, and Pm for expressions for the velocity field, the induced magnetic field the present analysis are 0.05, 1.667, 5.0 and 0.5. and the temperature field in non-dimensional form are given The expression for the temperature in equation (16), the by effects of rarefaction parameter ðbmKnÞ; and fluid wall interaction parameter (ln) on the temperature profile and rate of heat pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi A y 1 transfer which is expressed as the Nusselt number are exactly UðyÞ¼C2 coshðyM PmÞþC3 sinhðyM PmÞþ 2 M Pm the same as those given by Chen and Weng [31]. C1 Figs. 2–5 show the variation of the velocity profiles with the 2 ð14Þ M Pm parameters occurring in the governing equations. Fig. 2 illustrates the effects of rarefaction parameter ðbmKnÞ and wall- C A 1 HðyÞ¼ 4 1 ½A y þ 0:5A y2 ambient temperature difference ratio ðnÞ on velocity profiles M M3Pm M 0 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi for fixed values of ln = 1.667, M = 5.0, and Pm = 0.5. It is Pm½C2 sinhðyM PmÞþC3 coshðyM PmÞ ð15Þ noticed that, an increase in rarefaction parameter and wall- ambient temperature difference ratio causes a pronounced h ðyÞ¼A0 þ A1y ð16Þ enhancement in the velocity slip. This result yields an observ-

0.16 0.16 ξ =1 ξ=1 ξ=0 ξ=0 0.14 ξ=-1 0.14 ξ=-1

0.12 0.12 M=1,3,5

0.1 0.1

0.08 0.08

0.06 0.06 Velocity (u) Velocity (u)

0.04 0.04

0.02 0.02

0 β Kn=0,0.05,0.1 v 0

-0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y y

Figure 2 Variation of velocity with bmKn (ln = 1.667, M = 5.0, Figure 4 Variation of velocity with M (bmKn ¼ 0:05, ln = 1.667, Pm = 0.5). Pm = 0.5).

0.14 ξ=1 0.16 ξ=1 ξ=0 ξ 0.12 ξ=-1 =0 ln=0,5,10 0.14 ξ=-1

0.1 0.12

Pm=0.1,0.5,1.0 0.08 0.1

0.08 0.06

Velocity (u) 0.06

0.04 Velocity (u)

0.04 0.02 0.02 0 0 -0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.02 y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y

Figure 3 Variation of velocity with ln (bmKn ¼ 0:05, M = 5.0, b : Pm = 0.5). Figure 5 Variation of velocity with Pm ( mKn ¼ 0 05, ln = 1.667, M = 5.0). able increase in the fluid velocity. This effect can be explained from the fact that, as rarefaction parameter increases, the tem- where velocity profile is independent of fluid wall interaction perature jump increases and this reduces the amount of heat parameter. Also, the impacts of fluid–wall interaction parame- transfer from the microchannel surfaces to the fluid. The ter on the microchannel slip velocity become significant with reduction in velocity due to the reduction in heat transfer is the decrease of the wall-ambient temperature difference ratio. offset by the increase in the fluid velocity due to the reduction Figs. 4 and 5 exhibit the effects of wall-ambient tempera- in the frictional retarding forces near the microchannel sur- ture difference ratio ðnÞ, Hartmann number (M), and the mag- faces. Furthermore, as the wall-ambient temperature difference netic Prandtl number (Pm), respectively on velocity profiles for b : ratio ðnÞ increases, the effect of rarefaction parameter ðbmKnÞ fixed values of mKn ¼ 0 05, and ln = 1.667. It is evident from on the microchannel slip velocity becomes significant. these Figures that, an increase in Hartmann number and mag- Fig. 3 illustrates the effects of fluid wall interaction param- netic Prandtl number causes reduction in the fluid velocity. eter (ln) as well as wall-ambient temperature difference ratio Physically speaking, the presence of transverse magnetic field ðnÞ on velocity profiles for fixed values of bmKn ¼ 0:05, sets in a resistive type force (Lorentz force), which is a retard- M = 5.0, and Pm = 0.5. It is observed that, the effect of the ing force on the velocity field. It is further observed that, there fluid wall interaction parameter is to enhance the fluid velocity exist points of intersection inside the microchannel where at the microchannel wall (y = 0) and to reduce the fluid veloc- velocity field is independent of Hartmann number and mag- ity at the microchannel wall (y = 1). In addition, it is evident netic Prandtl number and this behaviour is observed in the that, there exist points of intersection inside the microchannel case of asymmetric heating ðn ¼1Þ. Also, the influence of

0.02 0.02 ξ=1 ξ=1 ξ=0 ξ=0 0.015 ξ=-1 0.015 ξ=-1

0.01 0.01

0.005 0.005 β Kn=0,0.05,0.1 v M=1,3,5 0 0 -0.005 -0.005 -0.01 Induced magnetic field (H) Induced magnetic field (H) -0.01 -0.015

-0.015 -0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.02 y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y Figure 8 Variation of induced magnetic ﬁeld with M (bmKn ¼ 0:05, ln = 1.667, Pm = 0.5). Figure 6 Variation of induced magnetic ﬁeld with bmKn (ln = 1.667, M =5,Pm = 0.5).

0.025 ξ=1 0.02 0.02 ξ=0 ξ=1 ξ=-1 ξ=0 0.015 0.015 ξ=-1

ln=0,5,10 0.01 0.01 0.005 0.005 0 Pm=0.1,0.5,1.0

0 -0.005

-0.01 -0.005 Induced magnetic field (H) -0.015

induced magnetic field (H) -0.01 -0.02 -0.015 -0.025 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.02 y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y Figure 9 Variation of induced magnetic ﬁeld with Pm b : Figure 7 Variation of induced magnetic ﬁeld with ln ( mKn ¼ 0 05, ln = 1.667, M = 5).

(bmKn ¼ 0:05, M =5,Pm = 0.5).

difference ratio ðnÞ for fixed values of bmKn ¼ 0:05, M = 5.0, Hartmann number and magnetic Prandtl number on the and Pm = 0.5. It is evident from Fig. 7 that, the fluid wall microchannel slip velocity becomes significant with the interaction parameter influences the flow excluding the case increase of the wall-ambient temperature difference ratio. of symmetric heating ðn ¼ 1Þ. For the case of asymmetric heat- Figs. 6–9 show the variation of induced magnetic field with ing ðn ¼ 0; 1Þ, it is found that, an increase in fluid wall inter- the parameters occurring in the governing equations. Fig. 6 action parameter causes a pronounced reduction in fluid depicts the distribution of induced magnetic field with respect velocity. Also, the impacts of fluid–wall interaction parameter to rarefaction parameter ðbmKnÞ and wall-ambient temperature on the induced magnetic field become significant with the difference ratio ðnÞ for fixed values of ln = 1.667, M = 5.0, decrease in wall-ambient temperature difference ratio. and Pm = 0.5. It is observed that, the rarefaction parameter The induced magnetic field profile is plotted in Figs. 8 and 9 influences the flow formation excluding the case of symmetric with various values of Hartmann number (M) and the mag- heating ðn ¼ 1Þ. For the case of asymmetric heating netic Prandtl number (Pm), respectively. It is interesting to ðn ¼ 0; 1Þ, it is found that, the induced magnetic field note that the strength of the induced magnetic field is directly increases with the increase in rarefaction parameter. It is fur- proportional to the strength of Hartmann number as well as ther noticed from Fig. 6 that, the role of rarefaction parameter magnetic Prandtl number near the microchannel wall at on induced magnetic field is more pronounced with reduction y = 0 while it is inversely proportional near the microchannel in the values of wall-ambient temperature difference ratio. wall at y = 1. In addition, it is observed that, there exist points Fig. 7 shows the induced magnetic field with respect to fluid of intersection inside the vertical microchannel where the wall interaction parameter (ln) and wall-ambient temperature induced magnetic field is independent of Hartmann number

0.15 0.15 ξ =1 ξ=1 ξ =0 ξ=0 0.1 ξ =-1 0.1 ξ=-1

0.05 0.05

0 β Kn=0,0.05,0.1 v 0 M=1,3,5

-0.05 -0.05 Induced current density (J)

Induced current density (J) -0.1 -0.1

-0.15 -0.15

-0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y

Figure 10 Variation of induced current density with bmKn (ln = 1.667, M =5,Pm = 0.5). Figure 12 Variation of induced current density with M (bmKn ¼ 0:05, ln = 1.667, Pm = 0.5).

0.15 ξ=1 0.15 ξ=0 ξ =1 ξ ξ =0 0.1 =-1 0.1 ξ =-1

0.05 0.05 ln=0,5,10 0 Pm=0.1,0.5,1.0

0 -0.05

-0.1 -0.05 -0.15

-0.1 -0.2 Induced current density (J) Induced current density (J) -0.25 -0.15 -0.3

-0.2 -0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y y

Figure 11 Variation of induced current density with ln Figure 13 Variation of induced current density with Pm (bmKn ¼ 0:05, M =5,Pm = 0.5). (bmKn ¼ 0:05, ln = 1.667, M = 5). and magnetic Prandtl number and these strongly depend on M = 5.0, and Pm = 0.5, is show in Fig. 11. The induced cur- the wall-ambient temperature difference ratio. Furthermore, rent density decreases with the increase in fluid wall interaction it is also observed from these Figures that, as the wall- parameter in one part of the microchannel and the reverse ambient temperature difference ratio increases, the effects of trend occurs in the other part. Hartmann number as well as magnetic Prandtl number on Figs. 12 and 13 exhibit the effects of wall-ambient temper- the induced magnetic field become significant. ature difference ratio ðnÞ, Hartmann number (M), and the Figs. 10–13 describe the behaviour of induced current den- magnetic Prandtl number (Pm), respectively on induced cur- sity profile with the parameters occurring in the governing rent density for fixed values of bmKn ¼ 0:05, ln = 1.667. It is equations. Fig. 10 presents the variation of induced current evident from these figures that, the effect of Hartmann number density with respect to rarefaction parameter and wall- and magnetic Prandtl number on the induced current density is ambient temperature difference ratio for fixed values of found to have a decreasing nature at the central region of the ln = 1.667, M = 5.0, and Pm = 0.5. It is seen that, an microchannel while reverse trend occurs at the microchannel increase in rarefaction parameter leads to an increase in the plates. Moreover, it is interesting to note that current density induced current density. It is also found that there exist points changes its behaviour with Hartmann number and magnetic of intersection inside the microchannel where induced current Prandtl number at two different locations inside the density is independent of rarefaction parameter. microchannel. The variation of the induced current density with respect to Figs. 14–16 describe the behaviour of volume flow rate with fluid wall interaction parameter (ln) and wall-ambient temper- the parameters occurring in the governing equations. The vol- ature difference ratio ðnÞ for fixed values of bmKn ¼ 0:05, ume flow rate variations are shown in Fig. 14 with respect to

0.12 fluid wall interaction parameter and rarefaction parameter for different values of wall-ambient temperature difference ratio. ξ=1 0.1 ξ=0 It is seen that, an increase in the value of rarefaction parameter ξ=-1 leads to enhancement in volume flow rate for both symmetric ln=0,5,10 0.08 and asymmetric heating. Also, it is observed that volume flow rate is a decreasing function of fluid wall interaction and wall- 0.06 ambient temperature difference ratio. Figs. 15 and 16 show the effects of rarefaction parameter,

0.04 Hartmann number and magnetic Prandtl number on volume ﬂow rate for different values of wall-ambient temperature dif-

Volume flow rate (Qm) 0.02 ference ratio. It is noticed that, increase in both Hartmann number and magnetic Prandtl number causes a reduction in volume flow rate. This may be attributed to the fact that, fluid 0 velocity decreases as the Hartmann number increases, which causes a reduction in the volume flow rate. In addition, it is -0.02 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 also interesting to note that the role of Hartmann number β Kn v and magnetic Prandtl number is insignificant for the case of asymmetric heating ðn ¼1Þ. Figure 14 Variation of volume flow rate versus bmKn with ln. Figs. 17 and 18 illustrate the effect of fluid wall interaction parameter, rarefaction parameter, and wall-ambient temperature difference ratio on the skin friction. It is observed that, the skin friction is enhanced with increase in fluid wall interac- 0.14 tion at the microchannel wall y = 0 while the reverse trend

0.12 occurs at the microchannel wall y = 1. Furthermore, it is found that, the magnitude of skin friction is higher in case of ξ 0.1 =1 asymmetric heating in comparison with symmetric heating. M=1,3,5 ξ=0 ξ=-1 Figs. 19–22 reveal the combined inﬂuences of rarefaction 0.08 parameter, Hartmann number and magnetic Prandtl number on skin friction for different values of wall-ambient tempera- 0.06 ture difference ratio at the microchannel walls y = 0 and y = 1, respectively. It is observed that, increase in Hartmann 0.04 number and magnetic Prandtl number causes a reduction in Volume flow rate (Qm) 0.02 skin friction at the microchannel walls y = 0 and y =1.Itis further observed that the magnitude of skin friction is higher 0 in case of asymmetric heating ðn ¼ 0; 1Þ in comparison with symmetric heating ðn ¼ 1Þ. -0.02 In Table 1, we compared the values of skin friction at 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 β Kn microchannel walls in the presence and absence of induced v magnetic ﬁeld. From this Table 1, we can observe clearly that

Figure 15 Variation of volume flow rate versus bmKn with M. for the same Hartmann number, the skin friction profiles in the presence of induced magnetic field are higher compared to the case when the induced magnetic field is neglected [11].

0.14 0.6

0.12 0.5 ξ =1 ξ =0 0.1 0.4 ξ =-1 Pm=0.1,0.5,1.0 ξ =1 ξ

=0 ) ξ 0 0.08 =-1 τ 0.3

0.06 0.2

0.04 0.1 Skin friction ( Volume flow rate (Qm) 0.02 0

0 -0.1 ln=0,5,10 -0.02 -0.2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 β Kn v β vKn

Figure 16 Variation of volume ﬂow rate versus bmKn with Pm. Figure 17 Variation of skin friction versus bmKn with ln (y = 0).

0.7 0.6 ξ =1 ξ =0 Pm=0.1,0.5,1.0 0.6 ξ =-1 0.5 ξ =1 ξ =0 ln=0,5,10 0.4 0.5 ξ =-1 ) ) 1 0 τ

τ 0.3 0.4

0.2 0.3 0.1 Skin friction ( Skin friction ( 0.2 0 0.1 -0.1

0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -0.2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 β vKn β vKn

Figure 18 Variation of skin friction versus bmKn with ln (y = 1). Figure 21 Variation of skin friction versus bmKn with Pm (y = 0).

0.6

M=1,3,5 0.5 0.55 ξ=1 ξ=0 0.5 Pm=0.1,0.5,1.0 0.4 ξ=-1 ξ =1 0.45 ξ =0 ξ =-1 )

0 0.3 0.4 τ ) 1 τ 0.35 0.2 0.3

skin friction ( 0.1 0.25 Skin friction ( 0 0.2

0.15 -0.1 0.1

-0.2 0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 βvKn β vKn

Figure 19 Variation of skin friction versus bmKn with M (y = 0). Figure 22 Variation of skin friction versus bmKn with Pm.

0.55

M=1,3,5 0.5 Table 1 Comparison of numerical values of skin friction on ξ=1 the vertical microchannel walls in the presence and absence of 0.45 ξ=0 ξ=-1 induced magnetic ﬁeld (I.M.F). 0.4 nb s s s s vKn 0 (without 0 (with I. 1 (without 1 (with I. )

1 I.M.F.) Jha M.F.) I.M.F.) Jha M.F.) τ 0.35 et al. [11] present et al. [11] present 0.3 work work

0.25 1 0.0 0.1973 0.5000 0.1973 0.5000

Skin friction ( 0.05 0.1583 0.5000 0.1583 0.5000 0.2 0.1 0.1321 0.5000 0.1321 0.5000

0.15 0 0.0 0.0373 0.1801 0.1600 0.3199 0.05 0.0372 1.1995 0.1211 0.3005 0.1 0.1 0.0355 0.2119 0.0966 0.2881

0.05 1 0.0 0.1227 0.1398 0.1227 0.1398 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.05 0.0839 0.1009 0.0839 0.1009 βvKn 0.1 0.0611 0.0763 0.0611 0.0763

Figure 20 Variation of skin friction versus bmKn with M.

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 electrically non-conducting infinite vertical parallel plates, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.06.030 10 B.K. Jha, B. Aina ffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi 5. Conclusions bmKnM Pm þ d3 p d ¼ ; d ¼ d d þ d bmKnM Pm; 8 M2Pm 9 3 4 1 pffiffiffiffiffiffiffi The effect of induced magnetic field on the MHD natural con- d10 ¼ d9 d6d7; d11 ¼ bmKnM Pmd2 þ d3; vection flow of a viscous, incompressible and electrically conducting fluid in the presence of transverse magnetic field in a d2 1 d ¼ ; d ¼ d d d ; d ¼ d d ; vertical microchannel formed by two electrically non- 12 M2Pm 13 1 2 4 14 5 6 conducting infinite vertical parallel plates has been investigated analytically. The effects of various parameters on the velocity, References induced magnetic field, induced current density and skin friction profiles have been shown in the line graphs. The main [1] K.R. Cramer, S. Pai, Magneto Fluid Dynamics for Engineers findings are as follows: and Applied Physicists, McGraw-Hill, New York, 1973, pp. 204–237. (1) Increasing the value of Hartmann number and magnetic [2] S.S. Chawla, Magnetohydrodynamics unsteady free convection, Prandtl number causes enhancement in induced mag- ZAMM 47 (1967) 499–508. netic field. [3] S. Das, R.N. Jana, O.D. Makinde, MHD boundary layer slip (2) There exist points of intersection inside the microchan- flow and heat transfer of nanofluid past a vertical stretching nel where the induced magnetic field is independent of sheet with non-uniform heat generation/absorption, Int. J. Hartmann number and magnetic Prandtl number and Nanosci. 13 (3) (2014) 1450019. these strongly depend on the wall-ambient temperature [4] M. Sheikholeslamia, M. Gorji-Bandpy, Free convection of difference ratio. ferro-fluid in a cavity heated from below in the presence of an external magnetic field, Powder Technol. 256 (2014) 490–498. (3) The effect of Hartmann number and magnetic Prandtl [5] M. Sheikholeslamia, M. Gorji-Bandpy, D.D. Ganji, P. Rana, number on the induced current density is found to have Soleimani Soheil, Magnetohydrodynamic free convection of a decreasing nature at the central region of the Al2O3–water nanofluid considering thermophoresis and microchannel. Brownian motion effects, Comput Fluid 94 (2014) 147–160. (4) It is found that, increase in Hartmann number and mag- [6] M. Sheikholeslamia, M. Gorji-Bandpy, D.D. Ganji, Lattice netic Prandtl number causes a pronounced reduction in Boltz-mann method for MHD natural convection heat transfer volume flow rate. using nanofluid, Powder Technol. 254 (2014) 82–93. (5) The magnitude of skin friction is higher in case of asym- [7] D. Chaughan, P. Rastogi, Heat transfer effects on a rotating metric heating ðn ¼ 0; 1Þ in comparison with symmet- MHD Couette flow in a channel partially by a porous medium with hall current, J. Appl. Sci. Eng. 15 (3) (2012) 281–290. ric heating ðn ¼ 1Þ. [8] W. Ibrahim, O.D. Makinde, Double-diffusive mixed convection (6) The effect of Hartmann number on the skin friction can and MHD stagnation point flow of nanofluid over a stretching be useful in mechanical engineering for modelling a sys- sheet, J. Nanofluids 4 (2015) 28–37. tem. We can obtain a suitable value of Hartmann num- [9] A. Farhad, M. Norzieha, S. SSharidan, I. Khan, Samiuthaq, On ber for which the value of the skin friction will be hydromagnetic rotating flow in a porous medium with slip optimum. condition and hall current, Int. J. Phys. Sci. 7 (10) (2012) 1540– (7) This study exactly agrees with the finding of Chen and 1548. Weng [31] in the absence of magnetic field. [10] Farhad Ali, Ilyas Khan, Samiulhaq, Norzieha Mustapha, Sharidan Shafie, Unsteady magnetohydrodynamic oscillatory flow of viscoelastic fluids in a porous channel with heat and mass transfer, J. Phys. Soc. Jpn. 81 (2012) 064402. Appendix [11] B.K. Jha, Babatunde Aina, A.T. Ajiya, MHD natural convection flow in a vertical parallel plate microchannel, Ain b n n Shams Eng. J. 6 (2015) 289–295. n mKn lnð1 Þ ; ð1 Þ ; A0 ¼ þ A1 ¼ [12] B.K. Jha, Babatunde Aina, A.T. Ajiya, Role of suction/injection 1 þ 2bmKn ln 1 þ 2bmKn ln on MHD natural convection flow in a vertical microchannel, d13d14 d10d11 d10d12 d8d13 Int. J. Energy Technol. 7 (2015) 30–39. C1 ¼ ; C3 ¼ ; d12d14 d8d11 d12d14 d8d11 [13] B.K. Jha, Babatunde Aina, Sani Isa, Transient magnetohydrodynamic free convective flow in vertical micro- ffiffiffiffiffiffiffi ðbmKnA1 þ C1Þ p concentric-annuli, Proc. IMechE Part N: J. Nanoeng. C ¼ þ bmKnM PmC ; 2 M2Pm 3 Nanosyst., doi:http://dx.doi.org/10.1177/1740349915578956. [14] B.K. Jha, Babatunde Aina, Sani Isa, Fully developed MHD A pffiffiffiffiffiffiffi A 1 ; 1 b ; natural convection flow in a vertical annular microchannel: an C4 ¼ 2 þ C3M Pm d1 ¼ 2 ð vKn 1Þ M Pm M Pm exact solution, J. King Saud Univ.-Sci. 27 (2015) 253–259. pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi [15] B.K. Jha, Babatunde Aina, MHD natural convection flow in a d ¼ coshðM PmÞþbmKnM Pm sinhðM PmÞ; vertical micro-porous- annulus in the presence of radial 2 ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi p p p magnetic field, J. Nanofluids 5 (2016) 292–301. d ¼ sinhðM PmÞþbmKnM Pm coshðM PmÞ; 3 [16] B.K. Jha, Babatunde Aina, Sani Isa, MHD natural convection pffiffiffiffiffiffiffi flow in a vertical micro-concentric-annuli in the presence of b vKnA1 ½1 coshðM PmÞ radial magnetic field: an exact solution, J. Ain Shams Eng. d4 ¼ ; d5 ¼ pffiffiffiffiffiffiffi ; M2Pm sinhðM PmÞ (2015), http://dx.doi.org/10.1016/j.asej.2015.07.010. ffiffiffiffiffiffiffi [17] G.S. Seth, Md.S. Ansari, Magnetohydrodynamics convective p 1 ½A0 þ 0:5A1 d6 ¼ bmKnM Pmd2 þ d3; d7 ¼ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ; flow in a rotation channel with Hall effect, Int. J. Theor. Appl. M Pm sinhðM PmÞ Mech. 4 (2) (2009) 205–222.

[18] G.S. Seth, N. Mahto, Md.S. Ansari, R. Nandkeodyar, [24] B.K. Jha, I. Sani, Computational treatment of MHD of Combined free and forced convection flow in a rotating transient natural convection flow in a vertical channel due to channel with arbitrary conducting walls, Int. J. Eng. Sci. symmetric heating in presence of induced magnetic field, J. Phys. Technol. 2 (5) (2010) 184–197. Soc. Jpn. 82 (2013) 084401. [19] G.S. Seth, R. Nandkeodyar, Md.S. Ansari, Unsteady MHD [25] S.K. Ghosh, O.A. Beg, J. Zueco, Hydromagnetic free convective flow within a parallel plate rotating channel with convection flow with induced magnetic field effects, Meccanica thermal source/sink in a porous medium under slip boundary 14 (2010) 175–185. conditions, Int. J. Eng. Sci. Technol. 2 (11) (2010) 1–16. [26] A. Kumar, A.K. Singh, Unsteady MHD free convective flow [20] G.S. Seth, J.K. Singh, Effects of Hall and wall conductance on past a semi-infinite vertical wall with induced magnetic field, mixed convection hydromagnetic flow in a rotating channel, Appl. Math. Comput. 222 (2013) 462–471. Appl. Math. Model. 40 (4) (2016) 2783–2803. [27] Sarveshanand, A.K. Singh, Magnetohydrodynamic free [21] G.S. Seth, B. Kumbhakar, R. Sharma, Unsteady hydromagnetic convection between vertical parallel porous plates in the natural convection flow of a heat absorbing fluid within a presence of induced magnetic field, SpringerPlus, doi:http:// rotating vertical channel in porous medium with Hall effect, J. dx.doi.org/10.1186/s40064-015-1097-1. Appl. Fluid Mech. 8 (4) (2015) 767–779. [28] M. Avci, O. Aydin, Mixed convection in a vertical parallel plate [22] G.S. Seth, R. Sharma, B. Kumbhakar, Effect of Hall current on microchannel, ASME J. Heat Transf. 129 (2007) 162–166. unsteady MHD convective Couette flow of heat absorbing fluid [29] E.R.G. Eckert, R.M. Drake Jr., Analysis of Heat and Mass due to accelerated movement of one of the plates of the channel Transfer, Mcgraw-Hill, New York, 1972, chapter 11. in a porous medium, J. Porous Media 19 (1) (2016) 13–30. [30] R. Goniak, G. Duffa, Corrective term in wall slip equations for [23] R.K. Singh, A.K. Singh, N.C. Sacheti, P. Chandran, On Knudsen layer, J. Thermophys. Heat Transf. 9 (1995) 383–384. hydromagnetic free convection in the presence of induced [31] C.K. Chen, H.C. Weng, Natural convection in a vertical magnetic field, Heat Mass Transf. 46 (2010) 523–529. microchannel, J. Heat Transf. 127 (2005) 1053–1056.