Alexandria Engineering Journal (2016) xxx, xxx-xxx

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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

Microchannel; Transverse magnetic field; Induced magnetic field; Velocity slip and temperature jump

Abstract The present work consists of theoretical investigation of MHD natural convection flow in vertical microchannel formed by two electrically non-conducting infinite vertical parallel plates. The influence of induced magnetic field arising due to the motion of an electrically conducting fluid is taken into consideration. The governing equations of the motion are a set of simultaneous ordinary differential equations and their exact solutions in dimensionless form have been obtained for 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.

© 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

Study associated with natural convection flow of an electrically conducting fluid in the presence of an external magnetic field has received considerable interest due to the enormous applications in various branches of industry, science and technology

* Corresponding author.

E-mail addresses: basant777@yahoo.co.uk (B.K. Jha), ainavicdydx@ gmail.com (B. Aina).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

such as, fire engineering, combustion modelling, geophysics, the cooling of nuclear reactors, operation of magnetohydrody-namic (MHD) generators, and plasma studies. Application of a magnetic field has been found to be effective in controlling the melt convection during crystal growth from melts under terrestrial conditions and has now been widely practises in the metals and semiconductor industries. Several studies have been reported on MHD convective flow under different physical situations. Record of such investigations can be found in the works of Cramer and Pai [1], Chawla [2], Das et al. [3], Sheikholesslami and Gorgi-Bandpy [4], Sheikholesslami et al.

http://dx.doi.org/10.1016/j.aej.2016.06.030

1110-0168 © 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 y 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

H constant strength of applied magnetic field

H dimensional induced magnetic field Greek letters

H dimensionless induced magnetic field ß coefficient of thermal expansion

ln fluid wall interaction parameter ßt; ßv dimensionless variables

J induced current density c ratio of specific heats (Cp/Cv)

ßvKn Knudsen number h dimensionless temperature

M Hartmann number q density

Pm magnetic Prandtl number le magnetic permeability

Pr Prandtl number v fluid kinematic viscosity

Qm dimensionless volume flow rate k molecular mean free path

T 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-

u 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-had et al. [9,10].

Although there are many studies on natural convection flow of an electrically conducting fluid in channels, there are only a few studies regarding natural convection flow of an electrically conducting fluid in microchannel and annular microchannel. In recent years, the present authors and their collaborators have carried out a number of studies on MHD natural convection covering several aspects. For instance, Jha et al. [11] analytically studied the fully developed steady natural convection flow of conducting fluid in a vertical parallel plate microchannel in the presence of transverse magnetic field. The effect of Hartmann number was reported to decrease the volume flow rate. The combined influence of externally applied transverse magnetic field and suction/injection on steady natural convection flow of conducting fluid in a vertical microchannel was carried out by Jha et al. [12]. In another work, Jha et al. [13] examined the effect of wall surface curvature on transient MHD free convective flow in vertical micro-concentric-annuli. Jha et al. [14] studied exact solution of steady fully developed natural convection flow of viscous, incompressible, and electrically conducting fluid in a vertical annular microchannel. Recently, Jha and Aina [15] presented the MHD natural convection flow in a vertical micro-porous-annulus (MPA) in the presence of radial magnetic field. Also, the MHD natural convection flow in vertical micro-concentric-annuli (MCA) in the presence of radial magnetic field has been analysed by Jha et al. [16].

Some recent works related to the present investigation are found in the literature [17-22]. Seth and Ansari [17] presented a study on magnetohydrodynamics convective flow in a rotation channel with Hall effect. Combined free and forced convection flow in a rotating channel with arbitrary conducting walls was conducted by Seth et al. [18]. Also, Seth et al. [19] studied the unsteady MHD convective flow within a parallel plate rotating channel with thermal source/sink in a porous medium under slip boundary conditions. More recently, Seth

and Singh [20] considered the effects of Hall and wall conductance on mixed convection hydromagnetic flow in a rotating channel. Seth et al. [21] studied the unsteady hydromagnetic natural convection flow of a heat absorbing fluid within a rotating vertical channel in porous medium with Hall effect. In another work, Seth et al. [22] investigated the effect of Hall current on unsteady MHD convective Couette flow of heat absorbing fluid due to accelerated movement of one of the plates of the channel in a porous medium.

The above studies on MHD natural convective heat and mass transfer in vertical microchannel and annular microchannel have been limited to the cases in which the induced magnetic field is neglected in order to facilitate the mathematical analysis of the problem as simple. However, the induced magnetic field also generates its own magnetic field in the fluid and as a result of which it modifies the applied magnetic field and motion of the fluid. Therefore, it is known that in several physical situations, it will be necessary to include the effect of induced magnetic field in the MHD equations when magnetic Reynolds number is large enough [1]. Singh et al. [23] presented numerical studies on the hydromagnetic free convective flow in the presence of induced magnetic field. Jha and Sani [24] presented the MHD natural convection flow of an electrically conducting and viscous incompressible fluid in a vertical channel due to symmetric heating in the presence of induced magnetic field. A study on hydromagnetic free convective flow in the presence of induced magnetic field has been carried out by Ghosh et al. [25]. In another related work, Kumar and Singh [26] studied the unsteady MHD free convective flow past a semi-infinite vertical wall by taking into account the induced magnetic field. Recently, Sarveshanand and Singh [27] analytically studied the MHD free convective flow between vertical parallel porous plates in the presence of induced magnetic field and found that the induced current density profile increases with increase in the magnetic Prandtl number.

The induced magnetic field has many important applications in the experimental and theoretical studies of MHD flow

due to its use in many scientific and technological phenomena, for example in MHD electrical power generation, geophysics, purification of crude oil, and glass manufacturing. The role of induced magnetic field is important when the magnetic Reynolds number is large enough [1].

The objective of this work is to present a comprehensive theoretical study of steady hydromagnetic fully developed natural convection flow in a vertical microchannel formed by two infinite vertical parallel plates in the presence of induced magnetic field. Both walls of the microchannel are electrically nonconducting and maintained at different temperature.

The mathematical model employed herein represents a generalization of the work discussed by Jha et al. [11] by incorporating the effects of induced magnetic field. The governing equations corresponding to the velocity, induced magnetic and temperature fields have been obtained in closed form and further, the expression for the induced current density, volume flow rate and skin friction has also been obtained.

2. Mathematical analysis

A steady laminar fully developed natural convection flow of an electrically conducting, viscous incompressible fluid in a vertical microchannel formed by two electrically non-conducting infinite vertical parallel plates is considered. The flow is assumed to be steady and fully developed, i.e., the transverse velocity is zero. The x'-axis is taken vertically upward along the plates and the y'-axis normal to it as presented in the Fig. 1. The distance between the plates is b. The plates are heated asymmetrically with one plate maintained at a temperature T1 while the other plate at a temperature T2 where T > T2. Due to temperature gradient between the plates, natural convection flow occurs in the microchannel. A magnetic field of uniform strength H0 is assumed to be applied in the direction perpendicular to the direction of flow. Both plates y' =0 and y = b are taken to be non-conducting. As stated earlier, our emphasis in this study is to investigate the effect of induced magnetic field on the flow formation inside the vertical microchannel. It is shown that fluid flow and heat transfer at microscale differ greatly from those at macroscale. At macroscale, classical conservation equations successfully cou-

Hx=0 Hx= 0

Figure 1 Flow configuration and coordinate system.

pled with the corresponding wall boundary conditions are valid only if the fluid flow adjacent to the surface is in thermal equilibrium. However, they are not valid for fluid flow at microscale. For this case, the fluid no longer reaches the velocity or the temperature of the surface and therefore a slip condition for the velocity and a jump condition for the temperature should be adopted. In this present study, the usual continuum approach is applied by the continuum equations with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. Velocity slip is defined as [28]

where us is the slip velocity, k is the molecular mean free path and rv is the tangential momentum accommodation coefficient, and the temperature jump is defined as [28]

2 — rt 2y k dT

rt c + 1 Pr dy

where Ts is the temperature of the fluid at the wall, Tw is the wall temperature, and rt is the thermal accommodation coefficient, which depends on the gas and surface materials. However, for air, it assumes typical values near unity [28]. For the rest of the analysis, rv and rt will be assumed to be 1.

By taking into account the conducting fluid, transverse magnetic field and induced magnetic field, the governing equations of the system are given in dimensional form:

d2u | ieH0 dHx

t + gß(T — To) — 0

dy '2 p J"

1 d2H'

+ Ho —, — 0

rie dy'2 0 dy

with the boundary conditions for the velocity, and induced magnetic field and temperature field are as follows:

u' (y')—^ k dr; H'x' (S) = 0. rv ay

2 — rt 2c k dT' , , s

T(y) — T + ^ TTT »W at y — 0 (6)

,, , 2 — rv . du , , ,s

u' (S) =--k -p , Hx (S)=0,

„ 2 — rt 2c k dT' ,

T (y') — T--i -c---J. at y — b (7)

rt c + 1 Pr dy

In the above Eqs. (3)-(7), T is the temperature of the fluid,

H[— (Hx, H0,0)] the magnetic field, v the kinematic viscosity, g the acceleration due to gravity, b the coefficient of thermal expansion, ie the magnetic permeability, q the density, and r is electrical conductivity of the fluid.

Using the following non-dimensional quantities

y — y, U — -

y gßb (T — T0y t — t>'

Pm — voße,

gßb2(T - To)

/E M = Hob [El

the governing equations in non-dimensional form have taken the form of

d2U dH n

+ M— + в = 0 dy2 dy

dH dU n

—2 + MPm— = 0

dy2 dy

(9) (10)

d^ = 0 dy2

with the boundary conditions in non dimensional form as

UM^bmKndy, H(y)—0, fl(y) = n + bmKn lndy at y = 0

U(y) — -bmKndU, H(y) — 0, h(y) —1 - bmKn ln^ at y — 1

where:

2 — rv

■n=ßv ;

2 - rt 2Cs

rt ys + 1 Pr' T2 - To

Kn = b ; b

Ti — To

Referring to the values of rv and rt given in Eckert and Drake [29] and Goniak and Duffa [30], the value of bv is near unity, and the value of bt ranges from near 1 to more than 100 for actual wall surface conditions and is near 1.667 for many engineering applications, corresponding to rv = 1, rt = 1, ys = 1.4 and Pr = 0.71 (bv = 1, bt = 1-667).

The physical quantities used in the above equations are defined in the nomenclature.

3. Method of solution

Eqs. (9)—(11) are coupled system of ordinary differential equations with constant coefficients. This system of linear ordinary differential equations has been solved in closed form by the theory of simultaneous ordinary differential equations. The expressions for the velocity field, the induced magnetic field and the temperature field in non-dimensional form are given by

U(y) — C2 cosWyMvPm) + C3 sinMyM^Pm) + Al

H(y)-^ MPn - M ^ + 0'5A> y2]

- \ZPm[C2 sinh(yM\fPm) + C3 cosh(yM\/Pm)] ( 15)

h(y)= Ao + Aly

The induced current density is given by

= M [Ci - Aiy] + M [Ao + Aiy] + MPmU(y) (18)

whereA0, Ab C1;..., C4 are all constants given in Appendix.

Two important parameters for buoyancy - induced micro-flow are the volume flow rate (Qm) and skin friction (s). The dimensionless volume flow rate is

Qm = U(y) dy

M Pm A\

[C2 sinh(M\/Pm) + C3{cosh(M\/Pm) - 1g]

2MPm M2Pm

Using expression (14), the skin - friction on both microchannel walls in dimensionless form are given by:

= M\f¥mC3 +

(21) (22)

: M^Pmn[C2 sinh + C3 cosh(M\ZPm)J

4. Results and discussion

The present study on magnetohydrodynamic natural convection flow in vertical microchannel formed by two electrically non-conducting infinite vertical parallel plates in the presence of induced magnetic field is controlled by a number of physical parameters such as the wall-ambient temperature difference ratio (П), rarefaction (bmKn), fluid wall interaction (ln), Hartmann number (M), and the magnetic Prandtl number (Pm). The effects of these parameters on the velocity profile, induced magnetic field profile, the induced current density profile and the skin friction are shown using the line graphs. The present parametric study has been carried out over reasonable ranges of 0 6 bmKn 6 0.1, 0 6 ln 6 10, 0 6 M 6 5, and 0 6 Pm 6 1 and the selected reference values of pvKn, ln, M, and Pm for the present analysis are 0.05, 1.667, 5.0 and 0.5.

The expression for the temperature in equation (16), the effects of rarefaction parameter (bmKn), and fluid wall interaction parameter (ln) on the temperature profile and rate of heat transfer which is expressed as the Nusselt number are exactly the same as those given by Chen and Weng [31].

Figs. 2-5 show the variation of the velocity profiles with the parameters occurring in the governing equations. Fig. 2 illustrates the effects of rarefaction parameter (bmKn) and wall-ambient temperature difference ratio (П) on velocity profiles for fixed values of ln = 1.667, M = 5.0, and Pm = 0.5. It is noticed that, an increase in rarefaction parameter and wall-ambient temperature difference ratio causes a pronounced enhancement in the velocity slip. This result yields an observ-

Figure 2 Variation of velocity with ßmKn (ln = 1.667, M = 5.0, Pm = 0.5).

Figure 3 Variation of velocity with ln (ßmKn = 0.05, M = 5.0, Pm = 0.5).

able increase in the fluid velocity. This effect can be explained from the fact that, as rarefaction parameter increases, the temperature jump increases and this reduces the amount of heat transfer from the microchannel surfaces to the fluid. The reduction in velocity due to the reduction in heat transfer is offset by the increase in the fluid velocity due to the reduction in the frictional retarding forces near the microchannel surfaces. Furthermore, as the wall-ambient temperature difference ratio (П) increases, the effect of rarefaction parameter (ßmKn) on the microchannel slip velocity becomes significant.

Fig. 3 illustrates the effects of fluid wall interaction parameter (ln) as well as wall-ambient temperature difference ratio (П) on velocity profiles for fixed values of ßmKn = 0.05, M = 5.0, and Pm = 0.5. It is observed that, the effect of the fluid wall interaction parameter is to enhance the fluid velocity at the microchannel wall (y = 0) and to reduce the fluid velocity at the microchannel wall (y = 1). In addition, it is evident that, there exist points of intersection inside the microchannel

Figure 4 Variation of velocity with M (ßmKn = 0.05, ln = 1.667, Pm = 0.5).

Figure 5 Variation of velocity with Pm (fivKn = 0.05, ln = 1.667, M = 5.0).

where velocity profile is independent of fluid wall interaction parameter. Also, the impacts of fluid-wall interaction parameter on the microchannel slip velocity become significant with the decrease of the wall-ambient temperature difference ratio.

Figs. 4 and 5 exhibit the effects of wall-ambient temperature difference ratio (П), Hartmann number (M), and the magnetic Prandtl number (Pm), respectively on velocity profiles for fixed values of pvKn = 0.05, and ln = 1.667. It is evident from these Figures that, an increase in Hartmann number and magnetic Prandtl number causes reduction in the fluid velocity. Physically speaking, the presence of transverse magnetic field sets in a resistive type force (Lorentz force), which is a retarding force on the velocity field. It is further observed that, there exist points of intersection inside the microchannel where velocity field is independent of Hartmann number and magnetic Prandtl number and this behaviour is observed in the case of asymmetric heating (П = —1). Also, the influence of

Figure 6 Variation of induced magnetic field with ßvKn (In = 1.667, M =5, Pm = 0.5).

Figure 7 Variation of induced magnetic field with ln (bmKn = 0.05, M =5, Pm = 0.5).

Hartmann number and magnetic Prandtl number on the microchannel slip velocity becomes significant with the increase of the wall-ambient temperature difference ratio.

Figs. 6-9 show the variation of induced magnetic field with the parameters occurring in the governing equations. Fig. 6 depicts the distribution of induced magnetic field with respect to rarefaction parameter (fivKn) and wall-ambient temperature difference ratio (£) for fixed values of ln = 1.667, M = 5.0, and Pm = 0.5. It is observed that, the rarefaction parameter influences the flow formation excluding the case of symmetric heating (n = 1). For the case of asymmetric heating (n = 0, — 1), it is found that, the induced magnetic field increases with the increase in rarefaction parameter. It is further noticed from Fig. 6 that, the role of rarefaction parameter on induced magnetic field is more pronounced with reduction in the values of wall-ambient temperature difference ratio.

Fig. 7 shows the induced magnetic field with respect to fluid wall interaction parameter (ln) and wall-ambient temperature

Figure 8 Variation of induced magnetic field with M (ßvKn = 0.05, ln = 1.667, Pm = 0.5).

Figure 9 Variation of induced magnetic field with Pm (fi,Kn = 0.05, ln = 1.667, M = 5).

difference ratio (n) for fixed values of pvKn = 0.05, M = 5.0, and Pm = 0.5. It is evident from Fig. 7 that, the fluid wall interaction parameter influences the flow excluding the case of symmetric heating (n = 1). For the case of asymmetric heating (n = 0, —1), it is found that, an increase in fluid wall interaction parameter causes a pronounced reduction in fluid velocity. Also, the impacts of fluid-wall interaction parameter on the induced magnetic field become significant with the decrease in wall-ambient temperature difference ratio.

The induced magnetic field profile is plotted in Figs. 8 and 9 with various values of Hartmann number (M) and the magnetic Prandtl number (Pm), respectively. It is interesting to note that the strength of the induced magnetic field is directly proportional to the strength of Hartmann number as well as magnetic Prandtl number near the microchannel wall at y = 0 while it is inversely proportional near the microchannel wall at y = 1. In addition, it is observed that, there exist points of intersection inside the vertical microchannel where the induced magnetic field is independent of Hartmann number

Figure 10 Variation of induced current density with ßvKn (ln = 1.667, M = 5, Pm = 0.5).

Figure 11 Variation of induced current density with ln (bvKn — 0.05, M =5, Pm = 0.5).

and magnetic Prandtl number and these strongly depend on the wall-ambient temperature difference ratio. Furthermore, it is also observed from these Figures that, as the wall-ambient temperature difference ratio increases, the effects of Hartmann number as well as magnetic Prandtl number on the induced magnetic field become significant.

Figs. 10-13 describe the behaviour of induced current density profile with the parameters occurring in the governing equations. Fig. 10 presents the variation of induced current density with respect to rarefaction parameter and wall-ambient temperature difference ratio for fixed values of ln = 1.667, M = 5.0, and Pm = 0.5. It is seen that, an increase in rarefaction parameter leads to an increase in the induced current density. It is also found that there exist points of intersection inside the microchannel where induced current density is independent of rarefaction parameter.

The variation of the induced current density with respect to fluid wall interaction parameter (ln) and wall-ambient temperature difference ratio (£) for fixed values of pvKn — 0.05,

Figure 12 Variation of induced current density with M (ßvKn — 0.05, ln = 1.667, Pm = 0.5).

Figure 13 Variation of induced current density with Pm (bvKn — 0.05, ln = 1.667, M = 5).

M = 5.0, and Pm = 0.5, is show in Fig. 11. The induced current density decreases with the increase in fluid wall interaction parameter in one part of the microchannel and the reverse trend occurs in the other part.

Figs. 12 and 13 exhibit the effects of wall-ambient temperature difference ratio (£), Hartmann number (M), and the magnetic Prandtl number (Pm), respectively on induced current density for fixed values of pvKn — 0.05, ln = 1.667. It is evident from these figures 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 while reverse trend occurs at the microchannel plates. Moreover, it is interesting to note that current density changes its behaviour with Hartmann number and magnetic Prandtl number at two different locations inside the microchannel.

Figs. 14-16 describe the behaviour of volume flow rate with the parameters occurring in the governing equations. The volume flow rate variations are shown in Fig. 14 with respect to

ß vKn

Figure 14 Variation of volume flow rate versus ßvKn with ln.

ß vKn

Figure 15 Variation of volume flow rate versus ßvKn with M.

fluid wall interaction parameter and rarefaction parameter for different values of wall-ambient temperature difference ratio. It is seen that, an increase in the value of rarefaction parameter leads to enhancement in volume flow rate for both symmetric and asymmetric heating. Also, it is observed that volume flow rate is a decreasing function of fluid wall interaction and wall-ambient temperature difference ratio.

Figs. 15 and 16 show the effects of rarefaction parameter, Hartmann number and magnetic Prandtl number on volume flow rate for different values of wall-ambient temperature difference 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 velocity decreases as the Hartmann number increases, which causes a reduction in the volume flow rate. In addition, it is also interesting to note that the role of Hartmann number and magnetic Prandtl number is insignificant for the case of asymmetric heating (П = —1).

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 interaction at the microchannel wall y = 0 while the reverse trend occurs at the microchannel wall y = 1. Furthermore, it is found that, the magnitude of skin friction is higher in case of asymmetric heating in comparison with symmetric heating.

Figs. 19-22 reveal the combined influences of rarefaction parameter, Hartmann number and magnetic Prandtl number on skin friction for different values of wall-ambient temperature difference ratio at the microchannel walls y = 0 and y = 1, respectively. It is observed that, increase in Hartmann number and magnetic Prandtl number causes a reduction in skin friction at the microchannel walls y = 0 and y = 1. It is further observed that the magnitude of skin friction is higher in case of asymmetric heating (П = 0, —1) in comparison with symmetric heating (П = 1).

In Table 1, we compared the values of skin friction at microchannel walls in the presence and absence of induced magnetic field. From this Table 1, we can observe clearly that 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].

0.02 0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

ß v Kn

?=1 -----?=-1

0 --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 ß vKn

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

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

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

Figure 21 Variation of skin friction versus bmKn with Pm

(y = 0).

Figure 22 Variation of skin friction versus bmKn with Pm.

Figure 20 Variation of skin friction versus bmKn with M.

Table 1 Comparison of numerical values of skin friction on the vertical microchannel walls in the presence and absence of induced magnetic field (I.M.F).

n PvKn s0 (without s0 (with I. s1 (without s1 (with I.

I.M.F.) Jha M.F.) I.M.F.) Jha M.F.)

et al. [11] present et al. [11] present

work work

1 0.0 0.1973 0.5000 0.1973 0.5000

0.05 0.1583 0.5000 0.1583 0.5000

0.1 0.1321 0.5000 0.1321 0.5000

0 0.0 0.0373 0.1801 0.1600 0.3199

0.05 0.0372 1.1995 0.1211 0.3005

0.1 0.0355 0.2119 0.0966 0.2881

-1 0.0 -0.1227 -0.1398 0.1227 0.1398

0.05 -0.0839 -0.1009 0.0839 0.1009

0.1 -0.0611 -0.0763 0.0611 0.0763

5. Conclusions

The effect of induced magnetic field on the MHD natural convection flow of a viscous, incompressible and electrically conducting fluid in the presence of transverse magnetic field in a vertical microchannel formed by two electrically nonconducting infinite vertical parallel plates has been investigated analytically. The effects of various parameters on the velocity, induced magnetic field, induced current density and skin friction profiles have been shown in the line graphs. The main findings are as follows:

(1) Increasing the value of Hartmann number and magnetic Prandtl number causes enhancement in induced magnetic field.

(2) There exist points of intersection inside the microchannel where the induced magnetic field is independent of Hartmann number and magnetic Prandtl number and these strongly depend on the wall-ambient temperature difference ratio.

(3) 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.

(4) It is found that, increase in Hartmann number and magnetic Prandtl number causes a pronounced reduction in volume flow rate.

(5) The magnitude of skin friction is higher in case of asymmetric heating (П = 0, —1) in comparison with symmetric heating (П = 1).

(6) The effect of Hartmann number on the skin friction can be useful in mechanical engineering for modelling a system. We can obtain a suitable value of Hartmann number for which the value of the skin friction will be optimum.

(7) This study exactly agrees with the finding of Chen and Weng [31] in the absence of magnetic field.

Appendix

A n . ßvKn ln(1 - n) . (1 - n)

A0 = n + 1 , rs ,-; A1 ="

ßvKnM\/Pm + d3

1 + 2ßvKn ln

d!3d!4 — d\od\\ d12 d14 - d8 d11

1 + 2ßvKn ln'

d10d12 - d8d13 d12 d14 - d8 d11

C2 = (ßvKnA' + C1 ) + ßv WP^

C4 = TT^d + C3MpPm, d1 = (-ßvKn - 1),

M2Pm M2Pm

d2 = cosh(M\/Pm) + ß^KnM\TPm sinh(M\f¥m), d3 = sinh(M\f¥m) + ßvKnM\pPm cosh(M\ÎPm),

_ ßvKnA, _ [1 - cosh(MiPm)] 4 = M2Pm ' 5 = sinh(MvPw) '

d6 = ßvKnMPPmd2 + d3; d7 =--i [Ao + ^^ ;

M\/Pm sinh(MvPm)

d8 = - .

d10 = d9 - d6d7, d,, = ßvKnM\JPmd2 + d3,

d9 = d3d4 + d, ßvKnM\J Pm,

d2 - 1

— ; d,3 = d, - d2d4, d,4 = d5d6,

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