Mixed convection hydromagnetic flow in a rotating

Applied Mathematical Modelling 40 (2016) 2783–2803

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Mixed convection hydromagnetic flow in a rotating channel with Hall and wall conductance effects G.S. Seth a, J.K. Singh b,∗ a b

Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826004, India Department of Mathematics, V. S. K. University, Bellary, Karnataka 583105, India

a r t i c l e

i n f o

Article history: Received 19 April 2013 Revised 19 March 2015 Accepted 12 October 2015 Available online 21 October 2015 Keywords: Hall current Rotation Magnetic field Modified Ekman boundary layer Modified Hartmann boundary layer Viscous and Joule dissipations

a b s t r a c t Mixed convection hydromagnetic flow of a viscous, incompressible, electrically and thermally conducting fluid in a rotating channel taking Hall current into account is studied. Fluid flow within the channel is induced due to an applied pressure gradient acting along the longitudinal axis of the plates of the channel. Exact solution of the governing equations is obtained in closed form. Expressions for the shear stress and critical Grashof number at the plates of the channel due to primary and secondary flows and mass flow rates in the primary and secondary flow directions are also derived. Asymptotic behavior of the solution for fluid velocity and induced magnetic field is analyzed for large values of rotation parameter K2 and magnetic parameter M2 to gain some physical insight into flow pattern. Heat transfer characteristics of fluid flow are considered taking viscous and Joule dissipations into account. Numerical solution of energy equation and numerical values of rate of heat transfer at the plates of the channel are computed with the help of MATLAB software. The numerical values of fluid velocity, induced magnetic field and fluid temperature are displayed graphically versus channel width variable η for various values of pertinent flow parameters whereas that of shear stress and critical Grashof number at the plates of the channel due to primary and secondary flows, mass flow rates in the primary and secondary flow directions and rate of heat transfer at the plates of the channel are presented in tabular form for various values of pertinent flow parameters. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Theoretical/experimental investigation of the problems of hydromagnetic flow of an electrically conducting fluid permeated by a magnetic field with heat transfer is of considerable interest because it has several practical applications in plasma aerodynamics [1], MHD power generator [2], nuclear engineering [3], astrophysical and geophysical fluid dynamics [4,5] and manufacturing process in industry [6, 7]. Problems of magnetohydrodynamic heat transfer may be roughly divided into two groups. In group one, heating is an incidental byproduct of electromagnetic fields. This group includes devices like MHD generators, MHD accelerators and to a lesser degree MHD pumps and flow meters. These are broadly mentioned as channel or duct flows. In group two, the primary use of electromagnetic fields is to control heat transfer. This group contains free convection flows and aerodynamic heating where geometric configurations are varied. It is worthy to note that, in MHD heat transfer problems, the usual Reynolds analogy between skin friction and heat transfer, as in non-conducting fluid, does not hold in general. This is due to the fact that, in addition to the viscous dissipation, there is a Joule dissipation of heat caused by flow of electric current ∗

Corresponding author. Tel.: +91 326 2235421; fax: +91 326 2296563. E-mail addresses: [email protected] (G.S. Seth), [email protected], [email protected] (J.K. Singh).

http://dx.doi.org/10.1016/j.apm.2015.10.015 S0307-904X(15)00660-5/© 2015 Elsevier Inc. All rights reserved.

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G.S. Seth, J.K. Singh / Applied Mathematical Modelling 40 (2016) 2783–2803

in the fluid. Hydromagnetic channel flow with heat transfer due to forced convection is studied by Yen [8], Jagadeesan [9] and Soundalgekar [10] under different conditions. The effects of thermal buoyancy force on MHD forced convection flow are of practical interest because fluids with low Prandtl number are electrically conducting and are more sensitive to the gravitational field than that with high Prandtl number. Hence the interplay of thermal buoyancy force with the electromagnetic forces determines the ultimate behavior of an electrically conducting fluid with low Prandtl number in the presence of magnetic field. Taking into consideration this fact Gill and Casal [11], Gupta [12], Das [13], Jana [14], Singh [15], Mazumder et al. [16], Datta and Jana [17], Kant [18], Seth and Ghosh [19], Ghosh and Nandi [20], Ghosh et al. [21] and Guria et al. [22] studied MHD mixed convection flow of a viscous, incompressible and electrical fluid within a parallel plate channel in the presence of magnetic field by considering different aspects of the fluid flow problem. Investigation of the problems of hydromagnetic flow of an electrically conducting fluid in a rotating environment assumes considerable importance due to the occurrence of various natural phenomena which are generated by the action of Coriolis and magnetic forces. An order of magnitude analysis shows that the effects of Coriolis force are more significant than that of inertial and viscous forces in the magnetohydrodynamic equations of motion in a rotating system whereas Coriolis and electromagnetic forces are comparable in magnitude. Investigation of such fluid flow problems may find application in several areas of astrophysics, geophysics and engineering viz. maintenance and secular variations of terrestrial magnetic field due to motion of Earth’s liquid core, internal rotation rate of sun, planetary and solar dynamo problems, structure of rotating magnetic stars, MHD Ekman pumping, turbo machines, rotating MHD generators, rotating drum type separators in closed cycle two phase MHD generator flow, etc. Combined effects of Coriolis and magnetic forces on hydromagnetic mixed convection flow of a viscous, incompressible and electrically conducting fluid in a rotating system in the presence of a uniform transverse magnetic field is studied by Mohan [23], Sarojamma and Krishna [24], Shivaprasad et al. [25], Prasad Rao et al. [26], Ghosh and Bhattacharjee [27], Seth and Singh [28], Seth and Ansari [29], Seth et al. [30] and Singh and Pathak [31] to analyze different aspects of this problem. It was pointed out by Gebhart [32] that the effect of viscous dissipation plays a vital role in natural convection in various devices that are subjected to large deceleration, or which operate at high rotating speeds and also in strong gravitational field processes on large scales (on large planets) and geological process. Under such situations if an electrically conducting fluid flows in the presence of magnetic field, the effects of viscous and Joule dissipations become significant because it act as volumetric heat source. Keeping in mind the importance of such study, Anjali Devi and Ganga [33] considered the effects of viscous and Joule dissipations on hydromagnetic natural convection heat and mass transfer flow past a stretching porous surface embedded in a porous medium. Seth and Singh [28] and Seth et al. [30] investigated combined free and forced convection MHD flow in a rotating channel with perfectly conducting and arbitrary conducting walls respectively taking viscous and Joule dissipations into account. When an ionized fluid with low density is subjected to a strong magnetic field then the electrical conductivity normal to the magnetic field is lowered because of free spiraling of electrons and ions about the magnetic lines of force just before the collision and a current, namely, Hall current is thereby generated which is mutually perpendicular to electric and magnetic fields (Sutton and Sherman [34]). Most important characteristic of Hall current is that it induces secondary flow in the flow-field which is also characteristics of Coriolis force. Therefore, it is of practical interest to study the combined effects of Hall current and rotation on such fluid flow problems which may find applications in MHD power generation, nuclear power reactors and underground energy storage system and in several areas of astrophysics and geophysics. Sivaprasad et al. [25] studied Hall effects on MHD free and forced convection flow in porous rotating channel whereas Singh and Pathak [31] considered the effects of Hall current and rotation on mixed convection oscillatory MHD flow through a porous medium filled in a vertical channel in the presence of thermal radiation. Shivprasad et al. [25] and Singh and Pathak [31] have neglected induced magnetic field produced by fluid motion. Due to this reason Joule dissipation is not taken into account in their study. Abo-Eldahab and Aziz [35] studied the effects of viscous and Joule dissipations on MHD free convection flow past a semi-infinite vertical flat plate in the presence of Hall current and ion slip. Seth and Ansari [29] extended the problem considered by Seth and Singh [28] to include the effects of Hall current. Present investigation deals with study of steady hydromagnetic mixed convection flow of a viscous, incompressible, electrically and thermally conducting fluid in a rotating channel with arbitrary conducting walls taking Hall current into account. Heat transfer characteristics of the flow are analyzed taking viscous and Joule dissipations into account. Lower wall of the channel is considered of finite thickness d1 and electrical conductivity σ 1 whereas upper wall of the channel is considered of finite thickness d2 and electrical conductivity σ 2 . To solve this problem, we have derived boundary conditions for magnetic field taking Hall current into account. The numerical values of fluid velocity, induced magnetic field and fluid temperature are displayed graphically versus channel width variable η for various values of pertinent flow parameters whereas that of shear stress and critical Grashof number at the plates of the channel due to primary and secondary flows, mass flow rates in the primary and secondary flow directions and rate of heat transfer at the plates of the channel are presented in tabular form for various values of pertinent flow parameters. Numerical values of above physical quantities are also obtained when wall conductance ratios φ 1 and φ 2 are zero (i.e. when the walls are non-conducting) and when φ 1 and φ 2 → ∞ (i.e. when the walls are perfectly conducting) which are the particular cases of our problem. Therefore, the problem under consideration represents the most general case and the problem investigated by Seth and Ansari [29] is a special case of the present one. 2. Mathematical analysis Consider flow of a viscous, incompressible, electrically and thermally conducting fluid between two infinite parallel arbitrary conducting plates z = 0 and z = L in the presence of a uniform transverse magnetic field H0 which is applied parallel to z-axis.


2785

Fig. 1. Physical model of the problem.

The thickness and electrical conductivity of lower and upper plates are d1 , σ 1 and d2 , σ 2 respectively. Both the fluid and channel rotate in unison in counter clockwise direction with a uniform angular velocity about z-axis. Fluid flow within the channel is induced due to a uniform pressure gradient applied along x-direction. The plates of the channel are cooled or heated by a uniform temperature gradient acting along x-direction so that there is an axial temperature variation within the plates of the channel. Physical model of the problem is presented in Fig. 1. Since plates of the channel are of infinite extent in x and y-directions and fluid flow is steady and fully developed so all induced magnetic field H, physical quantities, except pressure and temperature, depend on z only. Therefore, fluid velocity q current density J and electric field E are given by

≡ (Hx , Hy , H0 ), ≡ (Vx , Vy , 0), H q

J ≡ (Jx , Jy , 0) and E ≡ (Ex , Ey , Ez ),

(1)

which are compatible with the fundamental equations of magnetohydrodynamics in a rotating frame of reference. From Faraday’s law in steady state (i.e. ∇ × E = 0), we obtain

dEx = 0, dz

dEy = 0. dz

(2)

This implies that Ex and Ey are constant within the fluid as well as on interfaces of the plates. Taking into consideration the assumption made above, the governing equations for steady flow of a viscous, incompressible, electrically and thermally conducting fluid in a rotating system taking Hall current into account are presented in the following form

d2Vx μe H0 dHx 1 ∂ p∗ +υ 2 + , ρ ∂x ρ dz dz d2Vy μe H0 dHy , 2Vx = υ 2 + ρ dz dz 1 d μe 2 − p∗+ Hx + Hy2 + H02 − g(1 − β(T − T0 )) = 0, ρ dz 2 −2Vy = −

dHy dHx − = σ (Ex + μe H0Vy ), dz dz dHy dHx +m = σ (Ey − μe H0Vx ), dz dz m d 2 − H + Hy2 + H02 = σ (Ez + μe (Vx Hy − Vy Hx )), 2H0 dz x m

(3) (4) (5) (6) (7) (8)

where υ , σ , ρ , μe , m = ωe τe , ωe , τe , p∗, g, β , T and T0 are, respectively, kinematic coefficient of viscosity, electrical conductivity of the fluid, fluid density, magnetic permeability, Hall current parameter, cyclotron frequency, electron collision time, modified pressure including centrifugal force, acceleration due to gravity, coefficient of thermal expansion, fluid temperature and temperature in reference state. Boundary conditions for fluid velocity are given by

Vx = Vy = 0 at z = 0, Vx = Vy = 0 at z = L.

(9)

Differentiating Eqs. (6) and (7) with respect to z and making use of Eq. (2), we obtain

d2 Hy dVy d2 Hx , − = σ μe H0 2 dz dz dz2 2 2 d Hy d Hx dVx . +m = −σ μe H0 dz dz2 dz2

m

(10) (11)

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Since plates of the channel are cooled or heated by a uniform temperature gradient A1 acting along x-direction so fluid temperature may be considered in the following form

T − T0 = A1 x + θ1 (z),

(12)

where θ 1 (z) is arbitrary function of z. Integrating Eq. (5) with respect to z and using Eq. (12), we obtain

1 2 μe Hx + Hy2 + H02 − ρ gz + ρ gβ 2

p∗ = −

A1 x dz + ρ gβ

θ1 (z) dz + ρ K1 x,

(13)

where K1 is uniform pressure gradient acting along x-direction due to which fluid flow within the channel is induced. Eq. (3) with help of (13) reduces to

−2Vy = υ

d2Vx + dz2

μe H0 dHx − gβ A1 z − K1 . ρ dz

(14)

Combining Eqs. (14), (6) and (10) with (4), (7) and (11) respectively, we obtain

2iV = υ

d 2V + dz2

μe H0 dH − gβ A1 z − K1 , ρ dz

dH σ =− (iE + μe H0V ), dz (1 − im) d2 H σ μe H0 dV , =− dz2 (1 − im) dz

(15) (16) (17)

where V = Vx + iVy , H = Hx + iHy and E = Ex + iEy . Boundary conditions (9) for fluid velocity in, compact form, become

V = 0 at z = 0, V = 0 at z = L.

(18)

It is assumed that the plates of the channel are arbitrary electrically conducting. We have derived boundary conditions for the magnetic field. The procedure for derivation of boundary conditions is explained below. 2.1. Derivation of the boundary conditions for magnetic field It is evident from Eq. (2) that E is constant within the fluid as well as on the interfaces of the plates. Therefore, within the lower plate z = 0, we obtain from (16) as

dHd1 = −iσ1 Ed1 , dz

(19)

where subscript ‘d1 ’ denotes the quantities within the lower plate of the channel. Since tangential components of electric field are continuous at the interface of the lower plate, therefore from Eq. (19), we obtain

Ed1 = E = −

1 dHd1 . iσ1 dz

(20)

From Eqs. (16) and (20), we obtain

dH dz

=− z=0

dHd1 iσ E σ . = (1 − im) σ1 (1 − im) dz

(21)

The tangential component of magnetic field is continuous within and on interface of the lower plate. Therefore, we have

Hd1 (−d1 ) = 0 and

Hd1 (0) = H (0).

(22)

Integrating Eq. (19) and using conditions (22), we obtain

dHd1 H (0) = . dz d1

(23)

Using Eq. (23) in Eq. (21), we obtain

dH − dz

σH = 0 at z = 0. σ1 d1 (1 − im)

(24)

Similarly, within the upper plate z = L, we obtain from (16) as

dHd2 = −iσ2 Ed2 , dz

(25)


2787

where subscript ‘d2 ’ denotes the quantities within the upper plate of the channel. Since the tangential components of electric field is continuous at the interface of the upper plate also, therefore from Eq. (25), we obtain

Ed2 = E = −

1 dHd2 . iσ2 dz

(26)

From Eqs. (16) and (26), we obtain

dH dz

=− z=L

dHd2 iσ E σ = . (1 − im) σ2 (1 − im) dz

(27)

The tangential component of magnetic field is continuous within and on interface of the upper plate. Therefore, we have

Hd2 (L + d2 ) = 0

and Hd2 (L) = H (L).

(28)

Integrating Eq. (25) and using conditions (28), we obtain

dHd2 H (L) =− . dz d2

(29)

Using Eq. (29) in Eq. (27), we obtain

dH + dz

σH = 0 at z = L. σ2 d2 (1 − im)

(30)

Thus the boundary conditions for magnetic field are presented by Eqs. (24) and (30). We introduce following non-dimensional variables to represent Eqs. (15) and (17) in non-dimensional form

η = z/L, v = V L/υ and h = H/σ μe υ H0 .

(31)

Making use of (31), Eqs. (15) and (17), in non-dimensional form, become

d2 v dh + M2 − 2iK 2 v = Grη + R, dη dη2

(32)

1 d2 h dv + = 0, dη2 (1 − im) dη

(33)

where M2 = μ2e L2 H02 (σ /ρυ) is magnetic parameter which is square of Hartmann number and it represents the relative strength of magnetic force to the viscous force, K 2 = L2 /υ is rotation parameter which is reciprocal of Ekman number and it represents the relative strength of Coriolis force to the viscous force, Gr = gβ A1 L4 /υ 2 is Grashof number which represents the relative strength of thermal buoyancy force to the viscous force and R = K1 L3 /υ 2 is a non-dimensional pressure gradient which is a constant. Plates of the channel are cooled or heated by a uniform temperature gradient A1 acting along x-direction. Since expression for Gr containing A1 . So Gr is positive or negative according as the channel walls are heated or cooled in the axial direction. Boundary conditions (18) for fluid velocity, in non-dimensional form, become

v = 0 at η = 0, v = 0 at η = 1.

(34)

Boundary conditions (24) and (30) for magnetic field, in non-dimensional form, are given by

dh − dη dh + dη

1 h=0 φ1 (1 − im) 1 h=0 φ2 (1 − im)

at

⎫ η = 0,⎪ ⎬

at

⎭ η = 1,⎪

(35)

where φ1 = σ1 d1 /σ L and φ2 = σ2 d2 /σ L are wall conductance ratio of the lower and upper plate respectively. 2.2. Particular cases of boundary conditions for magnetic field Three particular cases of boundary conditions (35) for magnetic field may be considered. i) When φ1 = φ2 = 0 (i.e. non-conducting walls), conditions (35) reduce to h = 0 atη = 0, 1. ii) When φ 1 and φ 2 → ∞ (i.e. perfectly conducting walls), conditions (35) reduce to ddhη = 0 at η = 0, 1.

Seth and Ansari [29] has considered these boundary conditions when channel is symmetric about η = 0 (i.e.

dh dη

=

0 at η = ± 1) in their problem. iii) In absence of Hall current i.e. m = 0, conditions (35) reduce to ddhη − φh = 0 at η = 0; ddhη + φh = 0 at η = 1. 1 2 Seth et al. [30] used these boundary conditions when channel is symmetric about η = 0 (i.e. ddhη − φh = 0 at η = −1;

dh dη

+ φh = 0 at η = 1) to find solution of their problem. 1

2

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Thus the investigated problem provides the most general solution. Eqs. (32) and (33) subject to the boundary conditions, (34) and (35) represent the mathematical model of this fluid flow problem. Eqs. (32) and (33) with the help of boundary conditions (34) and (35) are solved and the solution for fluid velocity and induced magnetic field are presented in the following form

v(η) = vx + ivy = A sinh λ1 η − B(1 − cosh λ1 η) − Grη/λ21 ,

(36)

h(η) = hx + ihy = −

1 Grη2 A(1 − cosh λ1 η) − B(sinh λ1 η + λ1 φ1 (1 − im)) + 2λ1 λ1 (1 − im)

(2iK 2 B − R) M2

(η + φ1 (1 − im)),

where

⎡ 1 λ1 = α1 + iβ1 , α1 , β1 = √ ⎣

4

(37)

2

2 1/2

⎤1/2 2

M M mM ⎦ , + + 2K 2 ± (1 + m2 ) (1 + m2 ) (1 + m2 )2 ⎫ 2λ31 m∗1 (1 − cosh λ1 ) sinh λ1 R + m∗2 (2 + 2 cosh λ1 − λ1 sinh λ1 ) + 2λ1 m∗3 sinh λ1 Gr ⎪ ⎪ ,⎪ A= ⎪ ⎪ 2λ21 sinh λ1 2m∗2 + λ1 m∗3 sinh λ1 ⎪ ⎪ ⎪ ⎪ ⎬ 3 ∗ ∗ 2 2 2λ1 m1 sinh λ1 R − 2m2 + λ1 2iK − λ1 sinh λ1 Gr B= , ⎪ 2λ21 2m∗2 + λ1 m∗3 sinh λ1 ⎪ ⎪ ⎪ ⎪ ⎪ m∗1 = {1 + φ(1 − im)}, m∗2 = 2iK 2 − λ21 (1 − cosh λ1 ) , m∗3 = λ21 φ(1 − im) + 2iK 2 , ⎪ ⎪ ⎪ ⎭ φ = φ1 + φ2 , (vx , vy ) = (Vx , Vy )L/υ and (hx , hy ) = (Hx , Hy )/σ μe υ H0 . 2

(38)

It is noticed from Eqs. (36) to (38) that fluid velocity depends on the sum of wall conductance ratios φ 1 and φ 2 whereas induced magnetic field depends on their individual values. The solution (36)–(38) represents combined free and forced convection flow in a rotating channel with arbitrary conducting walls taking Hall current into account. In the absence of Hall current (i.e m = 0) the solution for fluid velocity and induced magnetic field given by (36)–(38), assumes the following form

v(η) = A1 sinh λ2 η − B1 (1 − cosh λ2 η) − Grη/λ22 , 2 2iK B1 − R 1 Grη2 h(η) = A1 (1 − cosh λ2 η) − B1 (sinh λ2 η + λ2 φ1 ) + − (η + φ1 ), λ2 2λ2 M2 where

⎫ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 3 ∗ ∗ ∗ ⎪ 2λ2 m4 (1 − cosh λ2 ) sinh λ2 R + m5 (2 + 2 cosh λ2 − λ2 sinh λ2 ) + 2λ2 m6 sinh λ2 Gr ⎪ ⎪ A1 = ,⎬ 2λ22 sinh λ2 2m∗5 + λ2 m∗6 sinh λ2 ⎪ ⎪ ⎪ 2λ32 m∗4 sinh λ2 R − 2m∗5 + λ2 2iK 2 − λ22 sinh λ2 Gr ⎪ ⎪ ⎪ B1 = , ⎪ ⎪ 2λ22 2m∗5 + λm∗6 sinh λ2 ⎪ ⎪ ⎪ ⎭ m∗4 = {1 + φ}, m∗5 = 2iK 2 − λ22 (1 − cosh λ2 ) , and m∗6 = λ22 φ + 2iK 2 . 1

λ2 = α2 + iβ2 ,

α2 , β2 = √ [{M4 + 4K 4 }

1/2

(39)

(40)

± M2 ]1/2 ,

(41)

This result agrees with the results obtained by Seth et al. [30] when channel is symmetric about η = 1/2 (i.e. lower plate is at

η = 0 and upper plate is at η = 1) in their problem. 3. Shear stress at the plates

The expressions for non-dimensional shear stress components τ x and τ y at the lower plate η = 0 and upper plate η = 1, due to primary and secondary flows respectively, are given by

(τx + iτy )η=0 = Aλ31 − Gr , (τx + iτy )η=1 =

1

λ21

λ31 (A cosh λ1 + B sinh λ1 ) − Gr .

(42) (43)


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4. Critical grashof number It may be noted that the value of Grashof number for which shear stress at the plates of the channel becomes zero, is called critical Grashof number. The expressions for critical Grashof numbers Gcx and Gcy at the lower plate η = 0 and upper plate η = 1, due to primary and secondary flows respectively, are given by

(Gcx + iGcy )η=0 = ⎡

m∗2

2λ41 m∗1 (cosh λ1 − 1) sinh λ1

R, λ1 m∗7 − 4 sinh λ1 + 2λ1 m∗3 (λ1 − sinh λ1 ) sinh λ1

(44)

⎤

⎢ ⎥ ⎢ ⎥ 2λ41 m∗1 (1 − cosh λ) sinh λ1 ⎥R, (Gcx + iGcy )η=1 = ⎢ ⎢ m∗ λ1 m∗ + λ2 (1 − cosh λ) − 4 sinh λ1 ⎥ 7 1 ⎣ 2 ⎦ 2 3 ∗ 2 2 +2λ1 m3 (λ1 cosh λ1 − sinh λ1 ) sinh λ1 − λ1 2iK − λ1 sinh λ1

(45)

where m∗7 = (2 + 2 cosh λ1 − λ1 sinh λ1 ). 5. Mass flow rate The expression for non-dimensional mass flow rates Qx and Qy , in the primary and secondary flow directions respectively, is given by

(Qx + iQy ) =

1 [2λ1 {A(cosh λ1 − 1) − B(λ1 − sinh λ1 )} − Gr]. 2λ21

(46)

6. Asymptotic solution To gain some physical insight into the flow pattern, we shall now analyze the asymptotic behavior of the solution (36)–(38) for large values of K2 and M2 . Case-I: K2 1 and M2 ∼ O(1) When K2 is large and M2 is of small order of magnitude, fluid flow becomes boundary layer type. For the boundary layer flow adjacent to the upper plate η = 1, introducing boundary layer coordinate ξ = 1 − η, the expressions for fluid velocity and induced magnetic field, which are obtained from (36) to (38), assume the following form

vx = − vy =

1 (R + Gr)e−α3 ξ sinh β3 ξ , 2K 2

1 [R + (1 − ξ )Gr − (R + Gr)e−α3 ξ cos β3 ξ ], 2K 2

1 + φ 1 − m2 Gr (1 + 2φ)Gr − φ1 (1 + φ)R − (φ1 + 1 − ξ ) φ R − (1 + m2 ) (1 + m2 ) 1 (R + Gr) −α3 ξ 2 e + m 1 − ξ Gr + 1 − m cos β ξ + 1 + m sin β ξ , ) ) ( (( ) ( ) 3 3 K 2(1 + m2 ) 1 + φ 1 − m2 Gr 1 1 (1 + 2φ)Gr 2 hy = + φ1 m φ R − φ1 R − 2 (φ1 + 1 − ξ ) (1 + φ)R − 2K 2 m1 (1 + m2 ) (1 + m2 ) 1 (R + Gr) −α3 ξ e − (1 − ξ )2 Gr − ((1 + m) cos β3 ξ − (1 − m) sin β3 ξ ) , K 2(1 + m2 ) 1 hx = 2K 2

where

(47) (48)

m m21

(m + 1)M2 (m − 1)M2 2 2 2 1/2 α3 = K 1 + 2 , β = K 1 + , and m = [ ( 1 + φ) + φ m ] . 3 1 4K (1 + m2 ) 4K 2 (1 + m2 )

(49)

(50)

(51)

It is revealed from the expressions (47) to (51) that a thin boundary layer of thickness O(α3−1 ) arises adjacent to the upper plate of the channel. This boundary layer may be identified as modified Ekman boundary layer and can be viewed as classical Ekman boundary layer modified by Hall current and magnetic field. It is seen from the expressions in (51) that the thickness of this boundary layer decreases on increasing either K2 or M2 or both and increases on increasing m. Similar type of boundary

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layer appears adjacent to the lower plate of the channel. Exponential terms in expressions (47) to (50) damp out quickly as ξ increases. When ξ ≥ 1/α 3 i.e. outside the boundary layer region, we obtain

1

vx ≈ 0, vy ≈ 2 (R + ηGr), 2K hx ≈

1 2K 2

m m21

(φ1 + η) φ R −

1 hy ≈ 2K 2

(52)

1 φ1 R − 2 m1

(1 + 2φ)Gr (1 + m2 )

− φ1

(φ1 + η) (1 + φ)R −

(1 + φ)R −

1 + φ 1 − m2

Gr

2

1+φ 1−m

Gr

(1 + m2 )

(1 + m2 )

+

mη2 Gr , 2(1 + m2 )

(53)

η2 Gr (1 + 2φ)Gr + φ1 m φ R − − . 2(1 + m2 ) (1 + m2 ) 2

(54) It is evident from the expressions (52) to (54) that, in a certain core given by ξ ≥ 1/α 3 i.e. outside the boundary layer region, fluid flows in the direction normal to the applied pressure gradient and axis of rotation which is in agreement with the Taylor– Proudman theorem (Greenspan [36]). Fluid velocity is unaffected by magnetic field and Hall current and varies linearly with channel width variable η due to presence of thermal buoyancy force. The induced magnetic field persists in both the primary and secondary flow directions due to influence of wall conductance and thermal buoyancy force and has considerable effects Hall current, rotation, wall conductance and thermal buoyancy force and are unaffected by magnetic field. However, in the absence of Hall current, primary induced magnetic field hx vanishes away whereas secondary induced magnetic fileld hy persists in the central core region. The variation in induced magnetic field components hx and hy is non-linear with η due to presence of thermal buoyancy force. Case II: M2 1 and K2 ∼ O(1) In this case also boundary layer type flow is expected. For the boundary layer flow near the upper plate η = 1, the expressions for the fluid velocity and induced magnetic field, obtained from (36) to (38), are given by

1 [R + Gr(1 − ξ ) − (R + Gr)e−α4 ξ (cos β4 ξ − m sin β4 ξ )], M2 1 = 2 [m(R + Gr(1 − ξ )) − (R + Gr)e−α4 ξ (m cos β4 ξ + sin β4 ξ )], M

vx = −

(55)

vy

(56)

hx =

α ∗ R Gr 2α ∗ φ φ (R + Gr) −α4 ξ ∗ e (α4 cos β4 ξ + β4∗ sin β4 ξ ) , (φ2 + 1 − ξ ) − 1 + 4 + (1 − ξ )2 − 1 1 − 4 − φ M 2 φ M M

1 R M2

(57)

1 φ1 Gr (R + Gr) −α4 ξ ∗ ∗ ∗ e hy = − 3 β4 R − + (β4 cos β4 ξ − α4 sin β4 ξ ) , φ M M where

M

α4 = √

2 mK 2 M ∗ ∗ K α 1 + − β , β4 = √ 4 4 2 2 2 M

1+m

1 !

α = √ [ 1+ ∗ 4

2

m2

+ 1]

1/2

M

and

1 !

β = √ [ 1+ ∗ 4

2

⎫ 2 mK 2 ⎪ ∗ ∗ K β 1 + + α ,⎪ ⎬ 4 4 2 2 2 M

1+m

m2

− 1]

(58)

1/2

.

M

⎪ ⎪ ⎭

(59)

The expressions (55)–(59) demonstrate the existence of a thin boundary layer of thickness O(α4−1 ) near the upper plate of the channel. This boundary layer may be recognized as modified Hartmann boundary layer and can be viewed as classical Hartmann boundary layer modified by Hall current and rotation. It is observed from the expressions in (59) that the thickness of this boundary layer decreases on increasing either K2 or M2 and increases on increasing m. Similar type of boundary layer appears adjacent to the lower plate of the channel. It may be noted that, in the absence of Hall current, classical Hartmann boundary layer of thickness O(M−1 )appears near the plates of the channel. Exponential terms in expressions (55)–(58) damp out quickly as ξ increases. When ξ ≥ 1/α 4 i.e. outside the boundary layer region, we obtain

1 (R + Grη), M2

m

vy ≈ 2 (R + Grη), M α4∗ R Gr 2 φ1 2α4∗ 1 φ1 + η − hx ≈ 2 R (φ2 + η) − + 1− , φ M 2 φ M M β∗ φ1 Gr hy ≈ − 43 R − . φ M vx ≈ −

(60) (61) (62)


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It is noticed from the expressions (60) to (62) that in a certain core given by ξ ≥ 1/α 4 i.e. outside the boundary layer region, fluid flows in both the directions due to presence of Hall current. Both the fluid velocity and induced magnetic field are independent of rotation. Both the primary and secondary fluid velocities vary linearly with η due to presence of thermal buoyancy force. Also induced magnetic field persists in both the directions due to presence of Hall current and has considerable effects on Hall current, magnetic field, wall conductance and thermal buoyancy force. The secondary induced magnetic field hy is weak in comparison to the primary induced magnetic field hx . Primary induced magnetic field hx varies non-linearly with η due to presence of thermal buoyancy force. 7. Heat transfer We shall now analyze heat transfer characteristics of steady fully-developed MHD mixed convection flow of a viscous, incompressible, electrically and thermally conducting fluid within two infinite parallel plates in a rotating system taking Hall current into account. The energy equation taking viscous and Joule dissipations into account is given by

∂(T − T0 ) ∂ 2 (T − T0 ) μe = α∗ Vx + ∂x ρCp ∂ z2

∂ Vx ∂z

2

∂ Vy + ∂z

2

1 + σ ρCp

∂ Hx ∂z

2

∂ Hy + ∂z

2

,

(63)

where α ∗ = K/ρC p is thermal diffusivity, Cp is specific heat at constant pressure and K is thermal conductivity of the fluid. Since plates of the channel z = 0 and z = L are cooled or heated by a uniform temperature gradient A1 along x-direction, therefore, the boundary conditions for temperature field are given by

T = T0 + A1 x + θ1W1 T = T0 + A1 x + θ1W2

at z = 0, at z = L,

(64)

where θ1W1 and θ1W2 are constant temperatures. Using the non-dimensional variables defined in (31), the Eq. (63), in dimensionless form, becomes

1 d2 θ ¯ − ErPr = GrPr(v + v) 2 dη2

dv dη

dv¯ dη

+ M2

dh dη

dh¯ dη

,

(65)

where θ = gβ L3 (θ1 − θ1W1 )/υ 2 is non-dimensional fluid temperature, Pr = υ /α ∗ is Prandtl number and Er = gβ L/C p is Eckert

number. v¯ and h¯ are complex conjugate of v and h respectively. Boundary conditions (64), in non-dimensional form, assume the form

θ = 0 at η = 0, θ = gβ L3 (θ1W2 − θ1W1 )/υ 2 = θ0

at

η = 1,

(66)

where θ 0 is a constant temperature. Making use of analytical expressions for fluid velocity and induced magnetic field given by (36)–(38) in Eq. (65), the resulting differential equation subject to the boundary conditions (66) is solved numerically using MATLAB software. The numerical values of rate of heat transfer at the plates of the channel are also computed with the help of MATLAB software for various values of pertinent flow parameters. 8. Results and discussion To study the effects of Hall current, rotation, magnetic field, thermal buoyancy force and wall conductance on the flow-field, the numerical values of primary and secondary fluid velocities and primary and secondary induced magnetic fields, computed from the analytical solution (36) to (38), are displayed graphically versus channel width variable η in Figs. 2 to 12 for various values of Hall current parameter m, rotation parameter K2 , magnetic parameter M2 , Grashof number Gr and wall conductance ratios φ 1 and φ 2 taking R = 1 and φ = φ1 + φ2 . Fig. 2 demonstrates the influence of Hall current on the primary velocity vx and secondary velocity vy . It is noticed from Fig. 2 that primary velocity vx decreases whereas the secondary velocity vy increases on increasing m. This implies that Hall current tends to retard fluid flow in the primary flow direction whereas it has reverse effect on fluid flow in the secondary flow direction. This is due to the fact that Hall current induces secondary flow into the flow field. Fig. 3 illustrates the effects of rotation on the primary velocity vx and secondary velocity vy . It is revealed from Fig. 3 that primary velocity vx decreases whereas the secondary velocity vy increases on increasing K2 . Since K2 represents the ratio of Coriolis force to viscous force, an increase in K2 implies the increase in the strength of Coriolis force. In a rotating system Coriolis force is generated due to rotation whose tendency is to suppress primary flow to induce secondary flow into the flow field. Thus, rotation tends to retard fluid flow in the primary flow direction whereas it has a reverse effect on fluid flow in the secondary flow direction which is also characteristics of Hall current. Fig. 4 shows the influence of magnetic field on the primary velocity vx and secondary velocity vy . It is observed from Fig. 4 that both the primary velocity vx and the secondary velocity vy decrease on increasing M2 . This implies that magnetic field tends to retard fluid flow in both the primary and secondary flow directions. This is due to the fact that when an electrically conducting fluid flows in the presence of a magnetic field, a

2792


Fig. 2. Velocity profiles when K 2 = 5, M2 = 20, Gr = 2.0 and φ = 2.

Fig. 3. Velocity profiles when m = 0.25, M2 = 20, Gr = 2.0 and φ = 2.

Fig. 4. Velocity profiles when m = 0.25, K 2 = 5, Gr = 2.0 and φ = 2.


2793

Fig. 5. Velocity profiles when m = 0.25, K 2 = 5, M2 = 20 and φ = 2.

Fig. 6. Velocity profiles when m = 0.25, K 2 = 5, M2 = 20 and Gr = 2.0.

mechanical force, known as Lorentz force, is generated in the flow field whose tendency is to resist fluid flow. Fig. 5 exhibits the effects of thermal buoyancy force on the primary velocity vx and secondary velocity vy . It is seen from Fig. 5 that both the primary velocity vx and the secondary velocity vy increase on increasing Gr. Since Gr measures the relative strength of thermal buoyancy force to viscous force, an increase in Gr implies an increase in the thermal buoyancy force. This implies that thermal buoyancy force tends to accelerate fluid flow in both the primary and secondary flow directions. Fig. 6 shows the effects of wall conductance on the primary velocity vx and secondary velocity vy . It is revealed from Fig. 6 that both the primary velocity vx and the secondary velocity vy decrease on increasing φ . This implies that wall conductance tends to retard fluid flow in both the primary and secondary flow directions which is also characteristics of the magnetic field. Fig. 7 illustrates the influence of Hall current on the primary induced magnetic field hx and secondary induced magnetic field hy . It is evident from the Fig. 7 that both the primary induced magnetic field hx and secondary induced magnetic field hy decrease on increasing m in the regions near both the lower and upper plates of the channel. This implies that Hall current tends to reduce both the primary and secondary induced magnetic fields in the regions near both the lower and upper plates of the channel. Fig. 8 demonstrates the influence of rotation on the primary induced magnetic field hx and secondary induced magnetic field hy . It is observed from the Fig. 8 that the primary induced magnetic field hx decreases whereas the secondary induced magnetic field hy increases on increasing K2 in the regions near both the lower and upper plates of the channel. This implies that rotation tends to reduce primary induced magnetic field whereas it has reverse effect on secondary induced magnetic field in the regions near both the lower and upper plates of the channel. Fig. 9 shows the effects of magnetic field on the primary induced magnetic field hx and the secondary induced magnetic field hy . It is seen from the Fig. 9 that both the primary induced magnetic field hx and the secondary induced magnetic field hy decrease on increasing M2 in the regions near both the lower and upper plates of the channel. This implies that magnetic field tends to reduce both the primary and secondary induced magnetic fields in the regions near both the lower and upper plates of the channel. Fig. 10 exhibits the influence of thermal buoyancy force on the primary induced magnetic field hx and secondary induced magnetic field hy . It is noted from the Fig. 10 that both the primary induced magnetic field hx and secondary induced

2794


Fig. 7. Induced magnetic field profiles when K 2 = 5, M2 = 20, Gr = 2.0 and φ1 = φ2 = 2.

Fig. 8. Induced magnetic field profiles when m = 0.25, M2 = 20, Gr = 2.0 and φ1 = φ2 = 2.

Fig. 9. Induced magnetic field profiles when m = 0.25, K 2 = 5, Gr = 2.0 and φ1 = φ2 = 2.

magnetic field hy increase on increasing Gr in the regions near both the lower and upper plates of the channel. This implies that thermal buoyancy force tends to enhance both the primary and secondary induced magnetic fields in the regions near both the lower and upper plates of the channel. Figs. 11 and 12 show the effects of wall conductance on the primary induced magnetic field hx and the secondary induced magnetic field hy . It is noticed from the Figs. 11 and 12 that, on increasing φ 1 , the primary induced magnetic field hx increases in the region near the lower plate of the channel and it decreases in the region near the upper plate of the channel whereas the secondary induced magnetic field hy increases in the region near the lower plate of the channel and


Fig. 10. Induced magnetic field profiles when m = 0.25, K 2 = 5, M2 = 20 and φ1 = φ2 = 2.

Fig. 11. Induced magnetic field profiles when m = 0.25, K 2 = 5, M2 = 20, Gr = 2.0 and φ2 = 2.

Fig. 12. Induced magnetic field profiles when m = 0.25, K 2 = 5, M2 = 20, Gr = 2.0 and φ1 = 2.

2795

2796


Fig. 13. Temperature profiles when K 2 = 5, M2 = 20, Gr = 2.0, Er = 2.0 and Pr = 7.0.

Fig. 14. Temperature profiles when m = 0.25, M2 = 20, Gr = 2.0, Er = 2.0 and Pr = 7.0.

it decreases in the region near the upper plate of the channel. On increasing φ 2 , the primary induced magnetic field hx decreases in the region near the lower plate of the channel and it increases in the region near the upper plate of the channel whereas the secondary induced magnetic field hy decreases in the region near the lower plate of the channel and it increases in the region near the upper plate of the channel. This implies that the wall conductance ratio of the lower plate tends to enhance primary induced magnetic field in the region near the lower plate of the channel and it tends to reduce primary induced magnetic field in the region near the upper plate of the channel whereas it has reverse effect on the secondary induced magnetic field in the region near the lower and upper plates of the channel respectively. Wall conductance ratio of the upper plate has reverse effect on the primary and secondary induced magnetic fields than that of wall conductance ratio of lower plate on it. It is also noticed from Figs. 11 and 12 that when lower wall is non-conducting the primary and secondary induced magnetic fields vanish at the lower plate of the channel whereas when the lower wall is perfectly conducting it vanish at the upper wall of the channel. Wall conductance of upper wall shows similar behavior on it. It is noted from Figs. 7 to 12 that for arbitrarily conducting walls both the primary and secondary induced magnetic fields vanish in the central region of the channel. This is due to the reason that the channel is symmetric about the line η = 1/2. The position of point, where both the primary and secondary induced magnetic fields vanish in the central region, depends on the influence of thermal buoyancy force and wall conductance ratios. To study the influence of Hall current, rotation, magnetic field and thermal buoyancy force on fluid temperature, the numerical solution of the energy equation, computed with MATLAB software, is presented graphically versus channel width variable η in Figs. 13 to 18 for various values of m, K2 , M2 , Gr, Prandtl number Pr and Eckert number Er taking R = 1, θ0 = 1 and φ1 = φ2 = 2. It is seen from Figs. 13 to 15 that fluid temperature θ decreases on increasing either m or K2 or M2 which implies that Hall current, rotation and magnetic field tend to reduce fluid temperature. It is observed from Fig. 16 that fluid temperature θ increases on increasing Gr which implies that thermal buoyancy force tends to enhance fluid temperature. It may be noted that in case of forced convection (i.e. Gr = 0) variation of fluid temperature θ is linear with channel width variable η. It is revealed from


Fig. 15. Temperature profiles when m = 0.25, K 2 = 5, Gr = 2.0, Er = 2.0 and Pr = 7.0.

Fig. 16. Temperature profiles when m = 0.25, K 2 = 5, M2 = 20, Er = 2.0 and Pr = 7.0.

Fig. 17. Temperature profiles when m = 0.25, K 2 = 5, M2 = 20, Gr = 2.0 and Pr = 7.0.

2797

2798


Fig. 18. Temperature profiles when m = 0.25, K 2 = 5, M2 = 20, Gr = 2.0 and Er = 2.0. Table 1 Primary and secondary shear stress at the lower plate when M2 = 20, φ = 2 and Gr = 2.0. K2 ↓m→

3 5 7

−(τx )η=0

(τy )η=0

0.25

0.50

0.75

0.25

0.50

0.75

0.3755 0.3397 0.3016

0.3633 0.3233 0.2843

0.3535 0.3090 0.2687

0.1010 0.1365 0.1582

0.1201 0.1489 0.1642

0.1412 0.1640 0.1733

Table 2 Primary and secondary shear stress at the upper plate when M2 = 20, φ = 2 and Gr = 2.0. K2 ↓m→

3 5 7

(τx )η=1

−(τy )η=1

0.25

0.50

0.75

0.25

0.50

0.75

0.6315 0.5918 0.5489

0.6230 0.5776 0.5325

0.6192 0.5681 0.5204

0.1291 0.1742 0.2046

0.1593 0.1977 0.2214

0.1889 0.2214 0.2393

Fig. 17 that fluid temperature θ increases on increasing Er which implies that viscous dissipation has tendency to enhance fluid temperature. It is evident from Fig. 18 that fluid temperature θ decreases on decreasing Pr. Since Pr is a measure of the strength of viscosity and thermal conductivity of the fluid, Pr decreases when thermal conductivity of the fluid increases. Therefore, we conclude from the above result that thermal diffusion tends to reduce fluid temperature. It is also noted that variation of fluid temperature θ is linear with channel width variable η for liquid metal whereas it is non-linear for air and water. The numerical values of non-dimensional shear stress at the lower and upper plates of the channel, computed from the analytical expressions (42) and (43), are presented in tabular form in Tables 1–6 while that of critical Grashof number and mass flow rate, computed from expressions (44)–(46), are displayed in tabular form in Tables 7–13 for various values of m, K2 , M2 , Gr and φ taking R = 1. It is revealed from Tables 1 and 2 that primary shear stress at the lower plate (τx )η=0 and that at the upper plate (τx )η=1 decrease on increasing either m or K2 whereas secondary shear stress at the lower plate (τy )η=0 and that at upper plate (τy )η=1 increase on increasing either m or K2 . This implies that both the Hall current and rotation have tendency to reduce primary shear stress at the lower and upper plates of the channel whereas it have reverse effect on the secondary shear stress at the lower and upper plates of the channel. It is evident from Tables 3 and 4 that both the primary and the secondary shear stress at the lower and upper plates of the channel decrease on increasing M2 which implies that magnetic field tends to reduce both the primary and secondary shear stress at the lower and upper plates of the channel. It is noticed from the Tables 5 and 6 that both the primary and secondary shear stress at the lower and upper plates of the channel increase on increasing Gr whereas both the primary and secondary shear stress at the lower and upper plates of the channel decrease on increasing φ . This implies that thermal buoyancy force tends to enhance both the primary and secondary shear stress at the lower and upper plates of the channel whereas wall conductance has reverse effect on it. It is revealed from Table 7 that both the critical Grashof number at the lower plate in primary flow direction (Gcx )η=0 and critical Grashof number at the lower plate in secondary flow direction (Gcy )η=0 increase on increasing


2799

Table 3 Primary and secondary shear stress at the lower plate when m = 0.25, φ = 2 and Gr = 2.0. K2 ↓M2 →

3 5 7

−(τx )η=0

(τy )η=0

20

25

30

20

25

30

0.3755 0.3397 0.3016

0.3461 0.3194 0.2900

0.3223 0.3018 0.2785

0.1010 0.1365 0.1582

0.0828 0.1132 0.1338

0.0695 0.0957 0.1149

Table 4 Primary and secondary shear stress at the upper plate when m = 0.25, φ = 2 and Gr = 2.0. K2 ↓M2 →

3 5 7

(τx )η=1

−(τy )η=1

20

25

30

20

25

30

0.6315 0.5918 0.5489

0.5903 0.5602 0.5267

0.5563 0.5327 0.5059

0.1291 0.1742 0.2046

0.1099 0.1485 0.1769

0.0957 0.1292 0.1551

Table 5 Primary and secondary shear stress at the lower plate when m = 0.25, K 2 = 5 and M2 = 20.

φ↓Gr→

−(τx )η=0

0 1 2 3 ∞

(τy )η=0

0

2

4

6

0

2

4

6

0.3431 0.2570 0.2329 0.2222 0.1948

0.5602 0.3879 0.3397 0.3183 0.2636

0.7772 0.5189 0.4465 0.4144 0.3323

0.9943 0.6498 0.5533 0.5105 0.4011

0.1977 0.0916 0.0777 0.0727 0.0628

0.3765 0.1643 0.1365 0.1265 0.1068

0.5553 0.2371 0.1953 0.1804 0.1508

0.7341 0.3098 0.2542 0.2342 0.1948

Table 6 Primary and secondary shear stress at the upper plate when m = 0.25, K 2 = 5 and M2 = 20.

φ↓Gr→

(τx )η=1

0 1 2 3 ∞

−(τy )η=1

0

2

4

6

0

2

4

6

0.3431 0.2570 0.2329 0.2222 0.1948

0.8123 0.6401 0.5918 0.5704 0.5157

1.2815 1.0231 0.9507 0.9186 0.8365

1.7506 1.4062 1.3097 1.2668 1.1574

0.1977 0.0916 0.0777 0.0727 0.0628

0.4141 0.2020 0.1742 0.1642 0.1445

0.6306 0.3124 0.2707 0.2557 0.2261

0.8471 0.4228 0.3672 0.3472 0.3078

Table 7 Critical Grashof number at the lower plate in primary and secondary flow directions when m = 0.25 and φ = 2. K2 ↓M2 →

3 5 7

−(Gcx )η=0

−(Gcy )η=0

20

25

30

20

25

30

3.9129 3.9598 4.0153

4.0590 4.1065 4.1614

4.1839 4.2310 4.2844

0.4446 0.7269 1.0072

0.3946 0.6550 0.9130

0.3458 0.5873 0.8263

K2 . Critical Grashof number at the lower plate in primary flow direction increases whereas critical Grashof number at the lower plate in secondary flow direction decreases on increasing M2 . This implies that rotation exerts a stabilizing influence on the flow in both the primary and secondary flow direction at the lower plate of the channel whereas magnetic field exerts a stabilizing effects in the primary flow direction and it exerts a destabilizing influence in the secondary flow direction at the lower plate of the channel. It is evident from Table 8 that, on increasing K2 , critical Grashof number at the upper plate in primary flow direction (Gcx )η=1 increases, attains a maximum, and then decreases in magnitude when M2 = 20 and 25 and it increases when M2 = 30 whereas critical Grashof number at the upper plate in secondary flow direction (Gcy )η=1 decreases, attains a minimum, and then increases in magnitude when M2 = 20 and 25 and it increases when M2 = 30. On increasing M2 , critical Grashof number at the upper plate in primary flow direction decreases when K 2 = 3 and 5 and it decreases, attains a minimum, and then increases in magnitude when K 2 = 7 whereas critical Grashof number at the upper plate in secondary flow direction decreases, attains a

2800

G.S. Seth, J.K. Singh / Applied Mathematical Modelling 40 (2016) 2783–2803 Table 8 Critical Grashof number at the upper plate in primary and secondary flow directions when m = 0.25 and φ = 2. K2 ↓M2 →

3 5 7

−(Gcx )η=1

−(Gcy )η=1

20

25

30

20

25

30

15.0627 20.9460 15.5474

13.8081 16.7670 14.1146

12.6092 15.1242 15.2422

–7.4846 1.0014 13.3361

–3.2047 3.3443 12.2645

2.2630 2.3829 9.4472

Table 9 Critical Grashof number at the lower plate in primary and secondary flow directions when K 2 = 5 and M2 = 20.

φ↓m→

0 1 2 3 ∞

−(Gcx )η=0

−(Gcy )η=0

0.25

0.50

0.75

0.25

0.50

0.75

2.7770 3.5933 3.9598 4.1636 4.8505

2.8555 3.7290 4.0736 4.2415 4.6472

2.9324 3.8619 4.1503 4.2597 4.3848

0.4666 0.5976 0.7269 0.8204 1.2759

0.4472 0.6095 0.8202 0.9690 1.6009

0.4492 0.6865 0.9795 1.1651 1.7951

Table 10. Critical Grashof number at the upper plate in primary and secondary flow directions when K 2 = 5 and M2 = 20.

φ↓m→

−(Gcx )η=1 0.25

0 1 2 3 ∞

1.8050 6.2235 20.9460 –0.3438 –5.4808

(Gcy )η=1 0.50 2.0910 6.4906 11.3681 8.0149 –6.7553

0.75

0.25

2.3135 5.5853 7.6359 7.7563 –7.7556

–1.1362 –2.3120 –1.0014 50.7522 2.7484

0.50 –1.0948 –0.0002 6.1530 16.1052 4.5696

0.75 –0.9676 1.1161 4.7095 8.6307 10.8760

minimum, and then increases in magnitude when K 2 = 3 and 5 and it decreases when K 2 = 7. Thus we conclude that rotation exerts a stabilizing influence on the flow in both the primary and secondary flow direction at the upper plate of the channel when M2 = 30 whereas magnetic field exerts a destabilizing effect on the flow in the primary flow direction when K 2 = 3 and 5 and it exerts a destabilizing influence on the flow in the secondary flow direction when K 2 = 7 at the upper plate of the channel. It may be noted that there is a flow separation in both the primary and secondary flow direction when M2 = 20 and 25 on increasing K2 at the upper plate of the channel. It is seen from Table 9 that both the critical Grashof number at the lower plate in primary and secondary flow directions increase on increasing φ . On increasing m, critical Grashof number at the lower plate in primary flow direction increases when φ = ∞ whereas critical Grashof number at the lower plate in secondary flow direction increases when φ = 0. This implies that wall conductance exerts a stabilizing influence on the flow in both the primary and secondary flow direction at the lower plate of the channel whereas Hall current exerts a stabilizing effects on the flow in the primary flow direction when φ = ∞ and it exerts a stabilizing influence on the flow in the secondary flow direction when φ = 0 at the lower plate of the channel. It is revealed from the Table 10 that, on increasing φ , critical Grashof number at the upper plate in primary flow direction increases, attains a maximum, and then decreases in magnitude when m = 0.25 and 0.50 and φ = ∞ and it increases when m = 0.75 and φ = ∞ whereas it is of oscillatory nature on increasing m when φ = ∞. Critical Grashof number at the upper plate in secondary flow direction is of oscillatory nature on increasing either m or φ . Thus we conclude that wall conductance exerts a stabilizing influence on the flow in primary flow direction at the upper plate of the channel when m = 0.75 and φ = ∞ whereas it is of oscillatory nature in secondary flow direction. It is observed from Tables 11 and 12 that primary mass flow rate Qx decreases whereas the secondary mass flow rate Qy increases on increasing either m or K2 . Both the primary and secondary mass flow rate decreases on increasing M2 . This implies that both the Hall current and rotation have tendency to reduce primary mass flow rate whereas it have reverse effect on the secondary mass flow rate. Magnetic field tends to reduce both the primary and secondary mass flow rates. It is revealed from Table 13 that both the primary and secondary mass flow rates increase on increasing Gr whereas both the primary and secondary mass flow rates decrease on increasing φ . This implies that thermal buoyancy force tends to enhance both the primary and secondary mass flow rates whereas wall conductance has reverse effect on it. The numerical values of rate of heat transfer at the lower and upper plates of the channel are computed with the help of MATLAB software and are displayed in tabular form in Tables 14–17 for various values of m, K2 , M2 , Gr, φ 1 , Er and Pr taking R = 1 , θ0 = 1 and φ2 = 2. It is seen from Tables 14 and 15 that rate of heat transfer at the lower plate ( ddθη )η=0 decreases on increasing


2801

Table 11 Primary and secondary mass flow rates when M2 = 20, φ = 2 and Gr = 2.0. K2 ↓m→

3 5 7

−Qx

Qy

0.25

0.50

0.75

0.25

0.50

0.75

0.0628 0.0558 0.0484

0.0613 0.0531 0.0452

0.0607 0.0513 0.0429

0.0218 0.0284 0.0323

0.0278 0.0330 0.0356

0.0335 0.0376 0.0390

Table 12 Primary and secondary mass flow rates when m = 0.25, φ = 2 and Gr = 2.0. K2 ↓M2 →

3 5 7

−Qx

Qy

20

25

30

20

25

30

0.0628 0.0558 0.0484

0.0559 0.0506 0.0450

0.0503 0.0463 0.0419

0.0218 0.0284 0.0323

0.0181 0.0235 0.0272

0.0154 0.0200 0.0232

Table 13 Primary and secondary mass flow rates when m = 0.25, K 2 = 5 and M2 = 20.

φ↓Gr→

−Qx

0 1 2 3 ∞

Qy

0

2

4

6

0

2

4

6

0.0395 0.0307 0.0279 0.0266 0.0234

0.0791 0.0613 0.0558 0.0533 0.0467

0.1186 0.0920 0.0837 0.0799 0.0701

0.1581 0.1227 0.1116 0.1065 0.0935

0.0314 0.0164 0.0142 0.0133 0.0116

0.0628 0.0328 0.0284 0.0267 0.0232

0.0941 0.0492 0.0425 0.0400 0.0348

0.1255 0.0655 0.0567 0.0534 0.0463

Table 14 Rate of heat transfer at the lower and upper plates when M2 = 20, Gr = 2.0, φ1 = 2.0, Er = 2.0 and Pr = 7.0. K2 ↓m→

3 5 7

( ddθη )η=0

−( ddθη )η=1

0.25

0.50

0.75

0.25

1.9989 1.8909 1.7770

1.9750 1.8434 1.7185

1.9707 1.8157 1.6791

0.3205 0.1933 0.0585

0.50 0.2973 0.1419 –0.0069

0.75 0.2990 0.1154 –0.0481

Table 15 Rate of heat transfer at the lower and upper plates when m = 0.25, Gr = 2.0, φ1 = 2.0, Er = 2.0 and Pr = 7.0. K2 ↓M2 →

3 5 7

( ddθη )η=0

−( ddθη )η=1

20

25

30

20

25

30

1.9989 1.8909 1.7770

1.8716 1.7933 1.7086

1.7729 1.7139 1.6491

0.3205 0.1933 0.0585

0.1630 0.0702 –0.0307

0.0398 –0.0306 –0.1083

either m or K2 or M2 . On increasing m, rate of heat transfer at the upper plate ( ddθη )η=1 decreases, attains a minimum, and then

increases in magnitude when K 2 = 3, it decreases when K 2 = 5 and again it decreases, attains a minimum, and then increases in magnitude when K 2 = 7. On increasing K2 , rate of heat transfer at the upper plate decreases when m = 0.25 and it decreases, attains a minimum, and then increases in magnitude when m = 0.50 and 0.75. On increasing M2 , rate of heat transfer at the upper plate decreases when K 2 = 3 and it decreases, attains a minimum, and then increases in magnitude when K 2 = 5 and 7. This implies that Hall current, rotation and magnetic field tend to reduce rate of heat transfer at the lower plate of the channel. It may be noted that there exist reverse flow of heat near upper plate of the channel on increasing m when K 2 = 3 and 7 and also reverse flow of heat takes place near upper plate of the channel on increasing K2 when m = 0.50 and 0.75. On increasing M2 , also there exist reverse flow of heat near upper plate of the channel when K 2 = 5 and 7. It is evident from Table 16 that rate of heat transfer at the lower and upper plates of the channel decrease on increasing φ 1 when Gr = 0. Rate of heat transfer at the lower and upper plates of the channel increase on increasing Gr. This implies that thermal buoyancy force tends to enhance rate of heat transfer at the lower and upper plates of the channel whereas wall conductance of lower plate has reverse effect on it

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G.S. Seth, J.K. Singh / Applied Mathematical Modelling 40 (2016) 2783–2803 Table 16 Rate of heat transfer at the lower and upper plates when m = 0.25, K 2 = 5, M2 = 20,Er = 2.0 and Pr = 7.0.

φ 1 ↓ Gr→

0 1 2 3 ∞

( ddθη )η=0

−( ddθη )η=1

0

2

4

6

0

2

4

6

1.1656 1.1639 1.1634 1.1632 1.1636

1.9255 1.9024 1.8909 1.8841 1.8584

3.2797 3.2152 3.1824 3.1626 3.0842

5.2282 5.1026 5.0380 4.9988 4.8412

–0.8344 –0.8361 –0.8366 –0.8368 –0.8364

0.2315 0.2061 0.1933 0.1857 0.1562

2.2487 2.1776 2.1409 2.1187 2.0288

5.2173 5.0784 5.0062 4.9620 4.7814

Table 17 Rate of heat transfer at the lower and upper plates when m = 0.25, K 2 = 5, M2 = 20, φ1 = 2 and Gr = 2.0. Er↓Pr→

0.50 1.0 1.50 2.0

( ddθη )η=0

( ddθη )η=1

0.03

0.71

3.0

7.0

0.03

0.71

3.0

1.0020 1.0026 1.0032 1.0038

1.0483 1.0623 1.0763 1.0904

1.2039 1.2632 1.3225 1.3818

1.4758 1.6141 1.7525 1.8909

0.9975 0.9966 0.9957 0.9949

0.9402 0.9198 0.8994 0.8790

0.7473 0.6610 0.5748 0.4886

7.0 0.4103 0.2091 0.0079 –0.1933

when Gr = 0. It is evident from Table 17 that, on decreasing Pr, rate of heat transfer at the lower plate decreases whereas rate of heat transfer at the upper plate increases when Er ≤ 1.50. On increasing Er, rate of heat transfer at the lower plate increases whereas rate of heat transfer at the upper plate decreases when Pr ≤ 3. This implies that thermal diffusion tends to reduce rate of heat transfer at the lower plate of the channel whereas it has reverse effect on rate of heat transfer at the upper plate of the channel when Er ≤ 1.50. Viscous dissipation tends to enhance rate of heat transfer at the lower plate of the channel whereas it has reverse effect on rate of heat transfer at the upper plate of the channel when Pr ≤ 3.

9. Conclusions A mathematical analysis has been presented for hydromagnetic mixed convection flow of a viscous, incompressible, electrically and thermally conducting fluid in a rotating channel with arbitrary conducting walls taking Hall current into account. The significant results are summarized below: Both the Hall current and rotation tend to retard fluid flow in the primary flow direction whereas it has reverse effect on fluid flow in the secondary flow direction. Magnetic field and wall conductance tend to retard fluid flow in both the primary and secondary flow directions whereas thermal buoyancy force has reverse effect on it. Hall current and magnetic field tend to reduce both the primary and secondary induced magnetic fields in the regions near both the lower and upper plates of the channel whereas thermal buoyancy force has reverse effect on it. Rotation tends to reduce primary induced magnetic field whereas it has reverse effect on secondary induced magnetic field in the regions near both the lower and upper plates of the channel. Wall conductance ratio of the lower plate tends to enhance primary induced magnetic field in the region near the lower plate of the channel and it tends to reduce primary induced magnetic field in the region near the upper plate of the channel whereas it has reverse effect on the secondary induced magnetic field in the region near the lower and upper plates of the channel respectively. Wall conductance ratio of the upper plate has reverse effect on the primary and secondary induced magnetic fields than that of wall conductance ratio of lower plate on it. Hall current, rotation and magnetic field tend to reduce fluid temperature whereas thermal buoyancy force has reverse effect on it. Viscous dissipation has tendency to enhance fluid temperature whereas thermal diffusion has reverse effect on it. Hall current and rotation have tendency to reduce primary shear stress at the lower and upper plates of the channel whereas they have reverse effect on the secondary shear stress at the lower and upper plates of the channel. Magnetic field and wall conductance tend to reduce both the primary and secondary shear stress at the lower and upper plates of the channel whereas thermal buoyancy force has reverse effect on it. Hall current and rotation have tendency to reduce primary mass flow rate whereas it have reverse effect on the secondary mass flow rate. Magnetic field and wall conductance tend to reduce both the primary and secondary mass flow rates whereas thermal buoyancy force has reverse effect on it. Hall current, rotation and magnetic field tend to reduce rate of heat transfer at the lower plate of the channel. Thermal buoyancy force tends to enhance rate of heat transfer at the lower and upper plates of the channel whereas wall conductance of lower plate has reverse effect on it when Gr = 0. Thermal diffusion tends to reduce rate of heat transfer at the lower plate of the channel and it tends to enhance rate of heat transfer at the upper plate of the channel when Er ≤ 1.50 whereas viscous dissipation has reverse effect on it when Pr ≤ 3.


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