Mixed convection of magneto hydrodynamic and viscous fluid in a vertical channel

International Journal of Non-Linear Mechanics 46 (2011) 278–285

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Mixed convection of magneto hydrodynamic and viscous fluid in a vertical channel J. Prathap Kumar a, J.C. Umavathi b,n,1, Basavaraj M. Biradar c a

Department of Mathematics, Gulbarga University, Gulbarga 585 106, Karnataka, India Department of Civil Engineering, National Chi Nan University, University Rd. Puli Nantou, Taiwan 545, ROC c Department of Mathematics, Rural Engineering College, Bhalki 585 328, Karnataka, India b

a r t i c l e in f o

a b s t r a c t

Article history: Received 17 November 2008 Received in revised form 16 June 2010 Accepted 17 September 2010

Mixed convective flow and heat transfer in a vertical channel with one region filled with conducting fluid and another region with non-conducting fluid is analyzed. The viscous and Ohmic dissipation terms are included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left–right walls of the channel are prescribed. Analytical solutions are found for the governing equations using the regular perturbation method. A selected set of graphical results illustrating the effects of various parameters involved in the problem are presented and discussed. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Mixed convection MHD Two fluid Perturbation method

1. Introduction Mixed convection flow in a vertical channel has been the subject of many previous investigations due its possible application in many industrial and engineering processes. These include cooling of electronic equipment, heat exchangers, chemical processing equipments, gas-cooled nuclear reactors and others. Tao [1] analyzed the laminar fully developed mixed convection flow in a vertical parallel-plate channel with uniform wall temperatures. Aung and Worku [2,3] discussed the theory of combined free and forced convection in a vertical channel with flow reversal conditions for both developing and fully developed flows. The case of developing mixed convection flow in ducts with asymmetric wall heat fluxes was analyzed by the same authors [4]. A combined free and forced convection flow of an electrically conducting fluid in a channel in the presence of a transverse magnetic field is of special technical significance because of its frequent occurrence in many industrial applications such as geothermal reservoirs, cooling of nuclear reactors, thermal insulation and petroleum reservoirs. This type of problem also arises in electronic packages, microelectronic devices during their operations. Oreper and Szekely [5] analyzed the buoyancy driven

n

Corresponding author E-mail address: [email protected] (J.C. Umavathi). 1 On leave from Department of Mathematics, Gulbarga University, Gulbarga, Karnataka, India. 0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2010.09.008

flow in a rectangular cavity under the action of externally imposed magnetic field. Alboussiere et al. [6] did an asymptotic analysis to study the buoyancy driven convection in a uniform magnetic field. For closed geometry Garandet et al. [7] studied the problem of free convective flow in a rectangular enclosure in the presence of transverse magnetic field. For rectangular vertical duct, Hunt [8] and Buhler [9] analyzed the fluid flow problem in magnetic field with or without buoyancy effect. For conducting fluid, Shercliff [10] analyzed the fluid flow characteristics in a pipe under transverse magnetic field. All these studies pertain to a single-fluid model. Most of the problems relating to petroleum industry, geophysics, plasmaphysics, magnetic-fluid dynamics, etc., involved multi-fluid flow situation. The study of the interaction of geomagnetic field with the fluid in the geothermal regions arises in geophysics. Once we know the interaction of the geomagnetic field with the flow we can easily determine, using the energy equation, the temperature distribution. The temperature is used to run the turbine across a magnetic field to generate electricity. Both theoretical and experimental work is found in the literature on a stratified laminar flow of two immiscible fluids in a horizontal pipe. Packham and Shail [11] analyzed a stratified laminar flow of two immiscible liquids in a horizontal pipe. The Hartmann flow of a conducting fluid in a channel with a layer of non-conducting fluid between the upper channel wall and the conducting fluid was studied by Shail [12]. He found that an increase of the order of 30% could be achieved in the flow rate for suitable ratios of depths and viscosities of the two fluids. Lohrasbi and Sahai [13]

J. Prathap Kumar et al. / International Journal of Non-Linear Mechanics 46 (2011) 278–285

Nomenclature A b B0 Br Cp g Gr GR h h1 h2 K K1 K2 M m n p Re T

defined in Eq. (2.8) thermal expansion coefficient ratio (b2/b1) magnetic field 2 Brinkman number ðm1 U0ð1Þ =K1 DTÞ specific heat at constant pressure acceleration due to gravity Grashoff number ðg b1 h31 DT=n21 Þ dimensionless parameter (Gr/Re) defined in Eq. (2.15) width ratio (h2/h1) width of the region-I width of the region-II ratio of the thermal conductivities (K1/K2) thermal conductivity of the fluid in region-I thermal conductivity of the fluid in region-II pffiffiffiffiffiffiffiffiffiffiffiffiffi Hartmann number ðh1 B0 se =m1 Þ ratio of the viscosities (m1/m2) ratio of the densities (r2/r1) dimensional pressure Reynolds number ðU0ð1Þ h1 =n1 Þ temperature

dealt with the two-phase MHD flow and heat transfer in a parallel-plate channel. Alireza and Sahai [14] studied the effect of temperature-dependent transport properties on the developing MHD flow and heat transfer in a parallel-plate channel whose walls were held at constant and equal temperatures. Following the work of Alireza and Sahai [14], Malashetty and Umavathi [15], Malashetty et al. [16–18] studied the two-phase MHD flow and heat transfer in an inclined channel. Recently Prathap Kumar et al. [19] studied mixed convection of composite porous medium in a vertical channel with asymmetric wall heating conditions. In all the papers referred in the previous paragraph and in the present paper, the Oberbeck–Boussinesq approximation is considered. The status of the Oberbeck–Boussinesq approximation for the Newtonian fluid has been the object of much discussion. The studies devoted to the Oberbeck–Boussinesq approximations are too many, a few in which the approximations are discussed are Rayleigh [20], Chandrasekhar [21] and Spiegel and Veronis [22]. Spiegel and Veronis [22] have discussed the Oberbeck–Boussinesq approximation for a compressible fluid by expressing the state variables as the sum of a mean value variation in the absence of motion and a fluctuation resulting from the motion. However, their approximation is not consistent and analysis requires that the thickness of the layer be very small as pointed out by Rajagopal et al. [23]. Rajagopal et al. [23] for the first time have presented a framework within which the status of the Oberbeck–Boussinesq approximation can be defined for the Newtonian fluid. They derived the Oberbeck equations using a non-dimensionalisation as suggested by Chandrasekhar and utilizing the ratio of two characteristic velocities as a measure of smallness. As suggested by Chandrasekhar [21], for moderate temperature gradients, the coefficient of volume expansion is in the range 10  3–10  4; variation in the density is negligibly small. Variation in other coefficients like viscosity, thermal conductivity, specific heat is also negligibly small. These investigations were carried out for free convection flow and the work is not found in the literature for mixed convection of two-fluid flow. Keeping in view the practical applications of mixed convection flow as mentioned earlier, it is the objective of the present work to analyze mixed convective flow and heat

U0ðiÞ u ui T1,T2 X,Y

279

reference velocity ððdP=dXÞðh2i =48mi ÞÞ velocity average velocity temperature of the boundaries space co-ordinates

Greek symbols

a se b

r v

m e DT yi

thermal diffusivity electrical conductivity coefficient of thermal expansion density of the fluid kinematic viscosity viscosity dimensionless parameter defined in Eq. (3.14) difference in temperature (T2  T1) non-dimensional temperature (Ti  T0/DT)

Subscripts 1 and 2 reference quantities for region-I and II, respectively.

transfer of conducting and non-conducting immiscible fluids in a vertical channel.

2. Mathematical formulation The geometry under consideration illustrated in Fig. 1 consists of two infinite parallel plates maintained at different or equal constant temperatures extending in the X and Z directions. The region 0 rY r  h1/2 is occupied by viscous fluid of density r1, viscosity m1, thermal conductivity K1, thermal expansion coefficient b1 and electrical conductivity se. A constant magnetic field of strength B0 is applied transversely to the flow field. It is assumed that the magnetic Reynolds number is sufficiently large so that the induced magnetic field can be neglected, and the induced electric field is assumed to be negligible. The region

X

B0 Region-I

Region-II

MHD

Viscous

↓ g

Y

Y=−

h1

Y=

2 Fig. 1. Physical configuration.

h2 2

280

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h2/2rY r0 is occupied by a different (immiscible) viscous fluid of density r2, viscosity m2, thermal conductivity K2, and thermal expansion coefficient b2. The fluids are assumed to have constant properties except the density in the buoyancy term in momentum equation. A fluid rises in the channel driven by buoyancy forces. The transport properties of both fluids are assumed to be constant. We consider the fluids to be incompressible and the flow is steady, laminar and fully developed. It is assumed that the only non-zero components of the velocity are the X-component Ui (i¼1,2). Thus, as a consequence of the mass balance equation, one obtains

Eqs. (2.6) and (2.10), (2.7) and (2.11) allow one to obtain differential equation for Ui namely Region-I   d4 U1 se B20 d2 U1 r1 g b1 dU1 2 se B20 g b1 2  ¼ þ U1 ð2:12Þ m1 dY 2 K1 dY K1 v1 dY 4

@Ui ¼0 @X

The boundary conditions on Ui are both no slip conditions and those induced by the boundary conditions on T and by Eqs. (2.6) and (2.7) are

ð2:1Þ

so that Ui depends only on Y. The stream wise and the transverse momentum balance equations yields [24] Region-I g b1 ðT1 T0 Þ

1 @P

r1 @X

þ v1

d2 U1 se B20  U ¼0 r1 1 dY 2

ð2:2Þ

and Y-momentum balance equation can be expressed as @P ¼0 @Y Region-II g b2 ðT2 T0 Þ

1 @P d2 U2 þ v2 ¼0 r2 @X dY 2

ð2:6Þ

d U2 ¼0 dY 2

dP ¼A ð2:8Þ dX On account of Eq. (2.8) by evaluating the derivatives of Eqs. (2.6) and (2.7) with respect to X, one obtains ð2:9Þ

so that the temperature also depends on Y. By taking into account the effect of viscous and Ohmic dissipations, the energy balance equation can be written as Region-I  2 se B20 2 d2 T v dU1 a1 21 þ 1 þ U ¼0 ð2:10Þ Cp dY r1 Cp 1 dY Region-II  2 d2 T2 v2 dU2 þ ¼0 2 Cp dY dY

  d2 U1 ð0Þ se B20 1 d2 U2 ð0Þ A 1  U1 ð0Þ ¼ þ 1 2 2 mnb dY nb m1 m1 dY d3 U1 ð0Þ se B20 dU1 ð0Þ 1 d3 U2 ð0Þ ¼  mnKb dY 3 m1 dY dY 3

dU1 ð0Þ dU2 ð0Þ ¼ m2 ; dY dY

Eqs. (2.12)–(2.14) can be written in a dimensionless form by employing the dimensionless quantities u1 ¼

U1 U0ð1Þ

u2 ¼

;

y2 ¼

Y2 ; h2

Gr ¼

GR ¼

Gr ; Re

RT ¼

ð2:7Þ

Let us assume that the walls of the channel are isothermal. In particular, the temperature of the boundary at Y ¼ h1/2 is T1, while the temperature at Y¼  h1/2 is T2, with T2 ZT1. These boundary conditions are compatible with Eqs. (2.6) and (2.7) if and if only dP/dX is independent of X. Therefore, there exists a constant A such that

dT2 ¼0 dX

d2 U2 A g b2 DT RT h2 ¼  at Y ¼ m2 2v2 2 dY 2

U2 U0ð2Þ

y1 ¼

;

T1 T0 ; DT

y2 ¼

U ð1Þ h1 g b1 DTh31 ; Re ¼ 0 ; 2 v1 v1 T2 T1 ; DT

M2 ¼

T2 T0 ; DT

Br ¼

m1 U0ð1Þ K1 DT

se B20 h21 m1

y1 ¼

Y1 ; h1

2

; ð2:15Þ

The reference velocity U0ðiÞ and the reference temperature T0 are given by

2

þ v2

h2 2

ð2:14Þ

Region-II 1 dP

U2 ¼ 0 at Y ¼

ð2:4Þ

ð2:5Þ

r2 dX

h1 ; 2

ð2:13Þ

d2 U1 A g b DT RT h1 ¼ þ 1 at Y ¼  ; m1 2v1 2 dY 2

m1

1 dP d2 U1 se B20 g b1 ðT1 T0 Þ þ v1  U ¼0 r1 dX r1 1 dY 2

a2

U1 ¼ 0 at Y ¼ 

U1 ð0Þ ¼ U2 ð0Þ;

where P¼p + r0gx (assuming P1 ¼P2 ¼P) is the difference between the pressure and the hydrostatic pressure. On account of Eqs. (2.3) and (2.5), P depends only on X so that Eqs. (2.2) and (2.4) can be rewritten as Region-I

dT1 ¼ 0; dX

  d U2 r g b dU2 2 ¼ 2 2 4 K2 dY dY

ð2:3Þ

@P ¼0 @Y

g b2 ðT2 T0 Þ

Region-II 4

ð2:11Þ

U0ð1Þ ¼ 

Ah21 ; 48m1

U0ð2Þ ¼ 

Ah22 ; 48m2

T0 ¼

T1 þ T2 2

ð2:16Þ

Moreover, the temperature difference DT is given by DT¼ T2  T1 if T1 o T2. Consequently, the dimensionless parameter RT can only take the values 0 or 1. RT is 1 for asymmetric heating (T1 o T2), while RT is 0 for symmetric heating (T1 ¼T2). Eq. (2.8) implies that A can be either positive or negative. If A40, then U0ðiÞ , Re and GR are negative, i.e. the flow is downward. On the contrary, if Ao0, the flow is upward, so that U0ðiÞ , Re and GR are positive. Using Eqs. (2.15) and (2.16), Eqs. (2.12)–(2.14) become Region-I  2 4 d u1 d2 u1 du1 M 2 ¼ GRBr þ GRBr M 2 u21 ð2:17Þ 4 2 dy dy dy Region-II d4 u2 ¼ mnbKh4 GRBr dy4



du2 dy

2 ð2:18Þ

The boundary and interface conditions become 1 1 u1 ¼ 0 at y ¼  ; u2 ¼ 0 at y ¼ , 4 4 d2 u1 GR RT 1 d2 u2 nb GR RT 1 at y ¼ ; at y ¼ ¼ 48 þ ¼ 48 4 dy2 4 2 2 dy2

J. Prathap Kumar et al. / International Journal of Non-Linear Mechanics 46 (2011) 278–285

u1 ð0Þ ¼ mh2 u2 ð0Þ;

  d2 u1 ð0Þ 1 d2 u2 ð0Þ 2 M u ð0Þ ¼ þ 48ð1nbÞ 1 nb dy2 dy2

du1 ð0Þ du2 ð0Þ ¼h ; dy dy

d3 u1 ð0Þ du1 ð0Þ 1 d3 u2 ð0Þ ¼ M 2 dy nbKh dy3 dy3

ð2:19Þ

281

Using Eqs (3.7) and (3.8) and solving Eqs. (3.9) and (3.10) we obtain Region-I

y1 ¼ BrðF1 Coshð2MyÞ þ F2 Sinhð2MyÞ þ F3 CoshðMyÞ þ F4 SinhðMyÞ þ F5 y2 Þ þPy þ Q

ð3:12Þ

Region-II

3. Solutions

y2 ¼ BrKmh4 ðF6 y4 þF7 y3 þ F8 y2 Þ þ Ry þ S

3.1. Special cases

ð3:13Þ

Case-I: The solution of Eqs. (2.17) and (2.18) using boundary and interface conditions (2.19) in the absence of viscous dissipation terms (Br¼0) is given by Region-I

3.2. Perturbation method

u1 ¼ B1 þ B2 y þB3 CoshðMyÞ þB4 SinhðMyÞ

ð3:1Þ

e¼

ð3:2Þ

and does not depend on the reference temperature difference DT. To this end the solutions are assumed in the form

Region-II u2 ¼ B5 þ B6 y þB7 y2 þ B8 y3

Using Eq. (2.15) in Eqs. (2.6) and (2.7), the energy balance equation becomes Region-I   1 d2 u1 48 þ y1 ¼  M 2 u1 ð3:3Þ 2 GR dy Region-II

y2 ¼ 

  1 d2 u2 48 þ nbGR dy2

ð3:4Þ

Using the expression obtained in Eqs. (3.1) and (3.2) the energy balance Eqs. (3.3) and (3.4) become Region-I 1 ð48M 2 B1 M 2 B2 yÞ GR Region-II

y1 ¼ 

ð3:5Þ

1 ð48 þ 2B7 þ 6B8 yÞ y2 ¼  ð3:6Þ nbGR Case-II The solutions of Eqs. (2.17) and (2.18) can be obtained when buoyancy forces are neglected (GR¼0) and viscous dissipations are dominating (Bra0), so that a purely forced convection occurs. For this case, solutions of (2.17) and (2.18), using boundary and interface conditions given by (2.19), the velocities are given by Region-I u1 ¼ D1 þD2 y þ D3 CoshðMyÞ þ D4 SinhðMyÞ

ð3:7Þ

ð3:8Þ

The energy balance Eqs. (2.10) and (2.11) in non-dimensional form can also be written as Region-I  2 2 d y1 du1 ¼ Br Br M 2 u21 ð3:9Þ 2 dy dy

1 X

en un ðyÞ

ð3:15Þ

n¼0

Substituting Eq. (3.15) into Eqs. (2.17) and (2.18) and equating the coefficients of like powers e to zero, we obtain the zeroth and first-order equations for the cases of isothermal–isothermal, isoflux–isothermal and isothermal–isoflux wall conditions as Case 1: Isothermal–isothermal (T1  T2) walls Region-I Zeroth-order equations d4 u10 d2 u10 M2 ¼0 dy4 dy2

ð3:16Þ

First-order equations  2 2 d4 u11 du10 2 d u11 M ¼ þ M2 u210 dy dy4 dy2

ð3:17Þ

Region-II Zeroth-order equations d4 u20 ¼0 dy4 First-order equations  2 d4 u21 4 du20 ¼ mnbKh dy dy4

ð3:18Þ

ð3:19Þ

1 1 u10 ¼ 0 at y ¼  ; u20 ¼ 0 at y ¼ 4 4 d2 u10 GR RT 1 d2 u20 nbGR RT 1 at y ¼  ; at y ¼ ¼ 48þ ¼ 48 4 4 2 2 dy2 dy2   d2 u10 ð0Þ 1 d2 u20 ð0Þ 2 2 M u ð0Þ ¼ þ 48ð1nbÞ u10 ð0Þ ¼ mh u20 ð0Þ; 10 nb dy2 dy2 d3 u10 ð0Þ du10 ð0Þ 1 d3 u20 ð0Þ ¼ M 2 dy nbKh dy3 dy3 ð3:20Þ

ð3:10Þ

The boundary and interface conditions for temperature are   1 RT dy1 ð0Þ 1 dy2 ð0Þ ¼ y2 ¼ ; y1 ð0Þ ¼ y2 ð0Þ; 4 dy Kh dy 2 ð3:11Þ

  1 RT ¼ ; y1  4 2

ð3:14Þ

uðyÞ ¼ u0 ðyÞ þ eu1 ðyÞ þ e2 u2 ðyÞ þ . . . ¼

du10 ð0Þ du20 ð0Þ ¼h ; dy dy

Region-II  2 d2 y2 du2 ¼ Br Kmh4 2 dy dy

Gr Br Re

The corresponding boundary and interface conditions given by Eq. (2.19) for zeroth and first order reduce to

Region-II u2 ¼ D5 þD6 y þ D7 y2 þ D8 y3

We solve Eqs. (2.17) and (2.18) using the perturbation method with a dimensionless parameter 9e9 ({1) defined as

1 1 u11 ¼ 0 at y ¼  ; u21 ¼ 0 at y ¼ 4 4 d2 u11 1 d2 u21 1 ; ¼ 0 at y ¼  ¼ 0 at y¼ 4 4 dy2 dy2 d2 u11 ð0Þ 1 d2 u21 ð0Þ 2 2 u11 ð0Þ ¼ mh u21 ð0Þ; M u11 ¼ nb dy2 dy2

282

J. Prathap Kumar et al. / International Journal of Non-Linear Mechanics 46 (2011) 278–285

du11 ð0Þ du21 ð0Þ ¼h ; dy dy

d3 u11 ð0Þ du11 ð0Þ 1 d3 u21 ð0Þ ¼ M2 3 dy nbKh dy3 dy

and u2 are ð3:21Þ

Solutions of zeroth-order Eqs.(3.16) and (3.18) using boundary and interface conditions (3.20) are u10 ¼ C1 þC2 y þ C3 CoshðMyÞ þ C4 SinhðMyÞ 2

3

u20 ¼ C5 þC6 y þ C7 y þ C8 y

ð3:22Þ ð3:23Þ

Solutions of first-order Eqs. (3.17) and (3.19) using boundary and interface conditions (3.21) are

d3 u2 1 ¼ GRnb at y ¼ 4 dy3

ð3:32Þ

The other boundary condition at the right wall can be shown to be the same as that given for the isothermal–isothermal wall with RT being replaced by Rtq such that d2 u1 GRRtq 1 at y ¼  ¼ 48 þ 4 2 dy2

ð3:33Þ

The integrating constants appeared in the expressions (3.22)–(3.25) are evaluated using boundary and interface conditions (3.20), (3.21), (3.22) and (3.33) and are not presented.

u11 ¼ C9 þC10 y þC11 CoshðMyÞ þ C12 SinhðMyÞ þ f10 Coshð2MyÞ

u21 ¼ f19 y8 þ f20 y7 þ f21 y6 þ f22 y5 þ f23 y4 þ

4. Results and discussion ð3:24Þ

C13 3 C14 2 y þ y þ C15 y þ C16 6 2 ð3:25Þ

Using velocities given by relations (3.22)—(3.25), the expressions for energy balance Eqs. (3.3) and (3.4) become Region-I

y1 ¼ ð48 þ M2 C1 þ M2 C2 yeðM2 C9 M2 C10 y þ M2 f10 Coshð2MyÞ þ 3M2 f11 Sinhð2MyÞ þ f12 ð4MySinhðMyÞ þ2CoshðMyÞÞ þ f13 ð4MyCoshðMyÞ þ 2SinhðMyÞÞ þ 2Mf14 SinhðMyÞ þ 2Mf15 CoshðMyÞ þ f16 ð12y2 M2 y4 Þ þ f17 ð6yM 2 y3 Þ þ f18 ð2M2 y2 ÞÞÞ=GR

ð3:26Þ

Region-II

y2 ¼ ð482C7 6C8 yeð56f19 y6 þ 42f20 y5 þ30f21 y4 þ 20f22 y3 þ 12f23 y2 þ C13 yþ C14 Þ=nbGR

ð3:27Þ

The constants appeared in Eqs. (3.22)–(3.27) are not presented. Case 2: Isoflux–isothermal (q1  T2) walls For this case, the thermal boundary conditions for the channel walls can be written in the non-dimensional form as     Rqt dy1 1 1 ¼ 1; y2 ¼  ð3:28Þ 4 4 dy 2 where Rqt ¼(T2  T0)/DT is the dimensionless thermal parameter for isoflux–isothermal walls. Other than the no-slip conditions at the channel walls, two more boundary conditions in terms of u1 and u2 are

Analytical solutions for the steady fully developed mixed convective flow and heat transfer in a vertical channel containing conducting and non-conducting fluid layers are obtained using regular perturbation technique. The product of GR Br, where GR is the mixed convection parameter and Br is the Brinkman number is used as the perturbation parameter. The flow field for the case of asymmetric heating (RT ¼1) are obtained and depicted in Figs. 2–13. Solutions (3.1) and (3.2) show the same result obtained by Barletta [25] for one fluid model for the velocity u in both the regions for different values of GR and M ¼2. Eqs. (3.12) and (3.13) for the dimensionless temperature profiles y are evaluated for different values of Br with M ¼2 and it is noticed that the temperature field is similar to Umavathi et al. [26]. Figs. 2 and 3 display the effect of GR and e on the velocity and temperature fields for both assisting and opposing flows. For positive values of GR and e, the flow reversal is near the cool wall and for negative values of GR and e, the flow reversal is near the hot wall. The effect of GR and e on temperature will not vary much as seen in Fig. 3. The effect of viscosity ratio m on velocity is shown is Fig. 4. It is observed that as viscosity ratio increases velocity increases for conducting fluid and decreases for viscous fluid. Flow reversal is observed for m¼0.1 near the cold wall. Effect of viscosity ratio m on temperature is very sensitive. Effect of width ratio h on the velocity and temperature is shown in Figs. 5 and 6, respectively. As the width ratio h increases, both the velocity and temperature decrease. That is, larger the width of the clear viscous fluid, smaller the velocity and the temperature field. The magnitude of suppression is large for conducting fluid compared to viscous fluid. Effect of conductivity ratio K suppresses the velocity and temperature fields as seen in Figs. 7 and 8 respectively. The magnitude of suppression is almost equal for both

3

d3 u1 du1 1 ¼ GR at y ¼  M 2 4 dy dy3

ð3:29Þ

GR Rqt d2 u2 1 at y ¼ ¼ 48 4 2 dy2

ð3:30Þ

The integrating constants appeared in Eqs (3.22)–(3.25) are evaluated using boundary and interface conditions (3.20), (3.21), (3.29) and (3.30) and are not presented. Case 3: Isothermal–isoflux (T1  q2) walls For this case, the thermal boundary conditions for the channel walls can be written in the non-dimensional form as     Rtq dy2 1 1 ¼ 1, y1  ¼ ð3:31Þ 4 dy 4 2 where Rqt ¼(T1  T0)/DT is the dimensionless thermal parameter for isothermal–isoflux walls. Other than the no-slip conditions at the channel walls, two more boundary conditions in terms of u1

2

u

þ f11 Sinhð2MyÞ þf12 y2 CoshðMyÞ þ f13 y2 SinhðMyÞ þ f14 yCoshðMyÞ þ f15 ySinhðMyÞ þf16 y4 þf17 y3 þf18 y2

GR = 500, ε = 8

RT = 1 m=1 K=1 b=1 n=1 h=1 M=2

GR = 500, ε = 0.1

1 GR = -500, ε = −0.1

0 GR = -500, ε = −8 Region-I

-0.2

-0.1

Region=II

0.0

0.1

y Fig. 2. Velocity profiles for different values of GR and e.

0.2

J. Prathap Kumar et al. / International Journal of Non-Linear Mechanics 46 (2011) 278–285

the regions. The effect of Hartmann number M on the velocity is shown in Fig. 9. As the Hartmann number increases velocity decreases in both the regions, which is the classical Hartmann result. It is also observed from this figure that an increase in M decreases the velocity in viscous fluid region also. The effect of Hartmann number M on temperature is almost invariant.

0.4

θ

0.2

283

Figs. 10–13 illustrate the effect of Hartmann number M on the flow for isoflux–isothermal and isothermal–isoflux wall conditions for asymmetric heating. The effect of M is to suppress the velocity and temperature for isoflux–isothermal wall conditions as seen in Figs. 10 and 11. It is also seen that its effect on temperature is highly significant near the cold wall. Effect of M on

GR = 500, -500, ε = 0.1, 8, −8

RT = 1 m=1 n=1 K=1 b=1 h=1 M=2

0.4

0.2

GR = -500, ε = −0.1

0.0

0.0

GR = 500 ε = 0.1 RT = 1 m=1 n=1 K=1 b=1 M=2

Region-I

Region-II

h=1 h=2 h=3

θ

-0.2

-0.2

Region-I

-0.4

Region-II -0.4

-0.2

-0.1

0.0

0.2

0.1

y

-0.2

-0.1

0.0

0.1

0.2

y

Fig. 3. Temperature profiles for different values of GR and e.

Fig. 6. Temperature profiles for different values of width ratio h.

3

m = 0.1

2.5

m=2

2.0 1.5 1.0

m=4

1

0.5 u

u

2

GR = 500 ε = 0.1 RT = 1 M=2 K=1 h=1 b=1 n=1

GR = 500 ε = 0.1 RT = 1 m=1 n=1 h=1 M=2

K=1

K=2

0.0

K=3

-0.5

0

-1.0

Region-I

Region-II

-1.5

-1 -0.2

0.0

-0.1

0.1

Region-I -0.2

-0.1

y

1.5

u

1.0

GR = 500 ε = 0.1 RT = 1 m=1 n=1 K=1 b=1 M=2

0.2

Fig. 7. Velocity profiles for different values of thermal conductivity ratio K.

h=1 0.4

0.2

h=2

0.0

h=3

0.5 0.0

GR = 500 ε = 0.1 RT = 1 m=1 n=1 h=1 b=1 M=2

Region-I

Region-II

K=1 K=2 K=3

-0.2

-0.5

0.1

θ

2.0

0.0 y

Fig. 4. Velocity profiles for different values of viscosity ratio m.

2.5

Region-II

-2.0

0.2

Region-I

Region-II

-0.4

-1.0 -0.2

-0.1

0.0

0.1

y Fig. 5. Velocity profiles for different values of width ratio h.

0.2

-0.2

-0.1

0.0

0.1

0.2

y Fig. 8. Temperature profiles for different values of thermal conductivity ratio K.

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J. Prathap Kumar et al. / International Journal of Non-Linear Mechanics 46 (2011) 278–285

GR = 500 ε = 0.1 RT = 1 m=1 K=1 b=1 n=1 h=1

2

Region-II

M=6

M=4

M=8

M=2 -4 u

u

1

Region-I

0

M=8 M=6

0

Region-I -0.2

0.0

0.1

M=4

-8

Region-II

-0.1

GR = 500 ε = 0.1 Rtq = 1 m=1 n=1 K=1 b=1 h=1

M=2

0.2

y

-0.2

-0.1

0.0

0.1

0.2

y

Fig. 9. Velocity profiles for different values of Hartmann number M.

Fig. 12. Velocity profiles for different values of M in isothermal–isoflux wall conditions.

GR = 500 ε = 0.1 Rqt = 1 m=1 n=1 K=1 b=1 h=1

M=2 12

M=4 M=6

8

-0.5 Region-I

Region-II

-0.6

u

M=8

-0.7 Region-I

θ

4 Region-II

-0.8

0 -0.2

-0.1

0.0

0.1

GR = 500 ε = 0.1 Rtq = 1 m=1 n=1 K=1 h=1 b=1

0.2

M = 2,4,6,8

-0.9

y Fig. 10. Velcoity profiles for different values of M in isoflux-isothermal wall conditions.

-1.0 -0.2

-0.1

0.0

0.1

0.2

y

Region-I

Region-II

1.0

θ

M = 2,4,6,8 0.8

GR = 500 ε = 0.1 Rqt = 1 m=1 n=1 K=1 b=1 h=1

0.6

-0.2

0.0

-0.1

0.1

0.2

y

Fig. 13. Temperature profiles for different values of M in isothermal–isoflux wall conditions.

in the presence of viscous and Ohmic dissipation is analyzed. Three different combinations of thermal left–right wall conditions were presented. Various analytical solutions on the flow for different special cases with isothermal–isothermal, isoflux– isothermal and isothermal–isoflux wall heating conditions were obtained. Graphical results were displayed for different parameters governing the flow. Considering equal values for viscosity, width and conductivity for fluids in both the regions we get back the results of Barletta [25] and Umavathi et al. [26] for one fluid model. Hartmann number decreases the velocity in both the regions; viscosity ratio increases the velocity for conduction region and decreases the velocity for viscous region. Width ratio and conductivity ratio suppresses the flow in both the regions.

Fig. 11. Temperature profiles for different values of M in isoflux–isothermal wall conditions.

velocity and temperature is shown in Figs. 12 and 13 for isothermal–isoflux wall conditions. As M increases velocity increases in the negative direction and temperature decreases. Its effect on temperature is significant near the hot wall.

5. Conclusions The problem of steady, laminar mixed convective flow in a vertical channel filled with conducting and non-conducting fluids

Acknowledgment One of the authors J.C. Umavathi, thank Prof. I.C. Liu, Department of Civil Engineering, National Chi Nan University, University Rd. Puli, Nantou, Taiwan 545, ROC for his academic support.

References [1] L.N. Tao, On combined free and forced convection in channel, ASME J. Heat Transfer 82 (1960) 233–238.

J. Prathap Kumar et al. / International Journal of Non-Linear Mechanics 46 (2011) 278–285

[2] W. Aung, G. Worku, Theory of fully developed combined convection including flow reversal, ASME J. Heat Transfer 108 (1986) 485–488. [3] W. Aung, G. Worku, Developing flow and flow reversal in a vertical channel with symmetric wall temperatures, ASME J. Heat Transfer 108 (1986) 299–304. [4] W. Aung and G..Worku, Mixed convection in ducts with asymmetric wall heat fluxes ASME J. Heat Transfer, 109, 947-951, (1987). [5] G.M. Oreper, J. Szekely, The effect of an external imposed magnetic field on buoyancy driven flow in a rectangular cavity, J. Cryst. Growth 64 (1983) 505–515. [6] T. Alboussiere, J.P. Garandet, R. Moreau, Buoyancy-driven convection with a uniform magnetic field, Part 1. Asymptotic analysis, J. Fluid Mech. 253 (1983) 545–563. [7] J.P. Garandet, T. Alboussiere, R. Moreau, Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field, Int. J. Heat Mass Transfer 35 (1992) 741–749. [8] J.C.R. Hunt, Magnetohydrodynamic flow in a rectangular duct, J. Fluid Mech. 21 (1965) 577–590. [9] L. Buhler, Laminar buoyant magnetohydrodynamic flow in a rectangular duct, Phys. Fluids 10 (1998) 223–236. [10] J.A. Shercliff, Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. Cambridge Philos. Soc. 49 (1953) 136–144. [11] B.A. Packham, R. Shail, Stratified laminar flow of two immiscible fluids, Proc. Cambridge Philos. Soc. 69 (1971) 443–448. [12] R. Shail, On laminar tow-phase flow in magnetohydrodynamics, Int. J. Eng. Sci. 11 (1973) 1103–1108. [13] J. Lohrasbi, V. Sahai, Magnetohydrodynamic heat transfer in two phase flow between parallel plates, Appl. Sci. Res. 45 (1987) 53–66. [14] S. Alireza and V. Sahai, Heat transfer in developing magnetohydrodynamic Poiseuille flow and variable transport properties, Int. J. Heat Mass Transfer, 33, 1711-1720, (1990).

285

[15] M.S. Malashetty, J.C. Umavathi, Two phase magnetohydrodynamic flow and heat transfer in an inclined channel, Int. J. Multiphase Flow 23 (1997) 545–560. [16] M.S. Malashetty, J.C. Umavathi, J. Prathapkumar, Two fluid magnetoconvection flow in an inclined channel, Int. J. Transp. Phenom. 3 (2000) 73–84. [17] M.S. Malashetty, J.C. Umavathi, J. Prathapkumar, Convective magneto hydrodynamic two fluid flow and heat transfer in an inclined channel, Heat Mass Transfer 37 (2001) 259–264. [18] M.S. Malashetty, J.C. Umavathi, J. Prathap Kumar, Magnetoconvection of two-immiscible fluids in vertical enclosure, Heat Mass Transfer 42 (2006) 977–993. [19] J. Prathap Kumar, J.C. Umavathi, B.M. Biradar, Mixed convection of composite porous medium in a vertical channel with asymmetric wall heating conditions, J. Porous Medium 13 (2010) 271–285. [20] Lord Rayleigh, On convection currents in a horizontal layer of liquid when the higher temperature is on the underside, Philos. Mag. 32 (1916) 529–546. [21] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover New York, 1981. [22] E.A. Spiegel, G. Veronis, On the Boussinesq Approximation for a compressible fluid, Astrophysical Journal 131 (1960) 442–447. [23] K.R. Rajagopal, M. Ruzicka, A.R. Srinivasa, On the Oberbeck–Boussinesq approximation, Math Models Methods Appl. Sci. 6 (1996) 1157–1167. [24] V.S. Arpaci, P.S. Larsen, Convection Heat Transfer, Pentice-Hall, Engle-Wood Cliffs, NJ, 1984 pp. 51–54. [25] A. Barletta, Analysis of combined forced and free flow in a vertical channel with viscous dissipation and isothermal–isoflux boundary conditions, ASME J. Heat Transfer 121 (1999) 349–356. [26] J.C. Umavathi, B.Patil Mallikarjun, I. Pop, On laminar mixed convection flow in a vertical porous stratum with symmetric wall heating conditions, Int. J. Transp. Phenom. 8 (2006) 1–14.