Hydromagnetic Mixed Convective Two-Phase Flow of Couple Stress and Viscous Fluids in an Inclined Channel Zaheer Abbas

Hydromagnetic Mixed Convective Two-Phase Flow of Couple Stress and Viscous Fluids in an Inclined Channel

Zaheer Abbas^a, Jafar Hasnain^a, and Muhammad Sajid^b

aDepartment of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

bTheoretical Physics Division, PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan Reprint requests to J. H.; E-mail:jafar_hasnain14@yahoo.com

Z. Naturforsch.69a, 553 – 561 (2014) / DOI: 10.5560/ZNA.2014-0048

Received January 22, 2014 / revised June 4, 2014 / published online July 30, 2014

An analysis is carried out to study magnetohydrodynamic (MHD) flow and heat transfer of two immiscible fluids in an inclined channel. The channel is filled with couple stress fluid in one region and a viscous fluid in the other region. The viscous fluid is assumed to be electrically conducting. The governing equations are modelled using the fully developed flow conditions. A closed form solutions of velocity and the temperature profiles are obtained by using perturbation method. The physical interpretation of the emerging parameters of interest on the velocity and temperature distributions are shown through graphs and discussed in detail.

Key words:Two-Phase Flow; MHD; Couple Stress; Viscous Fluids; Inclined Channel.

1. Introduction

The two phase flow of Newtonian and non- Newtonian fluids in channels or over moving surfaces have attained much attention in recent years due to their wide applications in the industry and engineering.

Applications of such flows can be found in the indus- trial processes, methods of transporatory multi-phase fluids through pipelines and wells. Furthermore, multi- phase petroleum wells have existed for a long time and multi-phase flow plays a very important role in the pro- cess industry and nuclear industry. The flow of oil–

water mixtures through pipes, channels, and liquid–

liquid solvent extraction mass transfer systems is an- other area of applications of two-phase flows. Packham and Shail [1] investigated the laminar two-phase flow of immiscible flow of fluids through a pipe. Shail [2]

extended the problem discussed in [1] by considering the magnetohydrodynamic (MHD) two-phase fluids through a rectangular channel. Lohrasbi and Sahai [3]

discussed two-phase MHD flow and heat transfer anal- ysis in a horizontal channel. Malashetty and Leela [4]

investigated the characteristics of heat transfer in the two-phase flow when the fluids are electrically con- ducting in both phases through a horizontal channel.

In another paper, Malashetty and Leela [5] studied the

MHD heat transfer analysis of two immiscible fluids in a horizontal pipe. Chamkha [6] discussed the ana- lytical solutions for flow of two immiscible fluids in porous and non-porous parallel plates. Recently, Uma- vathi et al. [7] studied the unsteady Hartmann flow of two immiscible fluids with time-dependent oscillatory wall transpiration velocity in a horizontal channel. In another paper, Umavathi et al. [8] discussed the un- steady oscillatory flow and heat transfer analysis in a horizontal composite porous channel analytically us- ing two term harmonic and non-harmonic functions.

In all the above studies the considered geometry is horizontal having zero inclination. Similarly flow regime maps can be drawn for vertical pipes/channels and pipes/channels with uphill and downhill incli- nations particularly in heat transfer analysis, e. g.

solar collective technology. First Raithby and Hol- lands [9] and Catton [10] gave the basic studies in this area. Rheault and Bilgen [11] presented the nu- merical solutions for the problem of steady mixed con- vection heat transfer in open-ended inclined channels with flow reversal using a control volume approach.

Malashetty and Umavathi [12] investigated two-phase MHD flow through an inclined channel making an an- gleφ with the horizontal axes analytically using per- turbation method. They obtained the velocity and tem-

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perature distributions for zero and first-order approx- imation and discussed several involved fluid parame- ters. An approximate solution by using regular pertur- bation method for MHD mixed convection two-phase flow through an inclined channel has been presented by Malashetty et al. [13,14]. Later on, Malashetty et al. [15] analyzed the MHD two-fluid convective flow and heat transfer in an inclined composite porous medium. The problem of combined free and forced convection MHD two-phase flow in a vertical chan- nel was discussed by Umavathi and Malashetty [16].

More recently, Sivaraj et al. [17] studied MHD mixed convective flow of two immiscible fluids in a vertical porous channel filled with a viscoelastic fluid in one region and a viscous fluid in the other region. They ob- tained an exact solution of the considered problem.

Due to various real world applications several in- dustrial processes involve the channel and heat trans- fer flow of non-Newtonian fluids. Among them the ex- trusion of polymer fluids, cooling of metallic plates in a bath, solidification of liquid crystals and colloidal solutions, etc. are a few well-studied problems con- cerning couple stress fluids, a special class of non- Newtonian fluids, and having an ability to describe various types of lubricants, blood, and suspension flu- ids. Stokes [18,19] did the pioneer work in develop- ing the theory of couple stress fluids. The theory rep- resents the generalization of classical Newtonian the- ory which permits the polar effects such as the pres- ence of couple stresses and body couples. In addition to polar effects, the couple stress fluids possess the large viscosity. The flow and heat transfer character- istics of Oberbeck convection of a couple stress fluid in a vertical porous stratum was investigated by Uma- vathi and Malashetty [20]. Sreenadh et al. [21] dis- cussed the MHD free convection flow of a couple stress fluid in a vertical porous layer. They presented the series solution using the perturbation method. Srini- vasacharya and Kaladhar [22] presented an analysis in which they examined the fully developed couple stress fluid flow between vertical parallel plates in the pres- ence of a temperature dependent heat source. Recently, Makinde and Eegunjobi [23] investigated the entropy generation in a couple stress fluid flow through a ver- tical channel filled with saturated porous substances and obtained the numerical solution using shooting method.

The aim of the present paper is to study the MHD mixed convection two-phase flow of two immisci-

Fig. 1. Geometry of the flow problem.

ble fluids in an inclined channel with an inclination φ. The upper phase of the channel contains a cou- ple stress fluid and the lower phase of the channel is filled with a viscous fluid which is electrically con- ducting. The approximate analytical solutions of the velocity and temperature distributions are obtained in the closed form using a perturbation method upto the first order. To the best of our knowledge the study of MHD flow of a couple stress and viscous fluids in an inclined channel has not been investigated be- fore.

2. Formulation of the Problem

Consider a fully developed hydromagnetic mixed convection and laminar flow of two immiscible flu- ids, namely a couple stress and a viscous fluid flowing through a channel inclined at an angleφwith horizon- tal axis. The geometry of the considered problem is presented in Figure1. A Cartesian coordinate system is chosen in such a way that thex-axis is along the chan- nel and they-axis is perpendicular to it. The Phase I is filled with a couple stress fluid and Phase II is occupied by a viscous fluid, which is electrically conducting. An external uniform magnetic field of strengthB₀ is ap- plied in they-direction. The walls of the channel are assumed to be at constant temperatures Tw₁ and Tw₂

withT_w1 >T_w2. The fluid properties, i. e. viscosities, densities, and conductivities of both fluids are differ- ent and assumed as constant. The flow is governed by a constant applied pressure gradient. Under these as-

sumptions, the flow equations for both phases are given as following:

for Phase I:

µ₁d²u₁

dy² +ρ₁gβ₁sinφ(T₁−T_w2)−ηd⁴u₁ dy⁴ =∂p

∂x, (1) d²T₁

dy² +µ1

K₁ du₁

dy 2

=0, (2)

for Phase II:

µ2

d²u₂

dy² +ρ2gβ₂sinφ(T₂−T_w2) +σB²₀u₂=∂p

∂x, (3) d²T₂

dy² +µ₂ K₂

du₂ dy

2

+σB²₀u2

K₂ =0, (4)

whereu_iare the velocities,T_iare the temperatures,µi

are the viscosities,ρiare the densities,K_iare the ther- mal conductivities,βiare the thermal expansion coeffi- cients of the fluids in the two phases, wherei=1 is for Phase I andi=2 is for Phase II,σis the electrical con- ductivity of the fluid in Phase II, gis the acceleration due of gravity, andηis the couple stress constant. The no-slip boundary condition is imposed on the walls of the channel. Furthermore, it is assumed that the cou- ple stresses vanish at the channel walls. Both the walls are kept at different temperatures. Moreover, the con- tinuity of fluid velocity, temperature, and shear stress is assured at the interface region. Hence the appropri- ate boundary and interface conditions for the velocity profiles are

u1(h₁) =0, u⁰⁰₁(h₁) =0, u⁰⁰₁(0) =0, u2(−h₂) =0, u1(0) =u2(0), µ1

du₁ dy =µ2

du₂

dy at y=0, (5) and for the temperature, they are

T₁(h₁) =T_w1, T₂(−h₂) =T_w2, T₁(0) =T₂(0), K₁dT₁

dy =K₂dT₂

dy at y=0. (6)

We are interested in the normalized form of the above equations. For this purpose, the following dimension- less variables and parameters have been introduced:

u^∗₁=u₁/u¯₁, u^∗₂=u₂/u¯₁, y^∗₁=y₁/h₁, y^∗₂=y₂/h₂, θ= (T−T_w2)/(T_w1−T_w2), m=µ1/µ₂, k=K₁/K₂, h=h2/h₁, n=ρ2/ρ₁, b=β2/β₁,

Gr=gβ₁h³₁(T_w1−T_w2)/ν₁², M=B₀h₂p σ/µ₂, Pr=µ1C_p/K₁, α²=µ1h₁/η, P= (h²₁/µ₁u¯₁)(∂p/∂x), Re=u¯1h1/ν₁, Ec=u¯²₁/C_p(T_w1−Tw₂),

(7)

where Gr is the Grashof number, Ec is the Eckert num- ber, Pr is the Prandtl number, Re is the Reynolds num- ber,Mis the Hartmann number or the magnetic param- eter,Pis the non-dimensional pressure gradient,α is the couple stress parameter, and ¯u1is the average ve- locity.

With the help of (7), the dimensionless form of the momentum and energy equations become

for Phase I:

d⁴u₁ dy⁴ −α²

d²u₁

dy² +Gr sinφ Re θ1−P

=0, (8) d²θ1

dy² +PrEc du₁

dy 2

=0, (9)

for Phase II:

d²u₂ dy² +Gr

Rebmnh²sinφ θ2−M²u2=mh²P, (10) d²θ₂

dy² +PrEc k

m du₂

dy 2

+ M²

m

PrEcku²₂=0. (11) The asterisks have been dropped with the understand- ing that all the quantities are now dimensionless.

The boundary conditions in new variables take the form

u₁(1) =0, u⁰⁰₁(1) =0, u⁰⁰₁(0) =0, u₂(−1) =0, u1(0) =u2(0), du₁

dy = 1

mh du₂

dy at y=0, (12) θ1(1) =1, θ2(−1) =0, θ1(0) =θ2(0),

dθ₁ dy =

1 kh

dθ₂

dy at y=0. (13)

3. Solution of the Problem

The governing equations of momentum (8) and (10), along with the energy equations (9) and (11) are solved subject to the boundary and interface conditions (12) and (13) for the velocity and temperature distributions.

The equations are coupled and nonlinear because of the

inclusion of the dissipation terms in the energy equa- tion. Following Malashetty and Umavathi [12], we can approximate the solution of (8) – (11) subject to the boundary conditions (12) and (13) valid for small val- ues ofε1(=PrEc), because in most of the practical problems, the Eckert number is very small and is of order 10⁻⁵. The solutions are assumed in the form

(u₁,θ₁) =

∞

∑

i=0

(u_1i,θ_1i)εⁱ, (14) (u₂,θ2) =

∞ i=0

∑

(u_2i,θ2i)εⁱ, (15) whereu_1i,θ1i,u_2i,θ2iare the perturbations inuandθ, respectively.

Using (14) – (15) in (8) – (13), after comparing the like powers ofεand neglecting the terms of O(ε²), we obtain the following system of equations:

for Phase I:

zeroth-order equations:

d⁴u10

dy⁴ −α² d²u10

dy² +Gr sinφ Re θ10−P

=0, (16) d²θ10

dy² =0 ; (17)

first-order equations:

d⁴u₁₁ dy⁴ −α²

d²u₁₁

dy² +Gr sinφ Re θ11

=0, (18) d²θ11

dy² + du₁₀

dy 2

=0 ; (19)

for Phase II:

zeroth-order equations:

d²u₂₀ dy² +Gr

Resinφbmnh²θ20−M²u₂₀=mh²P, (20) d²θ20

dy² =0 ; (21)

first-order equations:

d²u21

dy² +Gr

Resinφbmnh²θ₂₁−M²u₂=0, (22) d²θ21

dy² + k

m

du₂₀ dy

2

+M² k

m

u²₂₀=0. (23) The corresponding boundary conditions (12) and (13) reduce to

u₁₀(1) =0, u⁰⁰₁₀(1) =0, u⁰⁰₁₀(0) =0, u₂₀(−1) =0, u₁₀(0) =u₂₀(0),

du10

dy = 1

mh du20

dy at y=0 ;

(24)

θ₁₀(1) =1, θ₂₀(−1) =0, θ₁₀(0) =θ₂₀(0), dθ₁₀

dy = 1

kh dθ₂₀

dy at y=0 ; (25)

u₁₁(1) =0, u⁰⁰₁₁(1) =0, u⁰⁰₁₁(0) =0, u₁₁(0) =u₂₁(0), u₂₁(−1) =0,

du11

dy = 1

mh du21

dy at y=0 ;

(26)

θ₁₁(1) =0, θ₂₁(−1) =0, θ₁₁(0) =θ₂₁(0), dθ₁₁

dy = 1

kh dθ₂₁

dy at y=0. (27)

The exact solutions of (17), (21) and (16), (20) using boundary conditions (24) and (25) are

θ10= y+kh

(1+kh), (28)

θ20=kh(1+y)

(1+kh) , (29)

u₁₀=a₁y³+a₂y²+c₄y+c₃ + 1

α²(c₁coshαy+c₂sinhαy), (30) u₂₀=d₁coshMy+d₂sinhMy+f₁+f₂y, (31) where

G=Gr

Resinφ, a1=− G

6(1+kh)

, a₂=1

2

P− khG (1+kh)

, f₂= kGbmnh³ M²(1+kh), f₁=−mh²p

M² +f₂, c₁=−2a₂, c₂=2a₂coshα−6a₁−2a₂

sinhα

,

d₁= mh

α² mh(sinhM)+M(coshM)

f₂α²(M−sinhM)

−f₁²α²(M+mhsinhM) +c1mhsinhM(1−coshα) +c₂mhsinhM(α−sinhM)−(a₁+a₂)(α²mhsinhM)

, d₂=d₁coshM+f₁−f₂

sinhM , c₃=−c₂

α +(d₂M+f₂) mh , c2=−c1

α²+d1+f1.

Similarly, solution of (19), (23) and (18), (22) using boundary conditions (24) and (25) are

θ₁₁=g₁y⁶+g₂y⁵+g₃y⁴+g₄y³+g₅y²+g₆coshαy +g₇sinhαy+g₈ycoshαy+g₉ysinhαy +g₁₀y²coshαy+g₁₁y²sinhαy

+g₁₂cosh 2αy+g₁₃sinh 2αy+e₁y+e₂, (32)

θ21=j₁cosh 2My+j₂sinh 2My+j₃ycoshMy +j4ysinhMy+j5coshMy+j6sinhMy +j₇y⁴+j₈y³+j₉y²+e₃y+e₄,

(33) u₁₁=r₁y⁸+r₂y⁷+r₃y⁶+r₄y⁵+r₅y⁴+r₆y³+r₇y²

+r₈coshαy+r₉sinhαy+r₁₀ycoshαy+r₁₁ysinhαy +r₁₂y²coshαy+r₁₃y²sinhαy+r₁₄y³coshαy +r₁₅y³sinhαy+r₁₆cosh 2αy+r₁₇sinh 2αy + 1

α²(e₇coshαy+e₈sinhαy)e₆y+e₅,

(34)

u₂₁=e₉coshMy+e₁₀sinhMy+p₁cosh 2My +p2sinh 2My+p3y²coshMy+p4y²sinhMy +p₅ycoshMy+p₆ysinhMy+p7coshMy +p₈sinhMy+p₉y⁴+p₁₀y³+p₁₁y²+p₁₂y+p₁₃,

(35)

where g₁=−3a²₁

10 , g₂=−3a₁a₂

5 , g₃=−2a²₂+3a₁c₄

6 ,

g₄=−2a₂c₄

3 , g₅=−c²₄ 2 − c²₂

4α²+ c²₁ 4α², g₆=8a₂c₁

α⁴ −36a₁c₂

α⁵ −2c₂c₄ α³ , g7=8a₂c2

α⁴ −36a₁c1

α⁵ −2c₁c4

α³ , g8=24a₁c1

α⁴ −4a₂c2

α³ , g₉=24a₁c₂

α⁴ −4a₂c₁

α³ , g₁₀=−6a₁c₂

α³ , g₁₁=−6a₁c₁ α³ , g₁₂=−

c²₁+c²₂ 8α⁴

, g₁₃=−c₁c₂ 4α⁴, t1=−kM²

m (d₁²+d²₂), t2=−2kd₁d₂M²

m ,

t₃=−2kd₁f₂M²

m , t₄=−2kd₂f₂M²

m ,

t₅=−2kM

m (d₁f₁M+d₂f₂), t₆=−2kM²

m

d1f2

M +d2f1

, t7=−kM²f₂² m , t₈=−2kM²f₂f₁

m , t₉=−k

m(M²f₁²+f₂²),

j₁= t₁

4M², j₂= t₂

4M², j₃= t₃

M², j₄= t₄ M², j₅= t₅

M²−2t₄

M³, j₆= t₆ M²−2t₃

M³, j₇= t₇

12, j₈=t₈ 6, j₉=t₉

2, e₁= 1

(1+kh)

j₁cosh 2M−j₂sinh 2M

+ (j₅−j3)coshM+ (j4−j₆)sinhM−j1+2M j₂+j3

−j₅+M j₆+j7−j8+g₁₂+g₆−kh(αg7+g₈ +2αg₁₃)

− g₁+g₂+g₃+g₄+g₅+ (g₆+g₈ +g₁₀)coshα+ (g₇+g₉+g₁₁)sinhα+g₁₂cosh 2α +g₁₃sinh 2α

,

e₂=− g₁+g₂+g₃+g₄+g₅+ (g₆+g₈+g₁₀)coshα +(g₇+g₉+g₁₁)sinhα+g₁₂cosh 2α+g₁₃sinh 2α+e₁

, e₃=kh(αg₇+g₈+2αg₁₃+e₁)−2M j₂−j₃−M j₆, e₄=e₂+g₆+g₁₂−j₁−j₅,

r₁=−Gg₁ 56 , r₂=−Gg₂

42 , r₃=−G g₁

α²+g₃ 30

, r₄=−G

g₂ α²+g₄

20

, r₅=−G 30g₁

α⁴ +g₃ α²+g₅

12

, r₆=−G

20g₂ α⁴ +g4

α²+e1

6

, r₇=−G

360g₁ α⁶ +12g₃

α⁴ + g₅ α²+e₂

2

, r₈=G

−5g₆ 4α² +17g₉

8α³ −147g₁₀ 24α⁴

, r9=G

−5g₇ 4α² +17g₈

8α³ −147g₁₁ 24α⁴

, r₁₀=G

g₇ 2α−5g₈

4α²−51g₁₁ 12α3

, r₁₁=G

g₆ 2α−5g₉

4α²+51g₁₀ 12α³

, r₁₂=G g₉

4α−5g₁₀ 4α²

, r₁₃=G

g9

4α−5g11

4α²

, r₁₄=Gg11

6α , r₁₅=Gg10

6α , r₁₆=Gg₁₂

12α², r17= Gg₁₃ 12α²,

˜

p=bmnGh², p1=−p j˜ 1

3M², p2=−p j˜ 2

3M², p3=−p j˜ 4

4M, p₄=−p j˜ ₃

4M, p₅=−p˜ j₆

2M− j₃ 4M²

,

p₆=−p˜ j5

2M− j4

4M²

, p7=−p˜ j4

8M³− j5

4M²

, p₈=−p˜

j₃ 8M³− j₆

4M²

, p₉=p j˜ ₇

M², p₁₀= p j˜ ₈ M², p₁₁= p˜

M²

j₉+12j₇ M²

, p₁₂= p˜ M²

6j₈ M²+e₃

, p₁₃= p˜

M² 24j7

M⁴ +2j9

M²+e₄

, e₅=−

r1+r2+r3+r4+r₅+r₆+r₇+

r8+r₁₀+r₁₂ +r₁₄+e₇

a²

coshα+

r₉+r₁₁+r₁₃+r₁₅+e₈ α²

·sinhα+r₁₆cosh 2α+r₁₇sinh 2α+e₆ ,

e₆= −M

McoshM+mhsinhM

−

p₁cosh 2M−p₂sinh 2M +p₉−p₁₀+p₁₁−p₁₂+p₁₃(p₃−p₅+p₇)coshM + (p₆−p₄−p₈)sinhM

+coshM

r₁+r₂+r₃+r₄ +r₅+r₆+r₇

r₈+r₁₀+r₁₂+r₁₄+ e₇ α²

coshα +

r9+r11+r₁₃+r15+ e₈ α²

sinhα+r₁₆cosh 2α +r₁₇sinh 2α

+sinhM M

mhe8

α

+r₁₀+2αr₁₇+αr₉

−p5−2M p₂−M p8−p12

,

e₇=−(2r₇+α²r₈+2αr₁₁+2r₁₂+4α²r₁₆), e₈= −1

sinhα

(56r₁+42r₂+30r₃+20r₄+12r₅+6r₆ +2r₇) + α²(r₈+r₁₀+r₁₂+r₁₄) +2αr₁₁+2r₁₂ +4αr₁₃+6r₁₄+6αr₁₅+e₇

coshα+ α²(r₉+r₁₁ +r₁₃+r₁₅) +2αr₁₀+4αr₁₂+2r₁₃+6αr₁₄+6r₁₅

·sinhα+4α²r₁₆cosh 2α+4α²r17sinh 2α

, e₉=e₅+e₇

α²−p₁−p₇−p₁₃+r₈+r₁₆, e₁₀= 1

sinhM

p₉−p₁₀+p₁₁−p₁₂+p₁₃+p₁cosh 2M

−p₂sinh 2M+ (p₃−p₅+p₇+e₉)coshM + (p₆−p₄−p₈)sinhM

. 4. Results and Discussion

We computed the fluid velocity and the temperature distribution by solving the coupled nonlinear equa-

Fig. 2. Variation of velocity profiles for several values ofα with Gr=5,M=2,m=0.5,k=1,h=1,n=1.5,φ=π/6, Re=5,ε=0.05,P=5, andb=1; filled circles: Newtonian solution.

tions (8) – (11) with boundary conditions (12) and (13) analytically using regular perturbation method for small values ofεupto the order one. The velocity field and temperature distribution are plotted in order to ob- serve the effects of various involving parameters, for example, couple stress parameterα, magnetic parame- terM, Grashof number Gr, and the angle of inclination of channelφin Figures2–8.

Figure2exhibits the fluid velocityu(y)for several values of the couple stress parameter α by keeping Gr=5,M=2,m=0.5,k=1,h=1,n=1.5,φ=π/6, Re=5, ε =0.05,P=5, and b=1 fixed. It is ev- ident from this figure that the fluid velocityu(y)in- creases with an increase in couple stress parameterα in both phases (regions). However, it can be seen that in Phase I the fluid velocity is higher due to consider-

Fig. 3. Variation of velocity profiles for several values ofM with Gr=5,α=2.5,m=0.5,k=1,h=1,n=1.5,φ=π/6, Re=5,ε=0.05,P=5, andb=1.

Fig. 4. Variation of velocity profiles for several values of Gr withα=3,M=2,m=0.5,k=1,h=1,n=1.5,φ=π/6, Re=5,ε=0.05,P=5, andb=1.

ing a couple stress fluid in this region, and the velocity of the fluid near the wall of the upper phase has signifi- cant effects as compare to near the wall of lower phase.

Furthermore, as we increase the values of couple stress parameterα upto∞(α →∞), the case of Newtonian fluid is recovered, and our results are well agreed with the results of Malashetty and Umavathi [12]. Figure3 shows the change in the fluid velocityu(y)for several values of the magnetic parameterMwith other param- eters fixed. It is noted from this figure that the incre- ment of the magnetic parameter M decreases the ve- locity of the fluid throughout the channel. This is be- cause of that the application of a magnetic field normal to the flow direction has a tendency to slow down the movement of the fluids in the channel because it give rise to a resistance force called the Lorentz force which acts opposite to the flow direction. Figure4illustrates

Fig. 5. Variation of velocity profiles for several values ofφ with Gr=5,α=3,M=2,m=0.5,k=1,h=1,n=1.5, Re=5,ε=0.05,P=5, andb=1.

Fig. 6. Variation of temperature profiles for several values of M with Gr=5,α =2.5,m=0.5,k=1,h=1, n=1.5, φ=π/6, Re=5,ε=0.05,P=5, andb=1.

the influence of the Grashof number Gr on the fluid ve- locityu(y)by keeping all other parameters fixed. It is observed from this figure that increasing the Grashof number results in a decrease in the fluid velocityu(y).

Figure5gives the change in the fluid velocityu(y)for several values of an inclined angleφ. From this figure, it is evident that an increase in inclined angle results in the decrease in the velocity of the fluid in both phases.

However, the change or increment in Phase I contain- ing the couple stress fluid is large as compared with the Newtonian fluid in Phase II. In Figures2–5, it is noted that in the upper phase the couple stress fluid ve- locity is higher than the velocity of the viscous fluid (lower phase). This is due to the fact that the viscous fluid is electrically conducting and the MHD effects suppress the bulk motion and as a result the velocity in the lower phase has been reduced. Figure6depicts the

Fig. 7. Variation of temperature profiles for several values of Gr withα=3,M=2,m=0.5,k=1,h=1,n=1.5,φ= π/6, Re=5,ε=0.05,P=5, andb=1.

Fig. 8. Variation of temperature profiles for several values of φwith Gr=5,α=3,M=2,m=0.5,k=1,h=1,n=1.5, Re=5,ε=0.05,P=5,b=1.

effects of the magnetic parameterMon the temperature profileθ(y)by keeping all others parameters constant.

This figure indicates that the effect of magnetic param- eter is to decrease the temperature of the fluid through- out the channel. The effect of the Grashof number on the temperature profile θ(y)is presented in Figure7.

It is observed from the figure that the temperature pro- file decreases with an increase in the Grashof number.

The behaviour of the temperature profileθ(y)with the changes in the inclined angleφ is shown in Figure8.

The effect of the inclined angle φ is to decrease the temperature of the fluid in both regions of the channel.

It is further noted in all temperature profiles that the temperature of the viscous fluid is higher than the tem- perature of the couple stress fluid. This is due to the presence of Lorentz force and this force has the ten- dency to slow down the fluid motion and the resistance offered to the flow. Therefore, it is responsible for the

increase in temperature profiles. In the case of temper- ature distributions, only the first-order temperature is shown in the figures, since the zeroth-order temperatur profiles (28) and (29) are linear.

5. Conclusions

The problem of the MHD flow of viscous and couple stress fluids in an inclined channel is studied analyti- cally. The governing equations are obtained by using the dimensionless parameters. These nonlinear dimen- sionless equations are solved analytically using pertur- bation series method by keepingεas a perturbation pa- rameter. From the study, we draw the following conclu- sions:

• The fluid velocityu(y)in both phases increases with an increase in the couple stress parameterα.

• The fluid velocity u(y)decreases with an increase in the magnetic parameterM, Grashof number Gr, and the inclination of the channelφ throughout the channel.

• The temperature θ(y) profile also decreases with an increase in the magnetic parameterM, Grashof number Gr, and the inclination of the channelφ in the both phases.

Acknowledgements

We are thankful to the anonymous reviewer for his/her useful comments to improve the version of the paper. The financial support from Higher Education Commission (HEC) of Pakistan is also gratefully acknowledged.

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