Journal of Engineering Physics and Thermophysics, Vol. 84, No. 5, September, 2011

MHD FLOW AND HEAT TRANSFER IN A CHANNEL BOUNDED BY A SHRINKING SHEET AND A PLATE WITH A POROUS SUBSTRATE D. S. Chauhan and R. Agrawal

UDC 536.25+537.84

This research is concerned with heat transfer and an MHD flow of a viscous incompressible electrically conducting fluid in a channel bounded by a shrinking sheet and an impermeable plate. A fluid-saturated porous substrate of a very low permeability is attached to the impermeable plate. The flow in the channel is induced by the upper shrinking sheet, where a constant suction is imposed. By introducing the similarity transformations, the governing partial differential equations for the flow and heat transfer are transformed to ordinary differential equations which are solved analytically by using the perturbation series method for a small shrinking parameter. Expressions for the velocity distribution, temperature field, skin friction, and heat transfer rate are obtained, and numerical computations are carried out for various parameters. The results are displayed graphically and discussed. Keywords: MHD flow, heat transfer, shrinking sheet, porous substrate, permeability. Introduction. The study of a viscous fluid flow due to expanding or contracting surfaces is of importance because of its application in many industrial processes, such as glass wire production, extrusion of polymer sheets or filaments from a die, and hot rolling [1–4]. In particular, in polymer units, a polymer sheet is stretched, while it is extruded from a die, and it is pulled through a viscous fluid with a cooling effect to get the final product. The sheet stretching results in the motion of the adjacent fluid, or alternatively, the viscous fluid may be subjected to an independent forced convection flow parallel to the sheet. The problems in the presence of a magnetic field have also become of importance in many metallurgical processes which involve the cooling of continuous strips by drawing them through an electrically conducting fluid, because the effect of an applied magnetic field is to control the cooling rate. Sakiadis [5, 6] introduced the concept of such continuous surfaces and initiated the study of a flow over these surfaces. Crane [7] studied the flow past a stretching sheet and found a closed-form solution, when the velocity on the boundary is proportional to the distance from a fixed point. The flow inside a stretching channel was investigated by Brady and Acrivos [8]. The MHD flow and heat transfer aspects in a parallel plate channel with the lower plate being a stretching sheet were considered by Borkakoti and Bharali [9]. The stability of a viscous fluid flow past a stretching sheet was shown by Bhattacharya and Gupta [10], and the uniqueness of such a flow, by McLeod and Rajagopal [11] and Troy et al. [12]. The heat transfer effects on a viscoelastic fluid flow through a porous medium over a stretching sheet were studied by Abel and Veena [13], and such a problem over a nonisothermal stretching one, by Prasad, Abel, and Khan [14]. A two-dimensional non-Newtonian fluid flow in a channel bounded by a stretching sheet and a porous layer was considered by Chauhan and Jhakhar [15]. An MHD boundary layer flow of a viscoelastic fluid through a porous medium over a porous stretching sheet was studied by Khan and Sanjayanand [16]. Heat transfer and fluid flow through a porous medium over a stretching sheet with internal heat generation or absorption in the presence of suction or injection were investigated by Cortell [17]. Cortell [18] considered an MHD flow and mass transfer of the second grade fluid through a porous medium in the presence of chemical reaction. Thus, extensive researches have been conducted, including those cited above, on the flow and heat transfer over a stretching surface, where the velocity of the surface was assumed to be linearly proportional to the distance from a fixed point or otherwise dependent on it. However, much less emphasis has been given to the problems of a flow due to a shrinking sheet, where the velocity on the surface was directed towards a fixed point. It is the surface that is decreased to a certain area due to the heat supplied externally or to an imposed suction on it. One of the comDepartment of Mathematics, University of Rajasthan, Jaipur-302055, India; email: [email protected]. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 84, No. 5, pp. 961–971, September–October, 2011. Original article submitted January 25, 2010; revision submitted March 28, 2011. 1034

1062-0125/11/8405-10342011 Springer Science+Business Media, Inc.

Fig. 1. Schematic diagram of the problem. mon applications of such situations is the shrinking film which is useful in packaging the industrial products. The film shrinking can be achieved by the use of a shrink tunnel which is a chamber with recirculating hot air. The viscous fluid flow induced by a shrinking sheet is quite different from the stretching sheet-forwarded flow and it can hardly exist unless an adequate mass suction on the sheet is imposed. Miklavcic and Wang [19] considered a viscous fluid flow due to a shrinking sheet and studied the existence and uniqueness of exact solutions of the boundary-value problem. It was concluded that suction was required to maintain a viscous fluid flow over a shrinking sheet. Sajid and Hayat [20] studied an MHD viscous flow due to a shrinking sheet using the homotopy analysis method. An analytical solution for an MHD flow of the second grade fluid due to a shrinking sheet was obtained by Hayat, Abbas, and Sajid [21]. Hayat et al. [22] investigated a three-dimensional rotating flow due to a shrinking sheet with suction. Sajid, Javed, and Hayat [23] discussed an MHD rotating viscous fluid flow over a shrinking sheet. It was shown that the stable and convergent solution of the problem was possible only for magnetohydrodynamic flow. A stagnation fluid flow towards a shrinking sheet was considered by Wang [24]. Muhaimin, Kandasamy, and Khamis [25] investigated the effects of heat and mass transfer on an MHD boundary-layer fluid flow over a shrinking sheet with suction. Rahimpour et al. [26] obtained analytical solutions for a stagnation flow towards a stretching sheet. Fang, Zhang, and Yao [27] investigated viscous fluid flow and mass transfer over an unsteady shrinking sheet. Yao and Chen [28] analytically studied a viscous flow for the case of a flat plate continuously shrinking into a slot in a shrinking fluid with mass transfer employing the homotopy analysis method. Fang and Zhang [29] considered a magnetohydrodynamic flow over a shrinking sheet and obtained a closed-form exact solution. Fang and Zhang [30] obtained an analytical solution for the heat transfer with prescribed wall temperature and heat flux over a shrinking sheet. Noor, Awang Kechil, and Hashim [31] investigated a magnetohydrodynamic flow over a shrinking sheet and obtained a series solution using the Adomian decomposition method with the Padé approximants. The study of the heat transfer and flow adjacent to a shrinking sheet in the presence of a porous medium can be applied to investigate the capillary effects in pores and other hydraulic and mechanical properties of soils, which is essential for the development of environmental and soil mechanics. This can also be applied to examine the shrinking– swelling behavior of soil, filtration and water purification processes, transpiration cooling systems, separation processes in chemical industries, plastic behavior in the extruder for determining die dimensions, thin wall heat shrink tubing manufactured from polymer that exhibits extraordinary tensile strength and is designed for biological processing and to study the interaction of a fluid flow with the geomagnetic field in the geothermal regions and the Earth shrinking phenomena. Noor and Hashim [32] investigated an MHD boundary layer flow and heat transfer over a shrinking sheet embedded in a porous medium with suction. The present research is concerned with heat transfer and an MHD flow of a viscous incompressible electrically conducting fluid in a channel bounded by a shrinking sheet and an impermeable plate. Formulation of the Problem. A viscous incompressible electrically conducting fluid flow in a channel bounded by a horizontal shrinking sheet and an impermeable plate is considered (see Fig. 1). Constant suction is ap1035

plied at the upper shrinking sheet and a fluid-saturated porous substrate of a very low permeability is perfectly attached to the lower impermeable plate of the channel. We model the fluid flow in the porous substrate by Darcy’s equation. Thus, in the absence of any external pressure gradient, the velocity in the porous medium is assumed to be zero. The effect of the porous substrate is to introduce a slip at one boundary of the channel, and we model the interface between a clear fluid and a porous medium using the Saffman slip boundary condition [33]. A constant magnetic field of strength B0 is applied in the direction normal to the composite channel with parallel walls. The induced magnetic field is neglected, which is justified when the magnetic Reynolds number is small. Further no external electric field is applied, and the effect of polarization of the ionized fluid is negligible, so that it is assumed that the electric field is absent. The two walls of the channel are kept at constant but different temperatures. The Cartesian coordinate system x, y, z is taken, with the x and y axes being on the surface of the porous substrate and the z axis normal to it. The governing equations for the MHD flow and the temperature distribution in a clear fluid region (0 ≤ z ≤ h) are as follows: ∂u ∂v ∂w + + =0, ∂x ∂y ∂z

(1)

⎛ ∂2u ∂2u ∂2u ⎞ σB2 0 u +v +w =− + υ ⎜⎜ 2 + 2 + 2 ⎟⎟ − u, ⎟ ⎜ ∂x ∂x ρ ∂y ∂z ρ ∂x ∂y ∂z ⎠ ⎝

(2)

⎛ ∂2v ∂2v ∂2v ⎞ σB2 0 +v +w =− + υ ⎜⎜ 2 + 2 + 2 ⎟⎟ − v, u ⎜ ∂x ρ ∂y ∂z ⎟ ∂y ∂z ρ ∂y ∂x ⎝ ⎠

(3)

∂u

∂u

∂v

∂v

u

∂u

∂w ∂x

+v

1 ∂p

∂v

∂w ∂y

+w

1 ∂p

∂w ∂z

=−

⎛ ∂2w ∂2w ∂2w ⎞ + υ ⎜⎜ 2 + 2 + 2 ⎟⎟ , ⎜ ∂x ρ ∂z ∂y ∂z ⎟ ⎠ ⎝ 1 ∂p

⎛ ∂2t ∂2t ∂2t ⎞ ⎛ ∂t ∂t ∂t ⎞ ρCp ⎜u +v + w ⎟ = k ⎜ 2 + 2 + 2⎟ . ∂y ∂z ⎠ ⎝ ∂x ∂y ∂z ⎠ ⎝ ∂x

(4)

(5)

In the porous substrate (−d ≤ z ≤ 0), the fluid flow is taken to be zero, and here the energy equation is a simple heat conduction equation for the temperature T: 2

∂T 2

∂x

=0.

(6)

The boundary and matching conditions applicable to the present problem are as follows: z = h : u = − ax , v = − a (m − 1) y , w = w0 , t = T1 ; μα μα u , τzy = v, w=0, z = 0 : τzx = ⎯⎯k0 √ √⎯k0 ⎯ _ ∂t ∂T k =k , t=T; ∂z ∂z z = − d : T = T0 .

1036

(7)

Here, a > 0 is the shrinking constant and m is a constant. At m = 1 the sheet shrinks in the x direction only and at m = = 2 the sheet shrinks axisymmetrically. Method of Solution. Flow Field. Let us introduce the following similarity transformations: (8)

u = axf ′(η) , v = a (m − 1) yf ′(η) , w = − ahmf (η) ,

where η = z ⁄ h. We see that such a form of velocity components automatically satisfies the continuity equation (1), and Eq. (4) reduces to ∂w

1 ∂p

2

∂w

.

(9)

dw w p =υ − + const . dz 2 ρ

(10)

w

∂z

=−

ρ ∂z

+υ

2

∂z

Integrating (9), we obtain 2

Using similarity transformations (8) and eliminating the pressure from Eqs. (2) and (3), we obtain the following ordinary differential equation: 2

2

(11)

f ′′′− M f ′− R (f ′ − mff ′′) = A , where A is an integration constant to be determined by the boundary conditions, M =

√⎯ρσυ B0h is the Hartmann num⎯

ah2 is the nondimensional shrinking parameter. υ With the use of (8), boundary conditions (7) reduce to

ber, and R =

η = 1 : f ′(η) = − 1 , f (η) = − λ ; α η = 0 : f ′′= f ′, f = 0 , K

(12)

w0 √⎯k0 ⎯ is the permeability parameter and λ = is the suction parameter. h amh For small values of the parameter R (R << 1), we expand f and A in the series:

where K =

f = f0 + Rf1 + ... ,

(13)

A = A0 + RA1 + ... .

(14)

Substituting (13) and (14) into Eq. (12) and equating the coefficients at equal powers of R on both sides of the latter, we obtain the following equations and boundary conditions: for the zeroth order solution 2 f0′′′− M f0′ = A0 ;

(15)

η = 1 : f0′ = − 1 , f0 = − λ ;

1037

η = 0 : f0′′ =

α ′ f , f =0; K 0 0

(16)

for the first order solution 2

2 f1′′′− M f1′ = f0′ − mf0 f0′′ + A1 ;

(17)

η = 1 : f1′ = 0 , f1 = 0 ;

(18)

α η = 0 : f1′′ = f1′ , f1 = 0 . K Solving Eqs. (15) and (17), subject to boundary conditions (16) and (18), respectively, we obtain f0 = C1 + C2 cosh (Mη) + C3 sinh (Mη) −

f0′ = M (C2 sinh (Mη) + C3 cos (Mη)) −

A0η 2

(19)

,

M A0 M

2

(20)

,

f1 = C4 + C5 cosh (Mη) + C6 sinh (Mη) +F (η) −

f1′ = M (C5 sinh (Mη) + C6 cos (Mη)) + F ′(η) −

A1η M

2

A1 M

2

(21)

,

(22)

,

where F (η) =

3l6 ⎞ l2 ⎛ l3 sinh ( 2Mη) + cosh (2Mη) + − η cosh (Mη) ⎜ 3 3 2 3⎟ 6M 6M 4M ⎠ ⎝ 2M

l1

3l5 ⎞ l5 2 l6 2 l7 ⎛ l4 η sinh (Mη) + η cosh (Mη) + η sinh (Mη) − 2 η , +⎜ 2− 3⎟ 2 2 4M ⎠ 4M 4M M ⎝ 2M C1 =

d2 + λd4

λ , C2 = − C1 , C3 = − − d2 d2d3 − d1d4

d1 C , d2 1

λ d1 sinh M ⎞ ⎤ ⎛ 2⎡ A0 = M ⎢λ − sinh M + ⎜1 − cosh M − ⎟C ⎥, d2 sinh M − M ⎠ 1⎦ ⎝ ⎣ C4 =

d4 (d5 + d6) + d2 (d5 + d7)

d1d4 − d2d3

, C5 = − C4 −

l2 6M

3

, C6 = −

d5 + d6

−

d2

d1 sinh M ⎞ ⎤ ⎛ 2 ⎡ d5 + d6 sinh M − d6 + ⎜1 − cosh M − A1 = M ⎢ ⎟ C4⎥ . d d2 2 ⎣ ⎝ ⎠ ⎦

1038

d1 d2

C4 ,

The expressions for the coefficients dj (j = 1, 2, 3, ..., 10) and li (i = 1, 2, 3, ..., 7) are presented in Appendix. Temperature Field. We define the nondimensional parameters:

θ (η) =

t − T0

,

(23)

_ T − T0 θ (η) = , ΔT

(24)

ΔT

where ΔT = T1 – T2 = cx and c is a nonzero constant. Using Eqs. (23) and (24) and similarity transformations (8), we reduce equations (5), (6), and boundary conditions (7) for the temperature field to θ′′ = R (Pr f ′θ − Pr mfθ′) , _ 2 d θ 2

=0;

(25)

(26)

dη

η=1: θ=1;

_ dθ dθ η=0: θ=0, =φ ; dη dη _ ∗ η=−d : θ=0,

(27)

_ where d∗ = d ⁄ h, φ = k ⁄ k _ To obtain the temperature field, we expand θ and θ for small values of the parameter R into the series

θ = θ0 (η) + Rθ1 (η) + ... ,

(28)

_ _ _ θ = θ0 (η) + Rθ1 (η) + ... .

(29)

Substituting Eqs. (28) and (29) into Eqs. (25), (26), and boundary conditions (27), equating the coefficients at equal powers of R on both sides, and then solving the equations obtained, we find

θ0 = B1η + B2 ,

(30)

_ θ0 = B3η + B4 ,

(31)

C3B1 ⎡ C2B1 θ1 = B5η + B6 + Pr ⎢ η sinh (Mη) + η cosh (Mη) M ⎣ M ⎛ 2C3B1 MC2B2 − mC3B1 ⎞ + + ⎜− ⎟ sinh (Mη) 2 2 M ⎠ ⎝ M ⎛ 2C2B1 MC3B2 − mC2B1 ⎞ + ⎜− + ⎟ cosh (Mη) 2 2 M ⎝ M ⎠ 1039

3 2 ⎞ η ⎤⎥ ⎛ A0B1 mA0B1 ⎞ η ⎛ A0B2 + ⎜− + + − − m C B ⎥, ⎜ 1 1⎟ 2 2 ⎟ 2 M ⎠ 6 ⎝ M ⎝ M ⎠ 2 ⎥⎦ _ θ1 = B7η + B8 ,

(32)

(33)

where B1 =

∗

1 ∗

1 + φd

, B2 = B4 =

∗

B6 =

φd

∗

1 + φd

∗

φd d9 − d8 − φd d10 ∗

1 + φd

, B7 =

, B3 =

∗

φ

, B5 =

∗

1 + φd

φ (d8 − d10 + d9) ∗

1 + φd

d8 − d10 − φd d9 1 + φd

∗

,

∗

, B8 =

φd (d8 − d10 + d9) 1 + φd

∗

.

The physical quantities of interest are the skin-friction coefficient and the heat transfer rate which are given below. The local wall shear stress at the porous interface is given by ⎛ ∂u ⎞ μax τ0 = μ ⎜ ⎟ = f ′′(0) , h ∂z ⎝ ⎠z=0

(34)

′ = f ′′ (0) + Rf ′′ (0) . f ′(0) 0 1

(35)

where

Thus, the coefficient of the skin friction at η = 0 is (Cf)η=0 =

τ0 2

μ (υ ⁄ h )

= Rξf ′′(0) ,

(36)

where ξ = x ⁄ h. Similarly, the skin-friction coefficient at the shrinking sheet is (Cf)η=1 =

τ1 2

μ (υ ⁄ h )

=

− μ (∂u ⁄ ∂z)z=1 2

μ (υ ⁄ h )

= − Rξf ′′(1) ,

(37)

where f ′(′ 1) = f0′′ (1) + Rf1′′ (1) .

(38)

The heat transfer rate at the porous interface and at the shrinking sheet are given by ⎛ ∂θ ⎞ ⎛ ∂θ ⎞ = θ ′(0) = θ0′ (0) + Rθ1′ (0) , ⎜ ⎟ = θ ′(1) = θ0′ (1) + Rθ1′ (1) . ⎜ ∂η ⎟ ∂η ⎝ ⎠η=1 ⎝ ⎠η=0

(39)

Results and Discussion. In the present work, the effect of a porous substrate on the MHD flow and heat transfer in a channel with a shrinking boundary in the presence of suction is investigated. Numerical computations are performed for various values of the parameters: 1 ≤ M ≤ 3, 1 ≤ m ≤ 3, 1 ≤ λ ≤ 2, 0.0001 ≤ K ≤ 0.01, 0 ≤ R ≤ 0.5, 0.71 ≤ Pr ≤ 14, and the results are displayed in the figures presented. For all cases considered, α = 0.1 and φ = 1.67.

1040

Fig. 2. Velocity profiles for M = 1, λ = 1, m = 3, and different values of R and K.

Fig. 3. Velocity profiles for R = 0.2, K = 0.005, and different values of M, m, and λ. Typical velocity profiles are plotted in Figs. 2 and 3. Due to shrinking of the upper channel sheet, the flow velocity near this surface is negative. The fluid rushes from infinity towards the axis near the sheet and, due to continuity, in the remaining channel it flows with a positive velocity towards infinity. When K = 0, there is no slip at the porous medium interface; however, when K is very small and positive, as is seen in Fig. 2, there is a slip at the porous interface. This slip velocity increases with the permeability K. The flow in the region near the interface also increases with K and has a positive velocity. It decreases in the middle part of the channel and changes sign at certain η, which decreases with increase in K. Then, in the region near the upper shrinking sheet, the velocity is negative and its magnitude increases with K. It is found that the flow velocity in the lower part of the channel increases with increase in the nondimensional shrinking parameter R, while its magnitude decreases near the upper sheet. In Fig. 3, it is seen that a resistive type force due to the transverse magnetic field has a tendency to slow down the motion of the fluid in the channel, except for the region adjacent to the shrinking sheet, where the magnitude of the back flow increases with M. However, with increase in the suction parameter λ or the shrinking strength parameter m, the flow velocity increases, except for the region near the shrinking sheet, where its magnitude decreases. Figures 4 and 5 give the nondimensional temperature profiles θ vs. the coordinate η. It is found (see Fig. 4) that the permeability K of the porous substrate attached to the lower channel wall decreases the temperature throughout the channel. As expected, the temperature at a given point in the flow domain decreases with an increase in the Prandtl number Pr. It is also seen from these figures that the temperature throughout the channel decreases with increase in the value of each of the parameters: the magnetic field parameter M, shrinking parameter R, shrinking 1041

Fig. 4. Temperature profiles for M = 1, λ = 1, m = 3, and different values of R, K, and Pr.

Fig. 5 Temperature profiles for R = 0.2, K = 0.005, Pr = 7, and different values of λ, M, and m. The curves without indicated values of M and m relate to M = 1, m = 3. strength parameter m, and the suction parameter λ. Based on the results shown in Fig. 5, it can be concluded that the temperature in the channel for one-direction shrinking (m = 1) is greater than that for the case of axisymmetric shrinking (m = 2). Figure 6a shows that an increase in λ or R leads to an increase in the skin friction at the shrinking sheet. It is also seen that the skin friction for axisymmetric shrinking is greater than in the case of one-direction shrinking. However, when the value of the permeability parameter K or the magnetic field parameter M is increased, the skin friction decreases. This change in the skin friction is of importance from the industrial point of view, since in this case the power expenditure in shrinking the sheet decreases. Figure 6b displays the skin friction at the surface of the lower porous substrate. It is found that the skin-friction coefficient increases with each of the parameters: R, λ, and m, whereas it decreases with increase in the magnetic parameter M or the permeability K. The heat transfer rate at the shrinking sheet θ′(1) is plotted against R for different values of other parameters in Fig. 7a. It is observed that the permeability of the porous substrate increases the heat transfer rate. It is also seen that the shrinking strength parameter increases this rate too. For both one-direction and axisymmetric shrinking, θ′(1) decreases with increase in R; however, for m = 3 θ′(1) increases with R. It is also seen that an increase in each of the parameters Prandtl number Pr, suction parameter λ, and magnetic field parameter M increases the heat transfer rate at the shrinking sheet. Figure 7b gives the heat transfer rate at the porous interface against R. It is seen that θ′(1) decreases with increase in the permeability K. It is also observed that in the case of one-direction shrinking θ′(1) is greater than for the axisymmetric shrinking. For those both kinds of shrinking, θ′(1) increases with R, but it decreases

1042

Fig. 6. Skin-friction coefficient at the shrinking sheet (a) and at the porous interface (b) vs. R for different values of M, λ, K, and m. The curves without indicated values of m and K relate to m = 3, K = 0.0001. for m = 3. It is also seen that an increase in each of the parameters Pr, λ, and M reduces the heat transfer rate at the porous interface. Finally, it is found that the permeability K of the porous substrate attached to the lower wall of the channel and the shrinking strength parameter m play a decisive role in estimating the values of the skin friction and heat transfer rate in the channel with a shrinking wall. These results are of great importance from the point of view of industrial applications. The results of the work are of great interest for the packaging industry, extrusion processes in metal and plastic industries, biological processing, chemical engineering, and geophysics. The support of the Council of Scientific and Industrial Research through a Senior Research Fellowship to one of the authors, Rashmi Agrawal, is gratefully acknowledged.

Appendix 2

2

d1 = 1 − cosh M −

KM KM , d2 = sinh M − M , d3 = + M sinh M , α α

d4 = M (1 − cosh M) , d5 =

l1 3M

2

+

l3 2M

2

−

3l6 4M

3

−

l7 M

2

−

K ⎛ l2 l4 l5 ⎞ + − 2⎟ , ⎜ α ⎝ 2M M M ⎠

l6 ⎞ l3 3l6 l5 ⎞ 3l5 ⎛ l2 ⎛ l4 d6 = ⎜ 3 − + − cosh M − ⎜ 2 − + sinh M 2 3 2⎟ 3 2⎟ 2M 4M 4M ⎠ 4M 4M ⎠ ⎝ 6M ⎝ 2M

1043

Fig. 7. Heat transfer rate at the shrinking sheet (a) and at the porous interface (b) vs. R for different values of M, λ, K, m, and Pr. The curves without indicated values of m and Pr relate to m = 3, Pr = 7.

−

l2

l1 l7 cosh (2M) − sinh (2M) + , 3 3 2 6M 6M M

l3 l4 3l5 l1 l5 l6 ⎞ ⎛ l2 sinh M − d7 = ⎜ 2 − − + − + cosh (2M) 2 3 2⎟ 2 2M 2M 4M 4M 4M ⎠ 3M ⎝ 6M −

l6 ⎞ l4 l5 3l6 l7 ⎛ l3 sinh (2M) + ⎜− − + + − ⎟ cosh M + 2 , 2 2 3 3M 2M 4M 4M 4M ⎠ M ⎝ 2M l2

2

⎛ C3B1 MC2B2 − mC3B1 ⎞ ⎛ 2C2B1 MC3B2 − mC2B1 ⎞ d8 = Pr ⎜− + + ⎟ , d9 = Pr ⎜− ⎟, 2 2 M M ⎠ ⎠ ⎝ M ⎝ M ⎡1 d10 = Pr ⎢ ⎣6

⎞ ⎛ C2B1 2C3B1 MC2B2 − mC3B1 ⎞ ⎛ A0B1 mA0B1 ⎞ 1 ⎛ A0B2 + + ⎜− − mC1B1⎟ + ⎜ − + ⎟ sinh M ⎜− 2 2 ⎟ 2 2 2 M ⎠ 2⎝ M M M ⎠ ⎠ ⎝ M ⎝ M

⎤ ⎛ C3B1 2C2B1 MC3B2 − mC2B1 ⎞ − + +⎜ ⎟ cosh M⎥ ; 2 2 M M ⎠ ⎝ M ⎦ l1 = 1044

2

2 2 2 M (1 − m) (C2 + C3 ) , l2 = M (1 − m) C2C3 , 2

⎛ 2A0C3 ⎛ 2A0C2 ⎞ ⎞ 2 2 l3 = − ⎜ + mM C1C2⎟ , l4 = − ⎜ + mM C1C3⎟ , ⎝ M ⎝ M ⎠ ⎠

l5 = mA0C2 , l6 = mA0C3 , l7 =

A0

M

M

2

− 4

2 2

2

(1 + m) (C2 − C3 ) .

NOTATION –1

a, shrinking constant, s ; B0, applied magnetic field, A/m; c, nonzero constant; Cf, skin-friction coefficient; * Cp, specific heat at constant pressure, J/(kg⋅K); d, thickness of the porous substrate, m; d , nondimensional thickness of the porous substrate; h, distance between the porous interface and the upper shrinking sheet, _ m; K, nondimensional permeability of the porous layer; k, thermal conductivity of the clear fluid medium, W/(m⋅K); k, effective thermal con2 ductivity of the porous medium, W/(m⋅K); k0, permeability of the porous medium, m ; M, Hartmann number; m, shrinking strength parameter; p, pressure, Pa; Pr, Prandtl number; R, shrinking parameter; T, temperature in the porous region, K; t, temperature in the clear fluid region, K; T0, constant temperature of the lower impermeable channel plate, K; T1, constant temperature of the upper shrinking sheet, K; u, v, and w, velocity components in the clear fluid in the x, y, and z directions, respectively, m/s; w0, constant suction velocity, m/s; x, y, and z, coordinates, m; α, nondimensional constant; η, nondimensional coordinate in the z direction; θ, nondimensional temperature; λ, nondimensional suction parameter; μ, viscosity of the clear fluid, kg/(s⋅m); ξ, nondimensional coordinate in the x direction; ρ, fluid 3 2 2 density, kg/m ; σ, electrical conductivity, 1/(Ω⋅m); τ, shear stress, kg/(s ⋅m); υ, kinematic viscosity, m /s; φ, nondimensional conductivity ratio.

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