MHD Flow and Heat Transfer in a Channel Bounded by a Shrinking Sheet and a Porous Medium Bed: Homotopy Analysis Method

Abstract

MHD flow of viscous conducting fluid is considered between a shrinking sheet and a porous medium bed. Suction is applied at the upper shrinking sheet and its surface temperature is always maintained higher than the temperature of the lower porous bed surface. Similarity transformations and HAM are used to solve the governing equations for velocity and temperature fields. The effects of various pertinent parameters on the results are discussed graphically.

1. Introduction

The study of boundary layer viscous flow and heat transfer due to stretching/shrinking surfaces is very significant because of its several applications in engineering and industrial processes, such as extrusion of polymer sheets from a die, drawing of plastic films, polyester thin wall heat shrink tubing, shrink film, wire drawing, glass fiber, and paper production. It is very important to control drag and the rate of heat transfer from the sheet for a better quality of the final product. Crane [1] first examined such flow over a linearly stretching sheet and obtained an exact similarity solution for the steady boundary layer flow of a viscous fluid in a closed form. His work was extended under various situations and physical conditions, by several researchers, such as Rajagopal et al. [2], Dandapat and Gupta [3], Chiam [4], Cortell [5], Liu [6], Ariel et al. [7, 8], Abbas et al. [9], Abd El-Aziz [10], Chauhan and Rastogi [11], Chauhan and Agrawal [12], and Chauhan and Olkha [13].

Recently the study of boundary layer flow and heat transfer adjacent to a shrinking sheet is given attention because of its several industrial applications, particularly in packaging industry. Flow due to a shrinking sheet is relatively a new phenomenon, where the velocity on the sheet is directed towards a fixed point. Steady flow in such a case does not occur normally, since the vorticity generated due to the shrinking of the sheet is not confined in the boundary layer unless a proper suction is applied at the sheet. Wang [14] reported the flow development adjacent to a shrinking sheet. Miklavčič and Wang [15] studied the existence and uniqueness of steady flow due to shrinking sheet and reported that for specific suction values dual solutions exist. Wang [16] studied stagnation-point flow over a shrinking sheet. Nadeem and Awais [17] investigated fluid flow through porous medium due to unsteady shrinking sheet with variable viscosity. Fang et al. [18] obtained a new solution for the Blasius equation of a shrinking sheet problem. Muhaimin et al. [19] investigated effects of heat and mass transfer on MHD flow over a shrinking sheet with suction. Rahimpour et al. [20] obtained the analytical solution of axisymmetric stagnation flow towards a shrinking sheet. Using HAM, Hayat et al. [21] investigated MHD rotating flow over a shrinking surface. Sajid and Hayat [22] examined MHD viscous flow due to a shrinking sheet using homotopy analysis method. The boundary layer flow adjacent to a shrinking sheet with power-law surface velocity is investigated by Fang [23] with mass transfer. A closed form analytical solution was presented by Fang and Zhang [24] for MHD flow over a shrinking sheet. Fang et al. [25] investigated the unsteady flow over a shrinking sheet with mass suction. Yao and Chen [26] presented the new analytical solution of Blasius equation for a shrinking sheet problem. Mohd and Hashim [27] studied MHD flow and heat transfer over a shrinking sheet embedded in a porous medium with suction. Cortell [28] examined MHD flow due to shrinking sheet for axisymmetric and two-dimensional cases with suction. Noor et al. [29] obtained a series solution for MHD flow due to shrinking sheet by ADM (Adomian decomposition method). Hayat et al. [30] developed an analytical solution for a rotating flow due to a shrinking sheet. Fang and Zhang [31] presented an exact analytical solution for thermal boundary layer over a shrinking sheet with mass transfer effects. Fang et al. [32] investigated analytically flow adjacent to a shrinking-sheet with a second-order slip flow model. Ishak et al. [33] investigated stagnation-point flow of a micropolar fluid over a shrinking sheet. The channel flows with stretching/shrinking walls are also very important problems in fluid dynamics. Brady and Acrivos [34], Borkakoti and Bharali [35], Chauhan and Jakhar [36], Hayat et al. [37], and Chauhan and Agrawal [38] investigated flow and heat transfer problem in channels with one wall as stretching or shrinking sheet.

The purpose of the present investigation is to discuss the analytically solutions of electrically conducting viscous fluid flow and heat transfer in a channel bounded by a porous shrinking sheet and a porous medium bed of very small permeability, in the presence of a magnetic field. The equations governing velocity and temperature fields are simplified by a suitable similarity transformation, and then the reduced nonliner boundary value problem is solved analytically using homotopy analysis method. The physical effects of pertinent parameters on the flow and heat transfer are depicted graphically and discussed. The convergence of the solutions has also discussed by the so-called h-curves.

2. Formulation of the Problem

An electrically conducting, incompressible, viscous fluid flow in a horizontal channel bounded by a shrinking sheet and a porous medium bed is considered. A constant suction 𝑤₀ is applied at the upper shrinking sheet of the channel. The lower porous bed of the channel is of very small permeability 𝑘₀. We model flow in porous bed by Darcy’s law. So, the flow in the porous bed is assumed to be zero in the absence of any external pressure gradient. A slip boundary condition suggested by Saffman [39] is applied at the fluid-porous medium interface. The permeability of the porous bed affects the flow in the channel through this boundary condition. Thus the effect of the porous bed is to introduce slip at the lower boundary of the horizontal channel. A constant magnetic field of strength 𝐵₀ is applied to the normal direction of the horizontal channel. Magnetic Reynolds number is taken to be small; therefore, induced magnetic field is neglected. No external electric field is applied and the polarization effect of the ionized fluid is negligible. So the electric field 𝐸 is assumed to be zero. A Cartesian coordinate system (𝑥, 𝑦, 𝑧) is assumed with origin at the porous bed surface. The axis of 𝑥 and axis of 𝑦 are taken on the surface of the porous bed and the axis of 𝑧 is normal to it (see schematic diagram, Figure 1). (𝑢,𝑣,𝑤) are considered velocity components in 𝑥,𝑦, and 𝑧 directions, respectively, and 𝑇 is the temperature of the fluid in the channel. The width of the channel is assumed ℎ, so that walls of the channel are at 𝑧=0 and 𝑧=ℎ.

Figure 1 

Schematic diagram.

The governing equations for the present problem of MHD flow and heat transfer are given by 𝜕𝑢+𝜕𝑥𝜕𝑣+𝜕𝑦𝜕𝑤𝑢𝜕𝑧=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦+𝑤𝜕𝑢1𝜕𝑧=−𝜌𝜕𝑃𝜕𝜕𝑥+𝜐²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²+𝜕²𝑢𝜕𝑧²−𝜎𝐵²₀𝜌𝑢𝑢,𝜕𝑣𝜕𝑥+𝑣𝜕𝑣𝜕𝑦+𝑤𝜕𝑣1𝜕𝑧=−𝜌𝜕𝑃𝜕𝜕𝑦+𝜐²𝑣𝜕𝑥²+𝜕²𝑣𝜕𝑦²+𝜕²𝑣𝜕𝑧²−𝜎𝐵²₀𝜌𝑢𝑣,𝜕𝑤𝜕𝑥+𝑣𝜕𝑤𝜕𝑦+𝑤𝜕𝑤1𝜕𝑧=−𝜌𝜕𝑃𝜕𝜕𝑧+𝜐²𝑤𝜕𝑥²+𝜕²𝑤𝜕𝑦²+𝜕²𝑤𝜕𝑧²,𝜌𝐶_𝑝𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦+𝑤𝜕𝑇𝜕𝜕𝑧=𝑘²𝑇𝜕𝑥²+𝜕²𝑇𝜕𝑦²+𝜕²𝑇𝜕𝑧². (1)(2)(3)(4)(5)

The corresponding boundary conditions are given by at𝑧=ℎ,𝑢=−𝑎𝑥,𝑣=−𝑎(𝑐−1)𝑦,𝑤=𝑤₀,𝑇=𝑇₁,at𝑧=0,𝜏_𝑧𝑥=𝜇𝛼𝑘₀𝑢,𝜏_𝑧𝑦=𝜇𝛼𝑘₀𝑣,𝑤=0,𝑇=𝑇₀, (6) where 𝑃 is the pressure; 𝜌 is the density of the fluid; 𝜐 is the kinematic viscosity; 𝜎 is the electrical conductivity; 𝐶_𝑝 is the specific heat at constant pressure; 𝑘 is the thermal conductivity in the clear fluid medium; 𝑇₀ is the porous medium wall temperature; 𝑇₁ is the temperature of upper stretching wall; 𝑘₀ is the permeability of the porous medium; 𝜇 is the viscosity of the clear fluid; 𝛼 is a nondimensional constant that depends only on the structure of the porous matrix.

Here, 𝑎>0 is the shrinking constant. 𝑐 is a constant, where 𝑐=1 indicates one-direction shrinking and 𝑐=2 corresponds to axisymmetric shrinking.

3. Method of Solution

We introduce the following similarity transformations: 𝑢=𝑎𝑥𝑓^′𝜂,𝑣=𝑎(𝑐−1)𝑦𝑓^′𝜂𝜂,𝑧,𝑤=−𝑎ℎ𝑐𝑓𝜂=ℎ,𝜃𝜂=𝑇−𝑇₀𝑇₁−𝑇₀, (7) where 𝑇₁−𝑇₀=𝐷𝑥,𝐷 being a nonzero constant.

By using similarity transformations defined in (7), we find that the continuity equation (1) is automatically satisfied. Further, we see that (4) reduces to 𝑤𝜕𝑤1𝜕𝑧=−𝜌𝜕𝑃𝜕𝜕𝑧+𝜐²𝑤𝜕𝑧². (8)

On integration, we get 𝑤²2𝑃=−𝜌+𝜐𝑑𝑤𝑑𝑧+constant. (9)

Equations (2) and (3), using (9) and similarity transformations (7), are reduced to 𝑓^𝑖𝑣−𝑀²𝑓^′′−𝑅(2−𝑐)𝑓^′𝑓^′′−𝑐𝑓𝑓^′′′=0. (10)

Similarly, using (7), we see that (5) is reduced to 𝜃^′′=𝑅Pr𝑓^′𝜃−𝑐𝑓𝜃^′. (11)

The corresponding boundary conditions for the present problem are reduced to at𝜂=1,𝑓^′𝑤=−1,𝑓=−⁰ach=−𝜆,𝜃=1,at𝜂=0,𝑓^′′=𝛼𝐾𝑓^′,𝑓=0,𝜃=0, (12) where 𝑀=(𝜎/𝜌𝜐)𝐵₀ℎ, the Hartmann number; 𝑅=𝑎ℎ²/𝜐, the nondimensional shrinking parameter; Pr=𝜇𝐶_𝑝/𝑘, the Prandtl number; 𝐾=𝑘₀/ℎ, the permeability parameter.

4. Homotopy Analysis Method

Following Liao [40], we now solve the nonlinear system consisting of (10)–(12) by HAM. For HAM solutions, we choose the initial guesses 𝑓₀(𝜂) and 𝜃₀(𝜂) and the auxiliary linear operators 𝐿_𝑓 and 𝐿_𝜃 in the following forms: 𝑓₀𝜂=2𝐾(1−3𝜆)𝛼()4𝐾+𝛼𝜂+1−3𝜆𝜂4𝐾+𝛼²+𝛼(2𝜆−1)−2𝐾(1−𝜆)𝜂4𝐾+𝛼³,𝜃₀𝜂𝐿=𝜂,_𝑓𝑓=𝑑⁴𝑓𝑑𝜂⁴,𝐿_𝜃(𝜃)=𝑑²𝜃𝑑𝜂². (13)

Here, auxiliary linear operators have the property that 𝐿_𝑓𝑓₀=𝐿_𝑓𝑎_0,0+𝑎_0,1𝜂+𝑎_0,2𝜂²+𝑎_0,3𝜂³𝐿=0,_𝜃𝜃₀=𝐿_𝜃𝑏_0,0+𝑏_0,1𝜂=0. (14)

Let 𝑝∈[0,1] denote the embedding parameter and h₁, h₂ indicate the nonzero auxiliary parameters. Then we construct the following homotopies: 𝐻_𝑓𝑓𝜂;𝑝;𝑝=1−𝑝𝐿_𝑓𝑓𝜂;𝑝−𝑓₀𝜂−𝑝ℏ₁𝑁_𝑓𝑓,𝐻𝜂;𝑝_𝜃𝜃𝜂;𝑝;𝑝=1−𝑝𝐿_𝜃𝜃𝜂;𝑝−𝜃₀𝜂−𝑝ℏ₂𝑁_𝜃𝜃𝜂;𝑝, (15) where 𝑁_𝑓𝑓=𝜕𝜂;𝑝⁴𝑓𝜂;𝑝𝜕𝜂⁴−𝑀²𝜕²𝑓𝜂;𝑝𝜕𝜂²)𝜕−𝑅(2−𝑐𝑓𝜂;𝑝𝜕𝜕𝜂²𝑓𝜂;𝑝𝜕𝜂²−𝑐𝑓𝜕𝜂;𝑝³𝑓𝜂;𝑝𝜕𝜂³,𝑁_𝜃𝜃=𝜕𝜂;𝑝²𝜃𝜂;𝑝𝜕𝜂²×−𝑅Pr𝜃𝜕𝜂;𝑝𝑓𝜂;𝑝𝜕𝜂−𝑐𝑓𝜕𝜂;𝑝𝜃𝜂;𝑝.𝜕𝜂 (16)

When𝑝=0, we have 𝐻_𝑓𝑓𝜂;0;0=𝐿_𝑓𝑓𝜂;0−𝑓₀𝜂,𝐻_𝜃𝜃𝜂;0;0=𝐿_𝜃𝜃𝜂;0−𝜃₀𝜂. (17)

If𝑝=1, we have 𝐻_𝑓𝑓𝜂;1;1=−ℏ₁𝑁_𝑓𝑓,𝐻𝜂;1_𝜃𝜃𝜂;1;1=−ℏ₂𝑁_𝜃𝜃𝜂;1. (18)

4.1. Zeroth Order Deformation Equations

Thus, if we take 𝐻_𝑓𝑓𝐻𝜂;𝑝;𝑝=0,_𝜃𝜃𝜂;𝑝;𝑝=0, (19)

we obtain the family of equations as follows: 1−𝑝𝐿_𝑓𝑓𝜂;𝑝−𝑓₀𝜂=𝑝ℏ₁𝑁_𝑓𝑓,𝐿𝜂;𝑝1−𝑝_𝜃𝜃𝜂;𝑝−𝜃₀𝜂=𝑝ℏ₂𝑁_𝜃𝜃𝜂;𝑝,𝑓𝜂;𝑝. (20)

These equations are subjected to the boundary conditions: 𝑓^′1;𝑝=−1,𝑓1;𝑝=−𝜆,𝜃1;𝑝=1,𝑓^′′=𝛼0;𝑝𝐾𝑓^′0;𝑝,𝑓0;𝑝=0,𝜃0;𝑝=0. (21)

Obviously, when 𝑝=0, we see that 𝑓𝜂;0=𝑓₀𝜂,𝜃𝜂;0=𝜃₀𝜂, (22) and when 𝑝=1, we have 𝑓𝜂;1=𝑓𝜂,𝜃𝜂;1=𝜃𝜂. (23)

Thus, as the embedding parameter 𝑝 increases from 0 to 1, the mappings 𝑓(𝜂;𝑝) and 𝜃(𝜂;𝑝) vary smoothly from their known initial guesses, 𝑓₀(𝜂) and 𝜃₀(𝜂), to the required solutions, 𝑓(𝜂) and 𝜃(𝜂), respectively.

Expanding 𝑓(𝜂;𝑝) and 𝜃(𝜂;𝑝) by Taylor’s series in the embedding parameter 𝑝 and using (22), we can write 𝑓𝜂;𝑝=𝑓₀𝜂+^∞_𝑘=1𝑓_𝑘𝜂𝑝^𝑘, (24) where 𝑓_𝑘(𝜂)=[(1/𝑘!)(𝜕^𝑘𝑓(𝜂;𝑝)/𝜕𝑝^𝑘)]_𝑝=0 and 𝜃𝜂;𝑝=𝜃₀𝜂+^∞_𝑘=1𝜃_𝑘𝜂𝑝^𝑘, (25) where 𝜃_𝑘(𝜂)=[(1/𝑘!)(𝜕^𝑘𝜃(𝜂;𝑝)/𝜕𝑝^𝑘)]_𝑝=0.

Now assume that for properly chosen values of ℏ₁ and ℏ₂ the series (24) and the series (25) are convergent at 𝑝=1. Thus we obtain 𝑓𝜂=𝑓₀𝜂+^∞_𝑘=1𝑓_𝑘𝜂,𝜃𝜂=𝜃₀𝜂+^∞_𝑘=1𝜃_𝑘𝜂. (26)

4.2. 𝑚th Order Deformation Equations

The governing equations of 𝑓_𝑚(𝜂) and 𝜃_𝑚(𝜂) can be deduced from the zeroth order deformation equations (20) by differentiating these 𝑚 times with respect to the embedding parameter 𝑝, setting 𝑝=0 and finally dividing them by |𝑚. Thus we have 𝐿_𝑓𝑓_𝑚𝜂−𝜒_𝑚𝑓_𝑚−1𝜂=ℏ₁𝑅^𝑓_𝑚𝜂,𝐿_𝜃𝜃_𝑚𝜂−𝜒_𝑚𝜃_𝑚−1𝜂=ℏ₂𝑅^𝜃_𝑚𝜂, (27) subject to corresponding boundary conditions: 𝑓^′_𝑚(1)=0,𝑓_𝑚(1)=0,𝜃_𝑚(1)𝑓=0,_𝑚′′(0)=𝛼𝐾𝑓^′_𝑚(0),𝑓_𝑚(0)=0,𝜃_𝑚(0)=0, (28) where 𝑅^𝑓_𝑚𝜂=1𝑚−1𝜕^𝑚−1𝑁_𝑓𝑓𝜂;𝑝𝜕𝑝^𝑚−1_𝑝=0=𝑓^𝑖𝑣_𝑚−1−𝑀²𝑓^′′_𝑚−1+Re^𝑚−1_𝑟=0(2−𝑐)𝑓^′_{𝑚−1−𝑟}𝑓_𝑟^′′−𝑐𝑓_{𝑚−1−𝑟}𝑓_𝑟^′′′,𝑅^𝜃_𝑚𝜂=1𝑚−1𝜕^𝑚−1𝑁_𝜃𝜃𝜂;𝑝,𝑓𝜂;𝑝𝜕𝑝^𝑚−1_𝑝=0=𝜃^′′_𝑚−1−𝑅Pr^𝑚−1_𝑟=0𝑓^′_{𝑚−1−𝑟}𝜃_𝑟−𝑐𝑓_{𝑚−1−𝑟}𝜃^′_𝑟. (29)

Here, 𝜒_𝑚 is defined as 𝜒_𝑚=0,for𝑚≤1,1,for𝑚>1. (30)

The 𝑚th order (𝑚≥1) deformation equations (27) are linear, and using the initial guesses 𝑓₀(𝜂), 𝜃₀(𝜂), we can obtain 𝑓₁(𝜂), 𝑓₂(𝜂), 𝑓₃(𝜂), and so forth and𝜃₁(𝜂),𝜃₂(𝜂),𝜃₃(𝜂), and so forth using boundary conditions (28). With the help of the symbolic computation software MATHEMATICA, the solution series for (27) and (28) can be expressed in the following form: 𝑓𝜂=^∞_𝑚=0𝑓_𝑚𝜂=lim_𝑀→∞^4𝑀+3_𝑛=1^4𝑀+2_{𝑚=𝑛−1}𝑎_𝑚,𝑛𝜂^𝑛,𝜃𝜂=^∞_𝑚=0𝜃_𝑚𝜂=lim_𝑀→∞^4𝑀+1_𝑛=1^4𝑀_{𝑚=𝑛−1}𝑏_𝑚,𝑛𝜂^𝑛. (31)

The coefficient 𝑎_𝑚,𝑛 and 𝑏_𝑚,𝑛 can be obtained using the following recurrence relations: 𝑎_𝑚,0𝑎=0,_𝑚,1=𝜒_𝑚𝑎_𝑚−1,1+2𝐾4𝐾+𝛼^4𝑚+3_𝑘=4Γ_{𝑚,𝑘−4}𝑘(),𝑎𝑘−1)(𝑘−2_𝑚,2=𝜒_𝑚𝑎_𝑚−1,2+𝛼4𝐾+𝛼^4𝑚+3_𝑘=4Γ_{𝑚,𝑘−4}𝑘(),𝑎𝑘−1)(𝑘−2_𝑚,3=𝜒_𝑚𝑎_𝑚−1,3−^4𝑚+3_𝑘=4𝛼+2𝐾()Γ4𝐾+𝛼𝑘−3+1^{𝑚,𝑘−4}𝑘(),𝑎𝑘−1)(𝑘−2)(𝑘−3_𝑚,𝑘=𝜒_𝑚𝑎_{𝑚−1,𝑘}+Γ_{𝑚,𝑘−4}𝑘(𝑘−1)(𝑘−2)(𝑘−3),4≤𝑘≤4𝑚+3, (32) where Γ^𝑘_𝑚,1=ℏ₁𝑎4_{𝑚−1,𝑘}−𝑀²𝑎2_{𝑚−1,𝑘}−𝑅(2−𝑐)𝛿1_𝑚,𝑘−𝑐𝛿2_𝑚,𝑘,0≤𝑘≤4𝑚−1,𝛿1_𝑚,𝑛=^𝑚−1_𝑟=0^{min{𝑛,4𝑟+3}}_{{}𝑘=max0,𝑛−4𝑚+4𝑟+1}𝑎1_𝑟,𝑘𝑎2_{𝑚−𝑟−1,𝑛−𝑘},0≤𝑛≤4𝑚−1,𝛿2_𝑚,𝑛=^𝑚−1_𝑟=0^{min{𝑛,4𝑟+3}}_{{}𝑘=max0,𝑛−4𝑚+4𝑟+1}𝑎_𝑟,𝑘𝑎3_{𝑚−𝑟−1,𝑛−𝑘},0≤𝑛≤4𝑚−1,𝑎4_𝑚,𝑘=(𝑘+1)𝑎3_𝑚,𝑘+1,0≤𝑘≤4𝑚−1,𝑎3_𝑚,𝑘=(𝑘+1)𝑎2_𝑚,𝑘+1,0≤𝑘≤4𝑚,𝑎2_𝑚,𝑘=(𝑘+1)𝑎1_𝑚,𝑘+1,0≤𝑘≤4𝑚+1,𝑎1_𝑚,𝑘=(𝑘+1)𝑎_𝑚,𝑘+1𝑏,0≤𝑘≤4𝑚+2,_𝑚,0𝑏=0,_𝑚,1=𝜒_𝑚𝑏_𝑚−1,1−^4𝑚+1_𝑘=2𝛾_{𝑚,𝑘−2}𝑘(),𝑏𝑘−1_𝑚,𝑘=𝜒_𝑚𝑏_{𝑚−1,𝑘}+𝛾_{𝑚,𝑘−2}𝑘(𝑘−1),2≤𝑘≤4𝑚+1, (33) where 𝛾^𝑘_𝑚,1=ℏ₂𝑏2_{𝑚−1,𝑘}−𝑅PrΔ1_𝑚,𝑘−𝑐Δ2_𝑚,𝑘,0≤𝑘≤4𝑚−1,Δ1_𝑚,𝑛=^𝑚−1_𝑟=0^{min{𝑛,4𝑟+3}}_{{}𝑘=max0,𝑛−4𝑚+4𝑟+1}𝑎1_𝑟,𝑘𝑏_{𝑚−𝑟−1,𝑛−𝑘},0≤𝑛≤4𝑚−1,Δ2_𝑚,𝑛=^𝑚−1_𝑟=0^{min{𝑛,4𝑟+3}}_{{}𝑘=max0,𝑛−4𝑚+4𝑟+1}𝑎_𝑟,𝑘𝑏1_{𝑚−𝑟−1,𝑛−𝑘},0≤𝑛≤4𝑚−1,𝑏2_𝑚,𝑘=(𝑘+1)𝑏1_𝑚,𝑘+1,0≤𝑘≤4𝑚−1,𝑏1_𝑚,𝑘=(𝑘+1)𝑏_𝑚,𝑘+1,0≤𝑘≤4𝑚. (34)

5. Convergence of the HAM Solutions

Equations (31) are the series solutions of the present problem of MHD flow and heat transfer in a horizontal channel bounded by a shrinking sheet and a porous bed of very small permeability. As Liao [40] pointed out that the convergence region and convergence rate of the HAM solution series can be adjusted by means of the value of the auxiliary parameters ℏ₁and ℏ₂ here we have takenℏ₁=ℏ₂=ℏ. In order to find the convergence region for the series (31), we have plotted the so-called h-curves. We have plotted the curves of 𝑓^′′(0) versus ℏ and 𝜃^′(0) versus ℏin Figure 2, for 20th, 21st, and 22nd order approximation. It clearly indicates that the valid region of ℏ is about−1.8≤ℏ≤−0.4. In all numerical calculations for plotting graphs, the order of approximation is taken as 𝑚=22 andℏ=−1.1.

Figure 2 

h-curve for 𝑓′′(0) and 𝜃′(0) for 𝑀=1, 𝑐=3, 𝐾=0.001, 𝜆=0.5, 𝛼=0.1, Re=2, and Pr=2.

6. Discussion

In the present investigation, the effects of a porous bed and a shrinking wall on the MHD flow and heat transfer are examined in a horizontal channel. The governing nonlinear partial differential equations for flow and heat transfer are converted into ordinary differential equations using similarity transformations. Then series solutions are obtained for velocity and temperature fields using homotopy analysis method (HAM). The convergence region of the solution series by HAM is obtained by plotting the so-called h-curve graph in Figure 2. To ensure the steady flow near the shrinking sheet by confining the generated vorticity in the boundary layer, the opposite forces such as the wall mass suction and the magnetic field are taken as sufficiently high by assigning suitable values of 𝑀 or 𝜆.

Figures 3, 4, 5, 6, and 7 illustrate the variations of axial velocity for various values of the pertinent parameters. Figure 3 shows that magnitude of the axial velocity increases near the porous bed in the channel with the increase of the permeability parameter (𝐾), while it decreases near the upper shrinking sheet. Permeability is the measure of ease to flow and it enhances fluid flow near the porous bed. The effect of the fluid saturated porous medium bed is to introduce a slip at one wall of the channel, and therefore the flow velocity at this wall is no longer equal to zero. This slip depends on the parameters of structure and permeability of the porous matrix and affects the flow inside the channel through boundary condition suggested by Saffman [39]. In Figure 4, we observe that the magnitude of the axial velocity decreases with the increase of the magnetic parameter (𝑀) near the upper shrinking sheet since Lorentz force opposes the flow and leads to the deceleration of fluid flow near upper wall, while it increases near porous bed. Figure 5 illustrates that with increase in the value of suction parameter (𝜆), magnitude of the axial flow increases in the channel. Figure 6 shows that the effect of the shrinking parameter (𝑅) is to reduce the flow near the upper sheet while the reverse effect is observed near the lower porous bed. Figure 7 depicts that the axial flow near the upper wall is greater when the sheet shrinks axisymmetrically than that when it shrinks only in the 𝑥 direction, while near the porous bed reverse effect is observed.

Figure 3 

𝑓′(𝜂) for 𝑀=1, 𝑐=3, 𝛼=0.1, 𝜆=0.5, and 𝑅=2.

Figure 4 

𝑓′(𝜂) for 𝑐=3, 𝐾=0.001, 𝜆=0.5, 𝛼=0.1, and 𝑅=2.

Figure 5 

𝑓′(𝜂) for 𝑀=1, 𝑐=3, 𝐾=0.001, 𝛼=0.1, and 𝑅=2.

Figure 6 

𝑓′(𝜂) for 𝑀=1, 𝑐=3, 𝐾=0.001, 𝜆=0.5, and 𝛼=0.1.

Figure 7 

𝑓′(𝜂) for 𝑀=1, 𝐾=0.001, 𝛼=0.1, and 𝑅=2.

Figures 8, 9, 10, 11, 12, and 13 illustrate profiles of the nondimensional temperature against the coordinate𝜂. The effect of Prandtl number (Pr) on the temperature profiles can be seen in Figure 8. Increase in the Pr values leads to the decrease of the temperature in the channel and the thickness of the thermal boundary layer becomes smaller. As Pr value increases, the thermal diffusivity decreases which reduces the energy transfer ability; hence the temperature decreases. Note that very small values of Pr correspond to liquid metals, Pr∼1 corresponds to air and diatomic gases, and very large values of Pr correspond to highly viscous oils. At room temperature,Pr∼7 corresponds to water. Figure 9 depicts temperature distribution in the channel for various values of the permeability parameter (𝐾). It is found that the effect of the permeability of lower bed is to decrease the temperature throughout the channel. It is also observed in Figures 10, 11, and 12 that the temperature in the channel decreases with increase in the value of suction parameter (𝜆) or magnetic parameter (𝑀) or shrinking parameter (𝑅). Further, it is observed in Figure 13 that the temperature throughout the channel for the axisymmetric shrinking case (𝑐=2) is less than that for the case of one-direction shrinking (𝑐=1).

Figure 8 

𝜃(𝜂) for 𝑀=1, 𝑐=3, 𝐾=0.001, 𝜆=0.5, 𝛼=0.1, and 𝑅=2.

Figure 9 

𝜃(𝜂) for 𝑀=1, 𝑐=3, 𝜆=0.5, 𝛼=0.1, 𝑅=2, and Pr=2.

Figure 10 

𝜃(𝜂) for 𝑀=1, 𝑐=3, 𝐾=0.001, 𝑅=2, Pr=2, and 𝛼=0.1.

Figure 11 

𝜃(𝜂) for Pr=2, 𝑐=3, 𝐾=0.001, 𝜆=0.5, 𝛼=0.1, and 𝑅=2.

Figure 12 

𝜃(𝜂) for 𝑀=1, 𝑐=3, 𝜆=0.5, 𝛼=0.1, 𝐾=0.001, and Pr=2.

Figure 13 

𝜃(𝜂) for 𝑀=1, 𝜆=0.5, 𝐾=0.001, 𝑅=2, Pr=2, and 𝛼=0.1.

Many industrial processes, for example, in metallurgical and packaging units, involve the drawing and cooling of continuous strips, sheets, and so forth and there quality depends on the rate of cooling and shear stresses on them. Therefore the skin friction coefficient and the rate of heat transfer are the physical quantities of interest to engineers and these are plotted in Figures 14 and 15 for various values of the pertinent parameters. It is found that the magnetic field increases the surface shear stress for axial flow, that is, the local skin friction at the shrinking sheet𝜂=1, because an increase in the strength of magnetic field leads to a thinner boundary layer, thereby causing an increase in the velocity gradient 𝑓^′′(1) at the sheet. Shrinking parameter (𝑅) or permeability parameter (𝐾) also leads to an increase in the skin friction at the shrinking sheet, whereas the effect of the suction parameter (𝜆) is to reduce skin friction at the shrinking sheet. It is also noticed that the skin friction at shrinking sheet for axisymmetric shrinking is greater than that for the case of one-direction shrinking. It is observed in Figure 15 that the temperature gradient at the shrinking sheet increases as Prandtl number (Pr) increases, since a higher Pr fluid is more viscous and has a thinner thermal boundary layer which results at the sheet in a higher rate of heat transfer. It is also noticed that the temperature gradient at the shrinking sheet increases with the increase in the value of the shrinking parameter (𝑅), or permeability (𝐾), or suction parameter (𝜆), or magnetic parameter (𝑀). It is also observed that the temperature gradient at the shrinking sheet is greater for the axisymmetric shrinking case than to one-direction shrinking.

Figure 14 

−𝑓′′(1) versus 𝑅 for 𝑐=3, 𝐾=0.001, 𝜆=0.5, and 𝛼=0.1.

Figure 15 

𝜃′(1) versus 𝑅 for 𝑀=1, 𝑐=3, 𝐾=0.001, 𝜆=0.5, and 𝛼=0.1.

7. Conclusion

The present paper studies the flow and heat transfer in a channel bounded by a porous medium bed and a shrinking sheet in the presence of a magnetic field. Using similarity transformations and homotopy analysis method, analytical series solutions are obtained and presented graphically to discuss features of these solutions for various values of the pertinent parameters. Convergence of the HAM series solution is ensured by taking suitable value of the parameter ℎ. It is observed that the axial flow in the channel adjacent to upper wall is greater when it shrinks axisymmetrically to that when it shrinks in one direction, while reverse effect is observed at the other wall. Further, the effect of suction parameter is to increase the axial flow in the channel. Temperature in the channel decreases as the value of Prandtl number increases, and the effect of permeability parameter of the lower porous bed is to decrease temperature throughout the channel. It is also found that the temperature in the channel for the axisymmetric shrinking case is less than that for the case of one-direction shrinking. It is seen that the physical parameters such as suction and magnetic field control the rate of cooling and skin friction at the shrinking surface.

Acknowledgment

The support provided by the Council of Scientific and Industrial Research through Senior Research Fellowship to one of the authors, Rashmi Agrawal, is gratefully acknowledged.