Heat and mass transfer for Soret and Dufour’s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid

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Abstract

A mathematical model is analyzed in order to study the heat and mass transfer characteristics in mixed convection boundary layer flow about a linearly stretching vertical surface in a porous medium filled with a viscoelastic fluid, by taking into account the diffusion-thermo (Dufour) and thermal-diffusion (Soret) effects. The governing partial differential equations are transformed into a set of coupled ordinary differential equations, which are solved analytically using the homotopy analysis method (HAM) to determine the convergent series expressions of velocity, temperature and concentration. The physical interpretation to these expressions is assigned through graphs and a table for the wall shear stress f(0), Nusselt number -θ(0) and Sherwood number -ϕ(0). Results showed that the fields were influenced appreciably by the effects of the governing parameters: mixed convection parameter λ, Lewis number Le, Prandtl number Pr, viscoelastic parameter K, concentration buoyancy parameter N, porosity parameter γ, Dufour number Df and Soret number Sr. It was evident that for some kind of mixtures such as the light and medium molecular weight, the Soret and Dufour’s effects should be considered as well.

Introduction

Transport of heat through a porous medium has been the subject of many studies due to the increasing need for a better understanding of the associated transport processes. This interest stems from the numerous practical applications which can be modeled or can be approximated as transport through porous media such as packed sphere beds, high performance insulation for buildings, chemical catalytic reactors, grain storage, migration of moisture through the air contained in fibrous insulations, heat exchange between soil and atmosphere, sensible heat storage beds and beds of fossil fuels such as oil shale and coal, salt leaching in soils, solar power collectors, electrochemical processes, insulation of nuclear reactors, regenerative heat exchangers and geothermal energy systems and many other areas. Literature concerning convective flow in porous media is abundant. Representative studies in this area may be found in the recent books by Nield and Bejan [1], Ingham and Pop [2], Vafai [3], Pop and Ingham [4], Ingham et al. [5], Bejan et al. [6], Vadasz [7], etc.

It is well-known that in double-diffusive (e.g. thermohaline) convection the coupling between the transport of heat and mass takes place because the density ρ of the fluid mixture depends on both temperature T and concentration C. For sufficiently small isobaric changes in temperature and concentration the mixture density ρ depends linearly on both T and C (Nield and Bejan [1]). In some circumstances there is direct coupling between T and C. This is when cross-diffusion (Soret and Dufour effects) is not negligible. The energy flux caused by a composition gradient was discovered in 1873 by Dufour and was correspondingly reffered to the Dufour effect. It was also called the diffusion-thermo effect. On the other hand, mass flux can also be created by a temperature gradient, as was established by Soret. This is the thermal-diffusion effect. The Soret effect has been also used for isotope separation and in mixture between gases with very light molecular weight (H2,He) and of medium molecular weight (H2, air) (Postelnicu [8]). In many studies Soret and Dufour effects are neglected, on the basis that they are of a smaller order of magnitude than the effects described by Fourier’s and Fick’s laws. There are, however, exceptions. Eckert and Drake [9] have presented several cases when the Dufour effect cannot be neglected. Platten and Legros [10] state that in most liquid mixtures the Dufour effect is inoperate, but that this may not be the case in gases. Mojtabi and Charrier-Mojtabi [11] confirm this by noting that in liquids the Dufour coefficient is an order of magnitude smaller than the Soret effect. Benano-Molly et al. [12] have studied the problem of thermal diffusion in binary fluid mixture, lying within a porous medium and subjected to a horizontal thermal gradient and have shown that multiple convection-roll flow patterns can develop depending on the values of the Soret number. They conclude that for saturated porous media, the phenomenon of cross diffusion is further complicated because of the interaction between the fluid and the porous matrix and because accurate values of the cross-diffusion coefficients are not available. The onset of Soret-driven convection in an infinite cell filled with a porous medium saturated by a binary fluid was studied by Sovran et al. [13]. However, Soret and Dufour effects have been found to appreciably influence the flow field in free convection boundary layer over a vertical surface embedded in a fluid-saturated porous medium (Postelnicu [8] and Anghel et al. [14]).

Boundary layer behaviour over moving or stretched surfaces through an otherwise quiescent or moving medium is an important type of flow occurring in many engineering processes. Specifically, these include glass–fiber and paper production, hot rolling, spinning of laments, continuous casting, cooling of metallic sheets or electronic chips, crystal growing, polymer melts and solutions, the aerodynamic extrusion of plastic sheets, the cooling and/or drying of paper and textiles, etc. A vast body of knowledge encompassing analytical and numerical studies explaining various aspects is now available on the stretching flows in a viscous and incompressible fluid (Magyari and Keller [15], Sparrow and Abraham [16]). Much work has also been done for the forced convection flow and heat transfer due to a stretching surface in a viscoelastic fluid (Rajagopal et al. [17], Siddappa and Abel [18], Dandapat et al. [19], Pontrelli [20], Abel et al. [21], [22], Khan and Sanjayanand [23], Bujurke et al. [24], Dandapat and Gupta [25], Char and Chen [26], Bhattacharyya et al. [27], Vajravelu and Roper [28], Khan et al. [29]). The study of flow field and heat transfer is necessary for determining the quality of the final products of such processes as explained by Karwe and Jaluria [30].

In many practical situations the surface moves or stretches in a quiescent fluid, with the fluid flow being induced by the motion of the solid surface and by thermal buoyancy. Therefore, the resulting flow and thermal fields are determined by these two mechanisms, surface motion and buoyancy. It is well known that the buoyancy forces stemming from the heating or cooling of the continuous stretching sheets alter the flow and thermal fields and thereby the heat transfer characteristics of the manufacturing process. Effects of thermal buoyancy on heat transfer from a continuous moving or a stretched surface in a viscous and incompressible fluid were investigated by Daskalakis [31], Ali and Al-Yousef [32], Chen [33], [34], Ali [35], [36], Partha et al. [37], Ishak et al. [38], [39], [40], etc.

The aim of this investigation is to study the steady mixed convection boundary layer flow due to the combined effect of heat and mass transfer over a stretched vertical surface in a porous medium filled with a viscoelastic fluid under Soret and Dufour’s effects. Homotopy analysis method (HAM) proposed by Liao [41] is used in obtaining the analytical solution. This method has been also very successfully applied for other problems [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56] of fluid mechanics and heat transfer. To the best of the author’s knowledge, this problem has not been considered before. However, Tsai and Huang [57] have very recently presented a theoretical study of the steady stagnation point flow over a stretched vertical surface in the presence of species concentration and mass diffusion under Soret and Dufor’s effects. It was concluded that for some kinds of mixtures with the light and medium molecular weight, the Soret and Dufor’s effects should be considered as well.

Section snippets

Basic equations

Consider the heat and mass transfer flow due to the stretching of a heated or cooled vertical surface of variable temperature Tw(x) and variable concentration Cw(x) in a porous medium filled with a viscoelastic fluid of uniform ambient temperature T and uniform ambient concentration C. It is assumed that the surface is stretched in its plane with the velocity uw(x). The density variation and the buoyancy effects are taken into consideration, so that the Boussinesq approximation for both the

Homotopy analysis solutions

The velocity f(η), the temperature θ(η) and the concentration field ϕ(η) can be expressed by the set of base functionsηkexp(-nη)|k0,n0in the formf(η)=a0,00+n=0k=0am,nkηkexp(-nη),θ(η)=n=0k=0bm,nkηkexp(-nη),ϕ(η)=n=0k=0cm,nkηkexp(-nη),where am,nk, bm,nk and cm,nk are the coefficients based on the rule of solution expressions and the boundary conditions (10), one can choose the initial guesses f0, θ0 and ϕ0 of f(η),θ(η) and ϕ(η) asf0(η)=1-exp(-η),θ0(η)=exp(-η),ϕ0(η)=exp(-η).and the

Convergence of the homotopy solutions

The analytical series solutions (46), (47), (48) contain the non-zero auxiliary parameters f, θ, and ϕ which can adjust and control the convergence of the series solutions. In order to see the range of admissible values of f, θ, and ϕ of the functions f(0), θ(0), and ϕ(0) the f, θ, and ϕ- curves are displayed for 20th-order of approximations. It is obvious from Fig. 1 that the range for the admissible values of f, θ, and ϕ are -1.1f-0.2, -1.1θ-0.4, and -1.0ϕ-0.4. It is

Results and discussion

In order to see the effects of various parameters on f,θ and ϕ, we display the Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15, Fig. 16, Fig. 17, Fig. 18. The values of skin friction coefficient Rex1/2Cf, the local Nusselt number Rex-1/2Nux and the local Sherwood number Rex-1/2Sh for various values of embedded parameters are also given in Table 4. Fig. 2, Fig. 3, Fig. 4 show the effects of viscoelastic parameter K, the

References (59)

  • D.A. Nield et al.

    Convection in porous media

    (2006)
  • I. Pop et al.

    Convective heat transfer: mathematical and computational modeling of viscous fluids and porous media

    (2001)
  • A. Bejan et al.

    Porous and complex flow structures in modern technologies

    (2004)
  • A. Postelnicu

    Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects

    Int J Heat Mass Transfer

    (2004)
  • E.R. Eckert et al.

    Analysis of heat and mass transfer

    (1972)
  • J.K. Platten et al.

    Convection in liquids

    (1984)
  • A. Mojtabi et al.

    Double-diffusive convection in porous media

  • L.B. Benano-Melly et al.

    Modeling Soret coefficient measurement experiments in porous media considering thermal and solutal convection

    Int J Heat Mass Transfer

    (2001)
  • O. Sovran et al.

    Onset of Soret-driven convection in an infinite porous layer

    C R Acad Sci

    (2001)
  • M. Angel et al.

    Doufor and Soret effects on free convection over a vertical surface in a porous medium

    Studia Univ Babeş – Bolyai, ser Mathematica

    (2000)
  • E. Magyari et al.

    Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls

    Eur J Mech B/Fluids

    (2000)
  • E.M. Sparrow et al.

    Universal solutions for the streamwise variation f the temperature of a moving sheet in the presence of a moving fluid

    Int J Heat Mass Transfer

    (2005)
  • K.R. Rajagopal et al.

    Flow of a viscoelastic

    uid over a stretching sheet, Rheol Acta

    (1984)
  • B. Siddappa et al.

    Non-Newtonian flow past a stretching plate

    J Appl Math Phys (ZAMP)

    (1985)
  • B.S. Dandapat et al.

    Stability of flow of a viscoelastic fluid over a stretching sheet

    Arch Mech

    (1994)
  • G. Pontrelli

    Flow of a .uid of second grade over a stretching sheet

    Int J Non-Linear Mech

    (1995)
  • M.S. Abel et al.

    Convective heat and mass transfer in a viscoelastic fluid flow through a porous medium over a stretching sheet

    Int J Numer Methods Heat Fluid Flow

    (2001)
  • M.S. Abel et al.

    Study of visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity

    Int J Non-Linear Mech

    (2002)
  • S.K. Khan et al.

    Viscoelastic boundary layer MHD flow through a porous medium over a porous quadratic stretching sheet

    Arch Mech

    (2004)
  • N.M. Bujurke et al.

    Second order fluid flow past a stretching sheet with heat transfer

    J Appl Math Phys (ZAMP)

    (1987)
  • B.S. Dandapat et al.

    Flow and heat transfer in a viscoelastic fluid over a stretching sheet

    Int J Non-Linear Mech

    (1989)
  • M.I. Char et al.

    Temperature field in non-Newtonian flow over a stretching plate with variable heat flux

    Int J Heat Mass Transfer

    (1988)
  • S. Bhattacharyya et al.

    Heat transfer in the flow of a viscoelastic fluid over a stretching surface

    Heat Mass Transfer

    (1988)
  • K. Vajravelu et al.

    Flow and heat transfer in a second grade fluid over a stretching sheet

    Int J Non-Linear Mech

    (1999)
  • S.K. Khan et al.

    Visco-elastic MHD flow, heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work

    Heat Mass Transfer

    (2003)
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