shrinking sheet in a porous medium

International Communications in Heat and Mass Transfer 38 (2011) 1029–1032

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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t

Stagnation point flow and heat transfer over a stretching/shrinking sheet in a porous medium☆ Haliza Rosali a, Anuar Ishak b, Ioan Pop c,⁎ a b c

Department of Mathematics and Institute for Mathematical Research, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia Faculty of Mathematics, University of Cluj, CP 253, Romania

a r t i c l e

i n f o

Available online 19 May 2011 Keywords: Stagnation point flow Shrinking/stretching sheet Porous medium Forced convection Heat transfer

a b s t r a c t The steady stagnation point flow and heat transfer over a shrinking sheet in a porous medium is studied. A similarity transformation is used to reduce the governing system of partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using the Keller-box method. The behavior of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. The results indicate that dual solutions exist for the shrinking case. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Porous materials such as sand and crushed rock underground are saturated with water which, under the influence of local pressure gradients, migrate and transport the liquid through the material. The transport properties of fluid-saturated porous materials are very important in the petroleum and geothermal industries. Further examples of convection through porous media may be found in manmade systems such as fiber and granular insulations, winding structures for high-power density electric machines, and the cores of nuclear reactors (Bejan [1]), food processing and storage, thermal insulation of buildings, geophysical systems, electro-chemistry, metallurgy, the design of pebble bed nuclear reactors, underground disposal of nuclear or non-nuclear waste, cooling system of electronic devices, etc. Excellent reviews of the topic can be found in the books by Nield and Bejan [2], Pop and Ingham [3], Bejan et al. [4], Ingham and Pop [5], Vafai [6], Vadasz [7] and Vafai [8]. Vafai and Tien [9] analyzed the effects of a solid boundary and the inertial forces on flow and heat transfer through a porous medium and reported that the inertia effects increase with the higher permeability and the lower fluid viscosity. The steady stagnation point flow through a porous medium bounded by a vertical surface was investigated by Ishak et al. [10] and it was found that dual solutions exist for both assisting and opposing flows. Viscous fluid motion toward a stagnation point on a solid body has attracted the interest of many authors. Hiemenz [11] was the first to study the two-dimensional stagnation flow using a similarity transformation to reduce the Navier–Stokes equations to nonlinear

☆ Communicated by P. Cheng and W.-Q. Tao. ⁎ Corresponding author. E-mail address: [email protected] (I. Pop). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.04.031

ordinary differential equations. He developed an exact solution to the Navier–Stokes equations. Merril et al. [12] investigated the large time (final state flow) solutions for unsteady mixed convection boundary layer flow near a stagnation point on a vertical surface embedded in a Darcian fluid-saturated porous medium. Crane [13] was the first to study the problem of steady twodimensional boundary layer flow of an incompressible viscous fluid caused by a stretching plate whose velocity varies linearly with the distance from a fixed point on the sheet. The combination of both stagnation flow and stretching surface was considered by Mahapatra and Gupta [14,15]. The flow over a shrinking sheet was investigated by Miklavčič and Wang [16]. For this flow configuration, the sheet is shrunk toward a slot and the flow is quite different from the stretching case. It is also shown that mass suction is required to maintain the flow over the shrinking sheet. The flow induced by a shrinking sheet with constant or power-law velocity distribution was investigated recently by Fang [17] and Fang et al. [18]. Wang [19] studies the stagnation flow towards a shrinking sheet and found that solutions do not exist for larger shrinking rates and may be non-unique in the two-dimensional case. The flow over an unsteady shrinking sheet was studied by Fang et al. [20] and the solution is an exact solution of the unsteady Navier– Stokes equations. This shrinking sheet problem was extended to a second grade fluid [21], and MHD rotating flow of a viscous fluid [22]. The objective of this paper is to investigate the heat transfer characteristics caused by a shrinking sheet immersed in a fluidsaturated porous medium. The results for the skin friction coefficient, local Nusselt number, velocity profiles as well as the temperature profiles are obtained and discussed for different values of the governing parameters. We restrict our study to unit Prandtl number, taking Pr = 1. We expect our results are qualitatively similar with other values of Pr of O(1).

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Nomenclature a, b, c Cf f k K K1 Nux Pex Pr qw Rex T Tw T∞ u, v µe µw x, y

constants skin friction coefficient dimensionless stream function thermal conductivity permeability parameter permeability of the porous medium local Nusselt number local Péclet number Prandtl number surface heat flux local Reynolds number fluid temperature surface temperature ambient temperature velocity components along the x and y directions, respectively velocity of the external flow velocity of the stretching surface Cartesian coordinates along the surface and normal to it, respectively

Fig. 1. Physical model of two-dimensional stagnation point flow over a shrinking sheet.

u

∂T ∂T ∂2 T +v =α 2; ∂x ∂y ∂y

subject to the boundary conditions u = uw ðxÞ = b x; T = Tw at y = 0; u = ue ðxÞ = a x; T = T∞ as y→∞;

Greek letters α thermal diffusivity β thermal expansion coefficient η similarity variable μ dynamic viscosity ν kinematic viscosity θ dimensionless temperature ρ fluid density τw surface shear stress ψ stream function

ð3Þ

ð4Þ

where u and v are the velocity components along the x- and y-axes, respectively, T is the fluid temperature and the other physical quantities are defined in the Nomenclature. To obtain similarity solutions for the system of Eqs. (1)–(4), we introduce the following similarity variables (see Cheng [24] or Lai and Kulacki [25]) η=

u x1 = 2 y T−T∞ 1=2 e ; ψ = ðα x ue Þ f ðηÞ; θðηÞ = ; α x Tw −T∞

ð5Þ

where ψ is the stream function defined as u = ∂ ψ/∂ y and v = −∂ ψ/∂ x, which identically satisfy Eq. (1). Using the non-dimensional variables in Eq. (5), Eqs. (2) and (3) reduce to the following ordinary differential equations

Subscripts w condition at the surface ∞ condition away from the surface

Superscript ′ differentiation with respect to η


2 Pr f ‴ + f f ″−f ′ + 1 + K 1−f ′ = 0;

ð6Þ

θ″ + f θ′ = 0;

ð7Þ

subject to the boundary conditions 2. Problem formulation Consider a steady stagnation point flow over a shrinking sheet which is embedded in a porous medium as shown in Fig. 1. The Cartesian coordinates x and y are taken with the origin O at the stagnation point, and are defined such that the x-axis is measured along the stretching/ shrinking sheet and the y-axis is measured normal to it. It is assumed that the velocity of the external flow is given by ue(x) = a x, where a N 0 is the strength of the stagnation flow and the surface temperature Tw is a constant. It is also assumed that the velocity of the stretching/shrinking sheet is given by uw(x) = b x, where b is the stretching rate, with bN 0 and b b 0 are for stretching and shrinking cases, respectively. The boundary layer equations in a porous medium are given by [23] ∂u ∂v + = 0; ∂x ∂y ∂u ∂u d ue ∂ u ν +ν 2 + ðu −uÞ; +v = ue K1 e dx ∂x ∂y ∂y

ð8Þ

where primes denote differentiation with respect to η, Pr = ν/α is the Prandtl number and K = ν/(aK1) is the permeability parameter. It is worth mentioning that c N 0 and c b 0 correspond to stretching and shrinking sheets, respectively, while c = 0 is the planar stagnation flow towards a stationary sheet. Moreover, c = 1 corresponds to the flow with no boundary layer (uw = ue). The physical quantities of interest are the skin friction coefficient Cf and the local Nusselt number Nux, which are defined as Cf =

τw x qw ; Nux = ; kðTw −T∞ Þ ρ u2e = 2

ð9Þ

ð1Þ

where the surface shear stress τw and the surface heat flux qw are given by

ð2Þ

τw = μ

2

u

f ð0Þ = 0; f ′ ð0Þ = b = a = c; θð0Þ = 1; f ′ ðηÞ→1; θð∞Þ→0 as η→∞;

    ∂u ∂T ; qw = −k ; ∂y y = 0 ∂y y = 0

ð10Þ

H. Rosali et al. / International Communications in Heat and Mass Transfer 38 (2011) 1029–1032

3.5

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Table 1 Values of cc for various values of K when Pr = 1.

3 K = 0, 0.5, 0.8, 1

f (0)

2.5 2 1.5

K

cc

0 0.5 0.8 1

− 1.2465 − 1.6956 − 1.9727 − 2.1599

1 0.5 0 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

c Fig. 2. Variation of the skin friction coefficient f″(0) with c for K = 0, 0.5, 0.8 and 1 when Pr = 1.

with μ and k being the dynamic viscosity and thermal conductivity, respectively. Using the non-dimensional variables (5), we get 1 1=2 1=2 C Re = Pr = f ″ð0Þ; 2 f x

1=2

Nux = Pex

= −θ′ ð0Þ;

ð11Þ

where Rex = ue(x)x/ν is the local Reynolds number and Pex = ue(x)x/α is the local Péclet number. 3. Results and discussion

solutions exist for the shrinking case (c b 0). For all c ≥ 0 (stretching case) the solution is unique. From Fig. 2, it can be seen that the critical value of c (say cc) for which the solution exists increases as the permeability parameter K increases (permeability of the porous medium K1 decreases). It shows that lower porosity of the porous medium (large value of K) increases the range of existence of solution to Eqs. (6)–(8). These values of cc(b 0)are presented in Table 1. Notice that c = 0 corresponds to the Hiemenz [11] flow, and c = 1 is a degenerate inviscid flow, where the stretching matches the conditions at infinity (Chiam [27]). Fig. 4 presents the velocity profiles for the shrinking case (c b 0), with the corresponding temperature profiles being shown in Fig. 5. In these figures the solid lines are for the first solution and the dash lines are for the second solution. It is evident from these figures that both first and second solution profiles satisfy the far field boundary conditions asymptotically, thus support the validity of the numerical results obtained, besides supporting the dual nature of the solutions presented in Figs. 2 and 3.

Eqs. (6) and (7) subject to the boundary conditions (8) were solved numerically for some values of the permeability parameter K and the stretching or shrinking parameter c using a finite-difference method described in the book by Cebeci and Bradshaw [26]. In this study, we have considered K = 0, 0.5, 0.8 and 1. It is worth mentioning that K = 0 (K1 → ∞) corresponds to a shrinking sheet in a viscous (Newtonian) fluid and K ≠ 0 corresponds to a shrinking sheet in a porous medium saturated with a viscous fluid. Representative results for the skin friction coefficient f″(0) were obtained for some values of the permeability parameter K with the various values of c when Pr = 1. Further, in order to verify the accuracy of the present method, we have compared our results for the skin friction coefficient f″(0) with those reported by Wang [19] for K = 0, and the results are in excellent agreement. The variations of the skin friction coefficient f″(0) and the local Nusselt number − θ'(0) with parameter c for K = 0, 0.5, 0.8 and 1 are shown in Figs. 2 and 3 respectively. These figures show that dual Fig. 4. Velocity profiles f′(η) for different values of K and c when Pr = 1.

0.8 0.7 0.6

-θ (0)

0.5

K = 0, 0.5, 0.8, 1

0.4 0.3 0.2 0.1 0 -2.5

-2

-1.5

c

-1

-0.5

0

Fig. 3. Variation of the local Nusselt number − θ′(0) with c for K = 0, 0.5, 0.8 and 1 when Pr = 1.

Fig. 5. Temperature profiles θ(η) for different values of K and c when Pr = 1.

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4. Conclusions In the present paper the stagnation point flow and heat transfer over a linearly shrinking/stretching sheet immersed in a fluid-saturated porous medium was investigated. The governing boundary layer equations were solved numerically using the Keller-box method. The effects of the shrinking/stretching parameter c and the permeability parameter K on the skin friction coefficient and the local Nusselt number as well as the velocity and the temperature profiles were obtained and discussed. It was found that for the stretching case (c N 0), the solution is unique and exists for all values of c, whereas dual solutions were found to exist only for a certain range of c b 0, up to a certain critical value cc(K) b 0 for the shrinking case. Decreasing the porosity of the porous medium (increasing the values of K) is to widen the range of c for which the solution exists. Acknowledgements The authors would like to thank the editor and the reviewer for their comments and suggestions which led to the improvement of this paper. This work was supported by a research grant (Project Code: UKM-ST-07-FRGS0029-2009) from the Ministry of Higher Education, Malaysia. References [1] A. Bejan, Convection Heat Transfer, second ed. Wiley, New York, 1995. [2] D.A. Nield, A. Bejan, Convection in Porous Media, third ed. Springer, New York, 2006. [3] I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford, 2001. [4] A. Bejan, I. Dincer, S. Lorente, A.F. Miguel, A.H. Reis, Porous and Complex Flow Structures in Modern Technologies, Springer, New York, 2004. [5] D.B. Ingham, I. Pop, Transport Phenomena in Porous Media III, Elsevier, Oxford, 2005. [6] K. Vafai, Handbook of Porous Media, second ed. Taylor & Francis, New York, 2005. [7] P. Vadasz, Emerging Topics in Heat and Mass Transfer in Porous Media, Springer, New York, 2008.

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