Meccanica (2012) 47:293–299 DOI 10.1007/s11012-011-9439-8

Micropolar fluid flow over a shrinking sheet Nor Azizah Yacob · Anuar Ishak

Received: 19 November 2009 / Accepted: 24 May 2011 / Published online: 16 June 2011 © Springer Science+Business Media B.V. 2011

Abstract An analysis is carried out for the steady two-dimensional flow of a micropolar fluid over a shrinking sheet in its own plane. The shrinking velocity is assumed to vary linearly with the distance from a fixed point on the sheet. The features of the flow and heat transfer characteristics are analyzed and discussed. It is found that the solution exists only if adequate suction through the permeable sheet is introduced. Moreover, stronger suction is necessary for the solution to exist for a micropolar fluid compared to a classical Newtonian fluid. Dual solutions are obtained for certain suction and material parameters. Keywords Boundary layer · Micropolar fluid · Shrinking sheet · Dual solutions · Fluid Mechanics Nomenclature a, m constants Cf skin friction coefficient f dimensionless stream function h dimensionless microrotation

N.A. Yacob Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Pahang, 26400 Bandar Jengka, Pahang, Malaysia A. Ishak () School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia e-mail: [email protected]

microinertia density thermal conductivity suction parameter material parameter angular velocity Prandtl number fluid temperature surface temperature ambient temperature velocity components in the x and y directions, respectively Uw velocity of the shrinking sheet Vw transpiration velocity x, y Cartesian coordinates along the sheet and normal to it, respectively j k s K N Pr T Tw T∞ u, v

Greek letters α thermal diffusivity γ spin gradient viscosity η similarity variable θ dimensionless temperature κ vortex viscosity ν kinematic viscosity μ dynamic viscosity ρ fluid density ψ stream function Subscripts w condition at the solid surface ∞ ambient condition

294

Superscript  differentiation with respect to η

1 Introduction The boundary layer flow over moving or stretching surfaces has important applications in engineering processes, such as polymer extrusion, drawing of copper wires, continuous stretching of plastic films and artificial fibers, hot rolling, wire drawing, glassfiber, metal extrusion, and metal spinning [1]. The pioneering work on a moving surface was done by Sakiadis [2]. Thereafter, this problem has been studied extensively in various aspects in Newtonian fluids [3–11]. However, many engineering processes involve non-Newtonian fluid such as paints, lubricants, blood, polymers, colloidal fluids and suspension fluids which cannot be described by traditional Newtonian fluid. Therefore, to overcome this problem, Eringen [12, 13] has formulated the theory of micropolar fluids that is capable to describe those fluids by taking into account the effect arising from local structure and micro-motions of the fluid elements. This study has attracted many researchers to investigate the nonNewtonian fluid with various physical aspects. Takhar et al. [14] studied the effect of buoyancy force on the flow and heat transfer of a micropolar fluid over a stretching sheet. Xu and Liao [15] investigated the unsteady MHD viscous flows of non-Newtonian fluids caused by an impulsively stretching plate. The influence of temperature dependent thermal conductivity and thermal radiation on MHD flow and heat transfer of a micropolar fluid over a stretching surface with power-law variation in surface temperature was considered by Mahmoud [16]. Ishak et al. [17] studied the effect of buoyancy force on boundary layer flow of a micropolar fluid near the stagnation point on a stretching vertical sheet, and found that dual solutions exist for the opposing flow. Pal and Hiremath [18] analyzed the effect of viscous dissipation and internal heat generation/absorption in flow and heat transfer of an incompressible viscous fluid over an unsteady stretching sheet placed in a fluid saturated porous medium. Very recently, Ishak [19] investigated the radiation effect on a thermal boundary layer flow over a stretching sheet in a micropolar fluid. Different from the above studies, Miklavˇciˇc and Wang [20] investigated the shrinking flow where velocity of the boundary moves toward the fixed point.

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They found an exact solution of the Navier-Stokes equations and reported that an adequate suction is necessary to maintain the flow. To the best of our knowledge, there are only a few published papers on a flow due to a shrinking sheet, such as [20–38]. Hayat et al. [37] extended the idea of flow past a shrinking sheet to a steady two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid. Nadeem and Awais [38] have considered the unsteady case of the thin film flow of a shrinking sheet through porous medium with variable viscosity. The objective of the present study is to investigate the boundary layer flow and heat transfer over a shrinking sheet immersed in a micropolar fluid, which has not been considered before.

2 Mathematical formulation Consider the flow of an incompressible micropolar fluid in the region y > 0 driven by a shrinking sheet of uniform temperature Tw . The sheet velocity is assumed to vary proportional to the distance x from the fixed point on the sheet, i.e. Uw (x) = ax where a is a positive constant. The simplified two-dimensional equations governing the flow in the boundary layer of a steady, laminar and incompressible micropolar fluid are ∂u ∂v + = 0, ∂x ∂y

(1)

  ∂u ∂u κ ∂ 2 u κ ∂N u +v = ν+ , + ∂x ∂y ρ ∂y 2 ρ ∂y   ∂N γ ∂ 2N ∂u ∂N κ +v = 2N + , u − ∂x ∂y ρj ∂y 2 ρj ∂y u

∂T ∂ 2T ∂T +v =α 2 , ∂x ∂y ∂y

(2) (3) (4)

subject to the boundary conditions u = −Uw (x),

v = Vw ,

at y = 0, u → 0, N → 0,

N = −m

T → T∞

∂u ∂y

as y → ∞,

(5)

where u and v are the velocity components along the x and y axes, respectively, N is the microrotation or angular velocity whose direction of rotation is normal to the x–y plane, ν is the kinematic viscosity, ρ is the

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295

fluid density, j is the microinertia per unit mass, γ is the spin gradient viscosity, and κ is the vortex viscosity. Further, Vw is the mass flux through the permeable sheet with Vw < 0 for suction and Vw > 0 for injection. We notice that m is a constant such that 0 ≤ m ≤ 1 (see Ishak et al. [39]). Following Ahmadi [40], we assume that γ is given by γ = (μ + κ/2) j,

(6)

and we take j = ν/a as a reference length. Relation (6) is invoked to allow the field of equations to predict the correct behavior in the limiting case when the microstructure effects become negligible, and the total spin N reduces to the angular velocity [40]. The governing partial differential equations (1)–(4) subject to the boundary conditions (5) can be transformed into a set of ordinary differential equations by the following transformation: η=

 a 1/2 ν

N = ax

ψ = (νa)1/2 xf (η) ,

y,

 a 1/2 ν

h (η) ,

T − T∞ θ (η) = , Tw − T∞

(7)

where η is the similarity variable and ψ is the stream function defined in the usual way as u = ∂ψ/∂y and v = −∂ψ/∂x, which identically satisfies (1). Using (7), we obtain

f  (0) = −1,

f (0) = s,

h(0) = −mf  (0),

θ (0) = 1, f  (η) → 0,

h(η) → 0,

θ (η) → 0

(13)

as η → ∞. We notice that for K = 0 (viscous fluid), (10) subject to the corresponding boundary conditions (13) has an exact solution f (η) = s −

2 √ s ± s2 − 4

√

s± 2 e− + √ 2 s ± s −4

s 2 −4 η 2

,

(14)

which is identical to those found by Fang and Zhang [28] with M = 0 in their paper. The physical quantities of interest are the values of f  (0) and −θ  (0) which represent the skin friction coefficient and the heat transfer rate at the surface, respectively. Thus, our task is to investigate how the governing parameters K, Pr and s influence these quantities. For viscous fluid (K = 0), from (14), the quantity f  (0) is given by √ s ± s2 − 4 f  (0) = . (15) 2 3 Results and discussion



u = axf (η), v = −(νa)

1/2

(8) f (η).

From (8), the mass flux Vw can be defined as Vw = −(νa)1/2 s,

(9)

where s = f (0) is the suction parameter. The transformed nonlinear ordinary differential equations are (10) (1 + K) f  + ff  − f 2 + Kh = 0,     K 1+ h + f h − f  h − K 2h + f  = 0, (11) 2 1  θ + f θ  = 0, Pr

(12)

where K = κ/μ (≥ 0) is the material parameter and primes denote differentiation with respect to η. The boundary conditions (5) now become

The nonlinear ordinary differential equations (10)– (12) subject to the boundary conditions (13) have been solved numerically using an implicit finite-difference scheme known as the Keller-box method as described in the book by Cebeci and Bradshaw [41], for several values of K and s, while the Prandtl number Pr is fixed to unity and we take m = 0.5 (weak concentration). In order to support the validity of the numerical results obtained, we also have solved these equations by the Runge-Kutta-Fehlberg method with shooting technique, which was successfully used by the present authors to solve various two-point boundary value problems, and obtained dual solutions (see [42–45]). The results obtained by both methods for the present problem are in excellent agreement. For the Keller-box method, the dual solutions were obtained by setting two different boundary layer thicknesses η∞ , while for the Runge-Kutta-Fehlberg method with shooting technique, the dual solutions were obtained by “guessing”

296

Fig. 1 Variation of f  (0) with s for different values of K when m = 0.5

Fig. 2 Variation of −θ  (0) with s for different values of K when Pr = 1 and m = 0.5

two different values of f  (0) and −θ  (0), which produce two different profiles, where both of them satisfy the far field boundary conditions asymptotically. Figure 1 presents the variation of the skin friction coefficient f  (0) with s, while the respective heat transfer rate at the surface −θ  (0) is depicted in Fig. 2. These figures show that the values of f  (0) and −θ  (0) are always positive, which mean the fluid exerts a drag force on the sheet and the heat is transferred from the hot sheet to the cold fluid, respectively. It is observed that for Newtonian fluid (K = 0), solution exists when s ≥ 2, which is in agreement with the exact solution given by (15). Moreover, stronger suction

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Fig. 3 Velocity profiles f  (η) for different values of K when m = 0.5 and s = 3

Fig. 4 Temperature profiles θ(η) for different values of K when Pr = 1, m = 0.5 and s = 3

is necessary for the solution to exist for a micropolar fluid (K > 0) compared to a Newtonian fluid (K = 0). Figures 3, 4, 5, 6, 7 show the velocity, temperature and microrotation profiles that support the dual nature of the solutions presented in Figs. 1 and 2. From these figures, it is seen that the far field boundary conditions are satisfied asymptotically for both upper and lower branch solutions, and thus supporting the validity of the numerical results obtained. Figures 3 and 4 present the velocity and temperature profiles, respectively, for different values of material parameter K when s = 3. For the upper branch solutions, the velocity and the thermal boundary layer thicknesses increase with an increase in K, and in consequence decrease the ve-

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Fig. 5 Velocity profiles f  (η) for different values of s when m = 0.5 and K = 1

297

Fig. 7 Microrotation profiles h(η) for different values of s when m = 0.5 and K = 1

4 Conclusions

Fig. 6 Temperature profiles θ(η) for different values of s when Pr = 1, m = 0.5 and K = 1

locity and the temperature gradients at the surface, i.e the skin friction coefficient f  (0) and the heat transfer rate at the surface −θ  (0) decrease as K increases. The opposite trends are observed for the lower branch solutions. This observation is in agreement with the results presented in Figs. 1 and 2. We note that the Prandtl number Pr has no influence on the flow field, which is clear from (10)–(12). Finally, Fig. 7 presents the dimensionless microrotation or angular velocity profiles h(η) for different values of s when K = 1. As expected, the microrotation effects are more dominant near the wall.

The problem of steady two-dimensional micropolar fluid flow over a shrinking sheet was investigated numerically. The governing partial differential equations were transformed, using a similarity transformation, to a system of nonlinear ordinary differential equations, before being solved numerically. The results were presented graphically, and the effects of the material and suction parameters on the fluid flow and heat transfer characteristics were discussed. It was found that the solution exists only if adequate suction through the permeable sheet is introduced. The values of the suction parameter s for which the solution exists increase as the material parameter K increases. Thus, stronger suction effect is necessary for the solution to exist for a micropolar fluid (K > 0) compared to a classical Newtonian fluid (K = 0). It was also found that the heat transfer rate at the surface increases when the suction effect is increased. Acknowledgements The authors are indebted to the referees for their valuable comments and suggestions, which led to the improvement of the paper. The financial support received in the form of a research grant (Project Code: UKMST-07-FRGS0029-2009) from the Ministry of Higher Education Malaysia is gratefully acknowledged.

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