Meccanica (2010) 45: 367–373 DOI 10.1007/s11012-009-9257-4
Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect Anuar Ishak
Received: 9 December 2008 / Accepted: 2 October 2009 / Published online: 16 October 2009 © Springer Science+Business Media B.V. 2009
Abstract In the present paper, we study the effects of radiation on the thermal boundary layer flow induced by a linearly stretching sheet immersed in an incompressible micropolar fluid with constant surface temperature. Similarity transformation is employed to transform the governing partial differential equations into ordinary ones, which are then solved numerically using the Runge-Kutta-Fehlberg method. Results for the local Nusselt number as well as the temperature profiles are presented for different values of the governing parameters. It is found that the heat transfer rate at the surface decreases in the presence of radiation. Comparison with known results for certain particular cases is excellent. Keywords Boundary layer · Heat transfer · Micropolar fluid · Radiation · Stretching sheet · Fluids mechanics
microinertia density thermal conductivity mean absorption coefficient material parameter boundary parameter microrotation or angular velocity radiation parameter Prandtl number radiative heat flux fluid temperature surface temperature ambient temperature velocity components in the x- and y-directions, respectively Uw velocity of the stretching sheet x, y Cartesian coordinates along the sheet and normal to it, respectively
j k k∗ K m N NR Pr qr T Tw T∞ u, v
Greek Letters Nomenclature a, b cp f h
constants specific heat at constant pressure dimensionless stream function dimensionless microrotation
A. Ishak () School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia e-mail:
[email protected]
α β γ η θ κ ν μ ρ σ∗ ψ
thermal diffusivity thermal expansion coefficient spin gradient viscosity similarity variable dimensionless temperature vortex viscosity kinematic viscosity dynamic viscosity fluid density Stefan-Boltzmann constant stream function
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Subscripts w ∞
condition at the solid surface ambient condition
Superscript
differentiation with respect to η
in a micropolar fluid with the effect of thermal radiation is taken into consideration. The governing partial differential equations are transformed into ordinary ones using similarity transformation, before being solved numerically by the Runge-Kutta-Fehlberg method. The results obtained are then compared with those of Grubka and Bobba [3], Ali [6] and Chen [10] who reported the results for some special cases of the present study.
1 Introduction The development of boundary layer flow induced solely by a stretching sheet was first studied by Crane [1], who found an exact solution for the flow field. This problem was then extended by Gupta and Gupta [2] to a permeable surface. The flow problem due to a linearly stretching sheet belongs to a class of exact solutions of the Navier-Stokes equations. Thus, the exact solutions reported by Crane [1] and Gupta and Gupta [2] are also the exact solutions to the Navier-Stokes equations. The heat transfer aspects of similar problems were studied by Grubka and Bobba [3], Chen and Char [4], Dutta et al. [5], Ali [6, 7], Afzal and Varshney [8], Afzal [9] and many others. On the other hand, the effects of buoyancy force on the development of velocity and thermal boundary layer flows over a stretching sheet have been investigated by Chen [10], Ali and Al-Yousef [11], Daskalakis [12], Partha et al. [13], El-Aziz [14], Mahapatra et al. [15] and Ishak et al. [16–20], among others. The study of flow and heat transfer past a stretching sheet has gained tremendous interest among researchers due to its industrial and engineering applications. This include extrusion of plastic sheets, annealing and tinning of copper wire, paper production, crystal growing and glass blowing. The final products depend mainly on the stretching and cooling rates at the surface. Their studies are not restricted to Newtonian fluids, but also include non-Newtonian fluids such as micropolar fluids. Such studies have been carried out by Chiam [21], Heruska et al. [22], Agarwal et al. [23], Hassanian and Gorla [24], Kelson and Desseaux [25], Kelson and Farrell [26], Nazar et al. [27] and very recently by Hayat et al. [28] and Ishak et al. [29, 30]. Motivated by the above investigations, in the present paper we study the development of thermal boundary layer flow induced by a stretching sheet immersed
2 Mathematical model Consider the steady laminar boundary layer flow over a stretching sheet immersed in a quiescent and incompressible micropolar fluid with uniform surface temperature Tw . We assume that the sheet is stretched with a linear velocity Uw = ax, where a is a positive constant and x is the distance from the slit where the sheet is issued. The simplified two-dimensional equations governing the flow may be written as [27, 31] ∂u ∂v + = 0, ∂x ∂y ∂u κ ∂ 2 u κ ∂N ∂u +v = ν+ , + ∂x ∂y ρ ∂y 2 ρ ∂y ∂N ∂N γ ∂ 2N ∂u κ u +v = 2N + , − ∂x ∂y ρj ∂y 2 ρj ∂y u
u
∂T ∂T k ∂ 2T 1 ∂qr +v = , − 2 ∂x ∂y ρcp ∂y ρcp ∂y
(1)
(2)
(3)
(4)
where u and v are the velocity components in the xand y-directions, respectively, T is the fluid temperature inside the boundary layer, N is the microrotation or angular velocity, and j , γ , ν, κ, ρ, k and cp are the microinertia per unit mass, spin gradient viscosity, kinematic viscosity, vortex viscosity, fluid density, thermal conductivity and the specific heat at constant pressure, respectively. As it was shown by Ahmadi [32], the spin-gradient viscosity γ can be defined as γ = (μ + κ/2)j = μ(1 + K/2)j,
(5)
where μ is the dynamic viscosity, K = κ/μ is the dimensionless viscosity ratio and is called the material parameter, and we take j = ν/a as a reference length (cf. Nazar et al. [27]). Relation (5) is invoked to allow
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the field of equations predicts the correct behavior in the limiting case when the microstructure effects become negligible and the total spin N reduces to the angular velocity [32, 33]. We assume that the boundary conditions are as follows: u = Uw , T = Tw u → 0,
v = 0,
N = −m
at y = 0, N → 0,
∂u , ∂y
T → T∞
(6)
4σ ∗ ∂T 4 , 3k ∗ ∂y
(7)
where σ ∗ and k ∗ are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. We assume that the temperature differences within the flow such that the term T 4 may be expressed as a linear function of temperature. Hence, expanding T 4 in a Taylor series about T∞ and neglecting higher-order terms we obtain 3 4 T 4 ≈ 4T∞ T − 3T∞ .
(8)
Using (7) and (8), equation (4) reduces to u
∂T ∂T ∂ 2T +v = α(1 + NR ) 2 , ∂x ∂y ∂y
(9)
where α = k/(ρcp ) is the thermal diffusivity and 3 /(3kk ∗ ) is the radiation parameter [35]. NR = 16σ ∗ T∞ We introduce now the following similarity transformation: Uw 1/2 η= y, ψ = (νxUw )1/2 f (η), νx (10) Uw 1/2 T − T∞ N = Uw h(η), θ (η) = , νx Tw − T∞ where η is the similarity variable and ψ is the stream function defined as u = ∂ψ/∂y and v = −∂ψ/∂x, which identically satisfies the mass conservation equation (1). Substituting (10) into (2), (3) and (9) we obtain the following ordinary differential equations: (1 + K)f + ff − f + Kh = 0, 2
1+
K h + f h − f h − K(2h + f ) = 0, 2
1 (1 + NR )θ + f θ = 0, Pr
(11)
(12) (13)
where primes denote differentiation with respect to η and Pr = ν/α is the Prandtl number. The boundary conditions (6) now become f (0) = 0,
f (0) = 1,
h(0) = −mf (0),
as y → ∞,
where T∞ is the ambient fluid temperature and m is the boundary parameter with 0 ≤ m ≤ 1 [27]. Using the Rosseland approximation for radiation [34], the radiative heat flux is simplified as qr = −
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f (η) → 0,
θ (0) = 1,
h(η) → 0,
θ (η) → 0
(14)
as η → ∞. We note that K = 0 corresponds to viscous fluid. This case has been studied by Grubka and Bobba [3], but neglecting the effect of the thermal radiation. The quantities of physical 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 NR , m, K and Pr influence these quantities.
3 Results and discussion The system of coupled ordinary differential equations (11)–(14) has been solved numerically using RungeKutta-Fehlberg method in Maple package. To validate the accuracy of the numerical values obtained, we also have solved these equations by the Kellerbox method, which is very familiar to the present author (cf. [16–20]). The numerical results obtained by both methods are in favorable agreement with those reported by Grubka and Bobba [3], Ali [6] and Chen [10] for viscous fluid, as presented in Table 1. Thus, this lends confidence to the accuracy of the numerical results to be reported subsequently. Figure 1 presents the velocity profiles for various values of K when m = 0.5. We note that the parameters NR and Pr have no influence on the flow field, which is clear from (11)–(13). It is evident from this figure that the boundary layer thickness increases with K. The velocity gradient at the surface f (0) decreases (in absolute sense) as K increases. Thus, micropolar fluids show drag reduction compared to viscous fluids. The negative velocity gradient at the surface, f (0) < 0, as shown in Fig. 1 means the stretching sheet exerts a drag force on the fluid. This is not
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Meccanica (2010) 45: 367–373 Table 1 Values of NR
K
0
0
1
2
Pr
−θ (0)
for various values of NR , K and Pr when m = 0.5 Grubka and Bobba [3]
Ali [6]
Chen [10]
Present
0.01
0.0099
0.00991
0.0099
0.72
0.4631
0.4617
0.46315
0.4631
1.0
0.5820
0.5801
0.58199
0.5820
3.0
1.1652
1.1599
1.16523
1.1652
10
2.3080
2.2960
2.30796
2.3080
100
7.7657
7.76536
7.7657
1
0.3547
1
0.3893
2
0.4115
0
0.2588
1
0.2895
2
0.3099
Fig. 1 Velocity profiles f (η) for various values of K when m = 0.5
Fig. 2 Temperature profiles θ(η) for various values of NR when m = 0.5, K = 1 and Pr = 1
surprising since the development of the boundary layer is solely induced by it. Figure 2 shows the temperature profiles for different values of NR when m = 0.5, K = 1 and Pr = 1. It is seen that the temperature gradient at the surface θ (0) decreases (in absolute sense) as NR increases. This observation is in agreement with the results presented in Table 1, which shows that the values of −θ (0) are lower for NR > 0 compared to NR = 0.
Thus, the heat transfer rate at the surface is lower in the presence of radiation. Different behaviors are observed for the effects of K and Pr on the temperature profiles. Increasing K as well as Pr is to increase the heat transfer rate at the surface. The effects of m on velocity, angular velocity and temperature profiles, when the other parameters are fixed to unity, are depicted in Figs. 5, 6 and 7, respectively. As noted in [27], m = 0 represents concen-
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Fig. 3 Temperature profiles θ(η) for various values of K when m = 0.5, NR = 1 and Pr = 1
Fig. 4 Temperature profiles θ(η) for various values of Pr when m = 0.5, K = 1 and NR = 1
trated particle flows in which the microelements close to the wall surface are unable to rotate (strong concentration), m = 0.5 indicates the vanishing of antisymmetric part of the stress tensor (weak concentration) and n = 1 is used for modeling of turbulent boundary layer flows. Most of the previous investi-
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Fig. 5 Velocity profiles f (η) for various values of m when K =1
Fig. 6 Angular velocity profiles h(η) for various values of m when K = 1
gations considered the boundary parameter m = 0 or m = 0.5 (e.g. [21–24, 27–30]). Figures 5–7 present the variations of the velocity, angular velocity and temperature profiles, respectively, with boundary parameter m. Figure 5 shows that the velocity gradient at the surface is larger for larger values of m. Different behaviors are observed for the effect of m on the heat
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References
Fig. 7 Temperature profiles θ(η) for various values of m when K = 1, NR = 1 and Pr = 1
transfer rate at the surface as presented in Fig. 7. As expected, the couple stress h(0) is more dominant for larger values of m, as shown in Fig. 6.
4 Conclusions The steady two-dimensional laminar boundary layer flow and heat transfer due to a stretching sheet immersed in an incompressible micropolar fluid has been investigated. Different from previous investigations, the effect of thermal radiation on the development of the thermal boundary layer flow has been taken into consideration. We discussed the effects of the governing parameters NR , m, K and Pr on the fluid flow and heat transfer characteristics. The numerical results obtained are in favorable agreement with previously reported cases for viscous fluid (K = 0). We found that the heat transfer rate at the surface −θ (0) decreases as the radiation parameter NR increases. Different behaviors are observed for the effects of K and Pr on the temperature field. Acknowledgement This work is supported by a research grant (Science Fund 06-01-02-SF0610) from Ministry of Science, Technology and Innovation (MOSTI), Malaysia.
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