c Pleiades Publishing, Ltd., 2016. ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2016, Vol. 57, No. 4, pp. 681–689.  c K. Das, A. Sarkar. Original Russian Text 

EFFECT OF MELTING ON AN MHD MICROPOLAR FLUID FLOW TOWARD A SHRINKING SHEET WITH THERMAL RADIATION K. Dasa and A. Sarkarb

UDC 532.528

Abstract: The effect of melting on a steady boundary layer stagnation-point flow and heat transfer of an electrically conducting micropolar fluid toward a horizontal shrinking sheet in the presence of a uniform transverse magnetic field and thermal radiation is studied. A similarity transformation technique is adopted to obtain self-similar ordinary differential equations, which are solved numerically. The present results are found to be in good agreement with previously published data. Numerical results for the dimensionless velocity and temperature profiles, as well as for the skin friction and the rate of heat transfer are obtained. Keywords: micropolar fluid, melting, MHD, thermal radiation, shrinking sheet. DOI: 10.1134/S002189441604012X INTRODUCTION Many engineering processes involve non-Newtonian fluids (some exotic lubricants, paints, liquid crystals, colloidal fluids, suspension fluids, polymers, etc.) that cannot be described by the traditional Newtonian fluid model. Therefore, to overcome this problem, Eringen [1, 2] introduced a theory of micropolar fluids, which can undergo micro-rotation. During the last several decades, many researchers reported results on problems dealing with momentum and heat transfer in such fluids [3–11]. Various models of non-Newtonian fluids were successfully applied to solve boundary layer problems on a stretching sheet. However, works on the flow problem due to a shrinking sheet are scarce. Wang [12] was the first researcher who studied an unsteady stagnation-point flow toward a shrinking sheet. By using a similarity transform, the Navier–Stokes equations were reduced to a set of nonlinear ordinary differential equations, which were then solved numerically. Both two-dimensional and axisymmetric stagnation-point flows were considered. Miklavcic and Wang [13] proved the existence and uniqueness for a steady viscous flow caused by a shrinking sheet for a specific value of the suction parameter. Later on, the flow over a shrinking sheet was also considered by Wang [14], Hayat et al. [15], Yao and Chen [16], and Ishak et al. [17]. Recently Fan et al. [18] investigated an unsteady stagnation-point flow and heat transfer toward a shrinking sheet. Stagnation-point flows over a shrinking sheet have important practical applications in many industries, such as cooling of metallic plates and producing of solid transportable containers, which are widely used along with film packages. The study of an electrically conducting fluid in engineering applications is of considerable interest, especially in metallurgical and metal working processes or in separation of molten metals from non-metallic inclusions due to application of a magnetic field. The phase change problem occurs in casting, welding, melting purification of metals, and formation of ice layers on oceans as well as on aircraft surfaces. Epstein and Cho [19] studied the melting process on a flat plate in a steady laminar case, while Kazmierczak et al. [20, 21] investigated melting on

a A.B.N. Seal College, Cooch Behar, 736101 India; [email protected]. b Ramnagar High School, Nadia, 741502 India; [email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 57, No. 4, pp. 125–135, July–August, 2016. Original article submitted December 23, 2013; revision submitted July 21, 2014.

c 2016 by Pleiades Publishing, Ltd. 0021-8944/16/5704-0681 

681

y

ue

uw

ue

O

uw

x

Fig. 1. Physical model and coordinate system of the problem.

a vertical flat plate embedded in a porous medium for both natural and forced convection. Cheng and Lin [22] discussed the effect of melting on transient mixed convective heat transfer from a vertical plate in a liquid-saturated porous medium. Ishak et al. [23] studied a steady boundary-layer flow and heat transfer from a warm laminar liquid flow to a melting and moving surface. Very recently, Yacob et al. [24] investigated a steady boundary-layer stagnation-point flow of a micropolar fluid toward a horizontal linearly stretching/shrinking sheet. It should be noted that the effect of thermal radiation on the flow and heat transfer have not been taken into account in the majority of the above-cited investigations. However, if technology processes take place at high temperatures, thermal radiation cannot be neglected. Many process in engineering areas occur at high temperatures, and the knowledge of radiation heat transfer becomes very important for design of reliable equipment, nuclear plants, gas turbines, and various propulsion devices for aircraft, missiles, satellites, and space vehicles. Based on these applications, Cogley et al. [25] showed that, in the optically thin limit, the fluid does not absorb its own emitted radiation, but the fluid does absorb radiation emitted by the boundaries. Raptis [26] investigated a steady flow of a viscous fluid through a porous medium bounded by a porous plate subjected to suction with a constant velocity in the presence of thermal radiation. Makinde [27] examined transient free convection interaction with thermal radiation of an absorbing emitting fluid along a moving vertical permeable plate. Ibrahim et al. [28] discussed the case of a mixed convection flow of a micropolar fluid past a semi-infinite steady moving porous plate with a varying suction velocity normal to the plate in the presence of thermal radiation and viscous dissipation. Das [29] investigated the impact of thermal radiation on a magnetohydrodynamic (MHD) slip flow over a flat plate with variable fluid properties. Hayat et al. [30] studied a two-dimensional mixed convection boundary-layer MHD stagnation-point flow through a porous medium bounded by a stretching vertical plate with thermal radiation. Shit and Halder [31] studied the effect of thermal radiation on an MHD viscous fluid flow and heat transfer over a nonlinear shrinking porous sheet. The effects of suction/blowing on a steady boundary-layer stagnation-point flow and heat transfer toward a shrinking sheet with thermal radiation was examined by Bhattacharyya and Layek [32]. So far no attempt has been made to study the effect of melting on a boundary-layer flow of a non-Newtonian fluid over a shrinking sheet in the presence of a magnetic field and thermal radiation. Therefore, the focus of the present endeavor is threefold. First, we consider a micropolar fluid. Second, the MHD influence on a stagnationpoint flow over a shrinking sheet with thermal radiation is determined. Third, a numerical solution obtained by using the Mathematica 7.0 symbolic software is given. The classical model introduced by Cogley et al. [25] is used for the radiation effect as it has the merit of simplicity and enables us to introduce a linear term in the dependence on temperature in the analysis for optically thin media. The effects of various physical parameters on the velocity and temperature profiles, as well as on the local skin friction coefficient and local Nusselt number are discussed.

1. MATHEMATICAL FORMULATION OF THE PROBLEM Let us consider a steady two-dimensional laminar boundary-layer stagnation-point flow and heat transfer of an incompressible electrically conducting micropolar fluid toward a horizontal linearly shrinking sheet melting at a steady rate (Fig. 1). A uniform magnetic field of strength B0 is assumed to be applied along the y axis. The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field is assumed to be negligible in comparison with the applied magnetic field. It is assumed that the velocity of the external flow is ue (x) = ax, and the velocity of the shrinking sheet is uw (x) = cx, where c < 0 is a negative constant and x is the 682

coordinate measured along the shrinking sheet. It is also assumed that the temperature of the melting surface is Tm . The liquid phase far from the plate is maintained at a constant temperature T∞ higher than Tm . In addition, the temperature of the solid medium Ts far from the interface is constant (Ts < Tm ). Viscous dissipation and heat generation or absorption are assumed to be negligibly small. Under the foregoing assumptions, the governing boundary-layer equations have the following form [19, 23, 24]: ∂u ∂v + = 0; ∂x ∂y u

(1)

∂u μ + κ ∂ 2 u κ ∂ϕ σB02 ∂u due +v = ue + − (u − ue ), + ∂x ∂y dx ρ ∂y 2 ρ ∂y ρ  ∂ϕ  ∂ϕ  ∂2ϕ ∂u  ρj u +v =γ , − κ 2ϕ + ∂x ∂y ∂y 2 ∂y u

(2)

∂T ∂T ∂2T 1 ∂qr +v =α . − 2 ∂x ∂y ∂y ρcp ∂y

Here u and v are the velocity components along the x and y axes, κ is the vortex viscosity, ϕ is the micro-rotation component normal to the xy plane, γ is the spin gradient viscosity, σ is the electrical conductivity of the fluid, j is the micro-inertia density, T is the temperature of the fluid within the boundary layer, α is the thermal diffusivity of the fluid, cp is the specific heat at constant pressure p, ρ is the density of the fluid, qr is the radiative heat flux, and μ is the coefficient of dynamic viscosity of the fluid. The following boundary conditions are imposed for Eqs. (1) and (2) [19, 22, 23]: y = 0:

u = uw (x) = cx,

y → ∞: k

ϕ = −n

u = ue (x) = ax,

∂u , ∂y

ϕ = 0,

T = Tm , T = T∞ ;

 ∂T   = ρ[λ + cs (Tm − Ts )]v(x, 0).  ∂y y=0

(3)

(4)

Here k is the thermal conductivity, λ is the latent heat of the fluid, and cs is the heat capacity of the solid surface. Equation [4] states that the heat conducted to the melting surface is equal to the heat of melting plus the sensible heat required to raise the solid surface temperature to its melting temperature Tm [19]. For investigating the effect of different surface conditions, we choose a linear relationship between the micro-rotation variable ϕ and the surface stress ∂u/∂y in the boundary conditions (3). Here the micro-rotation parameter n ranges between 0 and 1. At n = 0, we have ϕ = 0, which is a generalization of the no-slip condition, i.e., the particle density is sufficiently large so that micro-elements close to the wall are not able to translate or rotate, as was stated by Jena and Mathur [8]. At n = 0.5, the case represents vanishing of the anti-symmetric part of the stress tensor. For this case, Ahmadi [5] suggested that the particle spin in a suspension of fine particles is equal to the fluid velocity at the wall. The value n = 1 is used for modeling the turbulent flow inside the boundary layer with micro-rotation, as suggested by Peddison and McNitt [33]. The radiative heat flux term is determined by using the Rosseland approximation as qr = −

4σ ∗ ∂T 4 , 3k ∗ ∂y

where σ ∗ is the Stefan–Boltzmann constant and k ∗ is the mean absorption coefficient. Assuming that T 4 can be expanded into the Taylor series about T∞ and neglecting higher-order terms, we obtain 3 4 T 4 = 4T∞ T − 3T∞ .

Thus, we have 3 ∗ 16T∞ σ ∂2T ∂qr =− . ∗ ∂y 3k ∂y 2

683

System (1), (2) can be transformed into a corresponding system of ordinary differential equations by using the following similarity transformations [19, 23, 24]:   √ T − Tm a a , ϕ(x, y) = ax ω(η), θ(η) = η=y , ψ(x, y) = aν xf (η). (5) ν ν T∞ − Tm Here η is the similarity variable, ν is the kinematic viscosity, f (η) is the dimensionless stream function, ω(η) is the dimensionless micro-rotation, and θ(η) is the dimensionless temperature. The stream function ψ is defined by the expressions u = ∂ψ/∂y and v = −∂ψ/∂x satisfying Eq. (1). Substitution of Eqs. (5) into Eqs. (2) yields the self-similar equations (1 + R)f  + f f  + 1 − f 2 + Rω  − Ha2 (f  − 1) = 0, (1 + R/2)ω  − R(2ω + f  ) − f  ω + f ω  = 0,

(6)

(1 + N )θ + Pr (f θ − θf  ) = 0,  where R = κ/μ is the vortex viscosity parameter, Ha = B0 σ/(ρa) is the Hartmann number (magnetic field 3 ∗ σ /(3k ∗ κ) is the thermal radiation parameter, and Pr = ν/α is the Prandtl number. parameter), N = 16T∞ The boundary conditions (3) and (4) turn to f  (0) = ε,

ω(0) = −nf  (0),

ω(∞) = 0,

θ(∞) = 1,

θ(0) = 0,

f  (∞) = 1,

Pr f (0) + M θ (0) = 0,

(7)

where ε = c/a < 0 is the shrinking parameter and M is the dimensionless melting parameter: cp (T∞ − Tm ) M= . λ + cs (Tm − Ts ) It should be noted that the melting parameter M is a combination of the Stefan numbers cp (T∞ − Tm )/λ and cs (Tm − Ts )/λ for the liquid and solid phases, respectively. The physical quantities of interest are the skin friction coefficient Cf (rate of shear stress) and the local Nusselt number Nu (rate of heat transfer). The equation defining the wall shear stress is   ∂u    τw = (μ + κ) + κ(ω) .  ∂y y=0 y=0 So the local skin friction coefficient on the surface can be expressed as τw Cf = 2 = Re−1/2 [1 + (1 − n)R]f  (0). x ρue The amount of heat transfer through the unit area of the surface is  ∂T  4σ ∗  ∂T 4   qw = −κ∞ − ∗   , ∂y y=0 3k ∂y y=0 so the rate of heat transfer can be written as  Nu = − Re1/2 x (1 + N )θ (0),

where Rex = xue (x)/ν is the local Reynolds number. It is worth mentioning that, in the absence of the magnetic field parameter Ha and the thermal radiation parameter N , the problem reduces to that considered by Yacob et al. [24].

2. METHOD OF THE SOLUTION The system of nonlinear differential equations is solved numerically in the Mathematica 7.0 symbolic computation software by using a finite difference code. Mesh selection and error control are based on the residual of the continuous solution. The system cannot be solved on an infinite interval, and it would be impractical to solve it even on a very large finite interval. Therefore, an attempt has been made to solve a sequence of problems posed on increasingly larger intervals to verify the consistent behaviour of the solution. The plot of each successive solution is superimposed over those of previous solutions, so that they can easily be compared for consistency. In these numerical computations, the infinity condition is imposed for a large, but finite value of η, where no considerable variations in velocity, micro-rotation, temperature, and other parameters occur. 684

Table 1. Values of f  (0) for Ha = 0, N = 0, and different values of ε, M , and R ε

M

R

0 0 0 −0.25 −0.25 −0.50 −0.50

0 0 1 0 1 0 1

0 1.0 1.0 0 0.5 0 0.5

f  (0) Data of Wang [14]

Data of Yacob et al. [24]

Present results

1.232588 — — 1.402240 — 1.495670 —

1.232588 1.006406 0.879324 1.402240 1.081368 1.495670 1.157496

1.232587 1.006400 0.879324 1.402243 1.081368 1.495674 1.157496

Table 2. Values of Cf and Nu for different values of ε, M , N , and Ha ε

M

N

Ha

Cf

Nu

0 −0.4 −0.8 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5

1 1 1 0 1 2 1 1 1 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5 0 0.5 1.0 0.5 0.5 0.5

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0 1.0 1.5

2.13899 2.73667 3.14785 2.94370 2.85842 2.79156 2.87043 2.85842 2.84970 −0.04424 2.85842 3.61683

−0.191475 −0.161724 −0.129993 −0.174682 −0.153993 −0.138643 −0.088053 −0.153993 −0.226597 −0.002460 −0.153993 −0.163304

3. NUMERICAL EXPERIMENT In this paper, the effect of melting on a steady two-dimensional MHD boundary-layer flow of an incompressible electrically conducting micropolar fluid with thermal radiation near the stagnation point on a horizontal shrinking sheet is investigated numerically by using the Mathematica 7.0 symbolic software. It can be seen that the solutions are affected by the vortex viscosity parameter R, Hartmann number Ha, Prandtl number Pr , shrinking parameter ε, melting parameter M , and thermal radiation parameter N . As experimental data of these physical parameters are not available in the literature, the choice of the parameter values in the present numerical simulations was dictated by the values chosen by the previous investigators. For the present investigation, the number of parameters is restricted to four: Ha, ε, M , and N . If the magnetic field parameter and the thermal radiation parameter are equal to zero, the dimensionless equations (6) with the boundary conditions (7) exactly coincide with the corresponding equations and conditions of Yacob et al. [24]. To assess the accuracy of the present code, the skin friction coefficient f  (0) and the local Nusselt number −θ (0) are calculated for different values of ε, M , and R. The values of f  (0) produced by the present code are compared in Table 1 with those of Wang [14] and Yacob et al. [24]. The results show excellent agreement.

4. NUMERICAL RESULTS AND DISCUSSION The effects of the melting parameter, magnetic field parameter, shrinking parameter, and thermal radiation parameter on the velocity and temperature profiles are depicted in Figs. 2–8, whereas the values of the skin friction coefficient and Nusselt number are listed in Table 2. In the present study, numerical computations are carried out for Pr = 0.71, n = 0.5, and R = 1. 4.1. Effect of the Melting Parameter M The effect of the melting parameter M on the streamwise velocity component f  versus the independent similarity variable η is illustrated in Fig. 2a. It is observed that the fluid velocity decreases with an increase in the melting parameter M near the shrinking surface and satisfies the far-field boundary conditions (7) asymptotically, 685

f0 1.0

(a)

(b)

o 1.0

0.8 0.8

0.6

1

0.4

2 1

0.2

4 4

0.4

3

0

2

0.6

_0.2

3

0.2

_0.4

0 0

1

2

3

4

5 n

0

1

2

3

4

5

n

5

n

Fig. 2. Velocity (a) and temperature (b) profiles for Ha = 1.0, ε = −0.5, N = 0.5, and M = 0 (1), 1 (2), 2 (3), and 3 (4).

f0 1.0 0.8

(a) 4 3

4

0.8

0.6 2

0.4

(b)

o 1.0 3

2

0.6

1

1

0.2

0.4

0

0.2

_0.2 _0.4

0 0

1

2

3

4

5 n

0

1

2

3

4

Fig. 3. Velocity (a) and temperature (b) profiles for M = 1.0, ε = −0.5, N = 0.5, and Ha = 0 (1), 0.5 (2), 1.0 (3), and 1.5 (4).

which supports the validity of the numerical results. An increase in M leads to a decrease in the boundary layer thickness. Figure 2b shows the effect of the melting parameter M on the temperature profile. As M increases, the temperature θ(η) decreases across the boundary layer; as a consequence, the thickness of the thermal boundary layer also decreases. Thus, the heat transfer rate increases with increasing melting parameter M . On the other hand, the skin friction coefficient Cf decreases with increasing M (see Table 2). The Nusselt number is negative for all values of M , which means that the heat flux is directed from the fluid to the solid surface. This fact is obvious because the fluid is hotter than the solid surface. Further, it is worth noticing from Table 2 that the rate of heat transfer is higher in the absence of melting than in the presence of melting. 4.2. Effect of the Magnetic Field Parameter Ha The influence of the magnetic field parameter Ha on the velocity distribution of the conducting fluid is illustrated in Fig. 3a. It can be easily seen that the velocity across the boundary layer region increases as Ha increases and satisfies the far field boundary conditions asymptotically. Thus, the momentum boundary layer thickness increases in the presence of a magnetic field. Figure 3b shows the temperature distributions for different values of the magnetic field parameter Ha. It is observed that the fluid temperature has the minimum value near the boundary layer region and increases with increasing boundary layer coordinate η to approach the free-stream value. In the presence of thermal radiation, the temperature increases slightly with increasing Ha; as a consequence, the thickness of the thermal boundary layer also increases. 686

1.0 1 2

0.8

4

0.6

3

0.4 0.2 0 0

1

2

3

4

5

Fig. 4. Temperature profiles for Ha = 1.0, ε = −0.5, M = 1.0, and N = 0 (1), 1 (2), 2 (3), and 3 (4).

f0

(a)

1.0 0.5

4 3 2

(b)

o 1.0 3

0.8 1

2

0.6

0

4 1

0.4 _0.5 0.2 _1.0 0 0

1

2

3

4

5 n

0

1

2

3

4

5 n

Fig. 5. Velocity (a) and temperature (b) profiles for Ha = 1.0, M = 1.0, N = 0.5, and ε = −1.2 (1), 0.8 (2), 0.4 (3), and 0 (4).

As is seen from Table 2, an increase in the magnetic field parameter Ha leads to reduction of the heat transfer rate, whereas the skin friction coefficient increases. It is worth mentioning that the presence of a magnetic field produces larger values of the Nusselt number as compared to the values where the magnetic field is absent. 4.3. Effect of the Thermal Radiation Parameter N The impact of the thermal radiation parameter N in the presence of a magnetic field on the temperature of an electrically conducting fluid is shown in Fig. 4. It can be easily seen that the temperature increases as the boundary layer coordinate η increases for a fixed value of N . For a fixed nonzero value of η (η = 0), the temperature distribution across the boundary layer decreases with increasing N ; therefore, the thickness of the thermal boundary layer also decreases. It follows from Table 2 that an increase in N leads to a minor decrease in the skin friction coefficient and to an increase in the temperature gradient at the surface. The heat transfer rate is enhanced due to the presence of thermal radiation. 4.4. Effect of the Shrinking Parameter ε Figure 5 depicts the variations of the streamwise velocity f  and temperature θ as functions of the coordinate η for different values of the shrinking parameter ε. The velocity distribution in the boundary layer region decreases with increasing ε and approaches the asymptotic value at infinity. Thus, the hydrodynamic boundary layer thickness decreases as the shrinking parameter increases. 687

It follows from Table 2 that the skin friction coefficient Cf increases with increasing shrinking parameter ε, whereas the local Nusselt number decreases, regardless of the value of M . Thus, increasing the shrinking parameter ε decreases the heat transfer rate at the solid-fluid interface.

CONCLUSIONS In this work, the effect of the melting phenomenon on a steady MHD boundary-layer stagnation-point flow and heat transfer of a micropolar fluid toward a horizontal shrinking sheet is investigated. A parametric study is performed to explore the effects of various governing parameters on the fluid flow and heat transfer characteristics. The following conclusion can be drawn from the present investigation. An increase in the melting and shrinking parameters leads to deceleration of the fluid velocity, but the reverse effect is observed as the magnetic parameter increases. The fluid temperature and the thermal boundary layer thickness increase as the thermal radiation and magnetic field parameters increase. On the other hand, the thermal boundary layer thickness decreases with increasing melting and shrinking parameters. The skin friction coefficient decreases with an increase in the melting and thermal radiation parameters, whereas the reverse effect is observed as the magnetic field and shrinking parameters increase. With an increase in the melting and shrinking parameters, the rate of heat transfer at the solid-fluid interface decreases, but the reverse effect is observed as the thermal radiation and magnetic field parameters increase.

REFERENCES 1. A. C. Eringen, “Theory of Micropolar Fluids,” J. Math. Mech. 16, 1–18 (1964). 2. A. C. Eringen, “Theory of Thermomicropolar Fluids,” J. Math. Anal. Appl. 138, 480–496 (1972). 3. T. Ariman, M. A. Turk, and N. D. Sylvester, “Microcontinuum Fluid Mechanics: A Review,” Int. J. Eng. Sci. 11, 905–930 (1973). 4. T. Ariman, M. A. Turk, and N. D. Sylvester, “Appls of Micro Fluid Mechs: A Review,” Int. J. Eng. Sci. 12, 9273–9293 (1974). 5. G. Ahmadi, “Self-Similar Solution of Incompressible Micropolar Boundary Layer Flow over a Semi-Infinite Plate,” Int. J. Eng. Sci. 14, 639–646 (1976). 6. G. Qukaszewicz, Micropolar Fluids: Theory and Application, (Birkh¨auser, Basel, 1999). 7. G. S. Guram and C. Smith, “Stagnation Flows of Micropolar Fluids with Strong and Weak Interactions,” Comput. Math. Appl. 6, 213–233 (1980). 8. S. K. Jena and M. N. Mathur, “Similarity Solution for Laminar Free Convection Flow of Thermo-Micropolar Fluid Past a Non-Isothermal Vertical Flat Plate,” Int. J. Eng. Sci. 19, 1431–1439 (1981). 9. Y. Y. Lok, I. Pop, and J. Chamkha, “Non-Orthogonal Stagnation Point Flow of a Micropolar Fluid,” Int. J. Eng. Sci. 45, 173–184 (2007). 10. A. Ishak, R. Nazar, and I. Pop, “Magnetohydrodynamic (MHD) Flow of a Micropolar Fluid Towards a Stagnation Point on a Vertical Surface,” Comput. Math. Appl. 56, 3188–3194 (2008). 11. M. Ashraf and M. M. Ashraf, “MHD Stagnation Point Flow of a Micropolar Fluid Towards a Heated Surface,” Appl. Math. Mech. 32, 45–54 (2011). 12. C. Y. Wang, “Liquid Film on an Unsteady Stretching Sheet,” Quart. Appl. Math. 48, 601–610 (1990). 13. M. Miklavcic and C. Y. Wang, “Viscous Flow due to a Shrinking Sheet,” Quart. Appl. Math. 64, 283–290 (2006). 14. C. Y. Wang, “Stagnation Flow Towards a Shrinking Sheet,” Int. J. Nonlinear Mech. 43, 377–382 (2008). 15. T. Hayat, Z. Abbas, T. Javed, and M. Sanjib, “Three Dimensional Rotating Flow Induced by a Shrinking Sheet for Suction,” Chaos Solitons Fractals 39, 1615–1626 (2009). 16. B. Yao and J. Chen, “A New Analytical Solution Branch for the Blasius Equation with a Shrinking Sheet,” Appl. Math. Comput. 215, 1146–1153 (2009). 17. A. Ishak, Y. Y. Lok, and I. Pop, “Stagnation Point Flow over a Shrinking Sheet in a Micropolar Fluid,” Chem. Eng. Comm. 197, 1417–1427 (2010). 688

18. T. Fan, H. Xu, and I. Pop, “Unsteady Stagnation Flow and Heat Transfer Towards a Shrinking Sheet,” Int. Comm. Heat Mass Transfer 37, 1440–1446 (2010). 19. M. Epstein and D. H. Cho, “Melting Heat Transfer in Steady Laminar Flow over a Flat Plate,” J. Heat Transfer 98, 531–533 (1976). 20. M. Kazmierczak, D. Poulikakos, and D. Sadowski, “Melting of a Vertical Plate in Porous Medium Controlled by Forced Convection of a Dissimilar Fluid,” Int. Comm. Heat Mass Transfer 14, 507–517 (1987). 21. M. Kazmierczak, D. Poulikakos, and I. Pop, “Melting from a Flat Plate in a Porous Medium in the Presence of Steady Convection,” Numer. Heat Transfer 10, 571–581 (1986). 22. W. T. Cheng and C. H. Lin, “Transient Mixed Convection Heat Transfer with Melting Effect from the Vertical Plate in a Liquid Saturated Porous Medium,” Int. J. Eng. Sci. 44, 1023–1036 (2006). 23. A. Ishak, R. Nazar, N. Bachok, and I. Pop, “Melting Heat Transfer in Steady Laminar Flow over a Moving Surface,” Heat Mass Transfer 46, 463–468 (2010). 24. N. A. Yacob, A. Ishak, and I. Pop, “Melting Heat Transfer in Boundary Layer Stagnation-Point Flow Towards a Stretching/Shrinking Sheet in a Micropolar Fluid,” Comput. Fluids. 47, 16–21 (2011). 25. A. C. Cogley, W. E. Vincenty, and S. E. Gilles, “Differential Approximation for Radiation in a Non-Gray Gas Near Equilibrium,” AIAA J. 6, 551–553 (1968). 26. A. Raptis, “Radiation and Free Convection Flow through a Porous Medium,” Int. Comm. Heat Mass Transfer 25, 289–295 (1998). 27. O. D. Makinde, “Free Convection Flow with Thermal Radiation and Mass Transfer Past a Moving Vertical Porous Plate,” Int. Comm. Heat Mass Transfer 32, 1411–1419 (2005). 28. F. S. Ibrahim, A. M. Elaiw, and A. A. Bakr, “Influence of Viscous Dissipation and Radiation on Unsteady MHD Mixed Convection Flow of Micropolar Fluids,” Appl. Math. Inform. Sci. 2, 143–162 (2008). 29. K. Das, “Impact of Thermal Radiation on MHD Slip Flow over a Flate Plate with Variable Fluid Properties,” Heat Mass Transfer 48, 767–778 (2011). 30. T. Hayat, Z. Abbas, I. Pop, and S. Asghar, “Effects of Radiation and Magnetic Field on the Mixed Convection Stagnation-Point Flow over a Vertical Stretching Sheet in a Porous Medium,” Int. J. Heat Mass Transfer 53, 466–474 (2010). 31. G. C. Shit and R. Haldar, “Effects of Thermal Radiation on MHD Viscous Fluid Flow and Heat Transfer over Nonlinear Shrinking Porous Sheet,” Appl. Math. Mech. 32, 677–688 (2011). 32. K. Bhattacharyya and G. C. Layek, “Effects of Suction/Blowing on Steady Boundary Layer Stagnation-Point Flow and Heat Transfer Towards a Shrinking Sheet with Thermal Radiation,” Int. J. Heat Mass Transfer 54, 302–307 (2011). 33. J. Peddison and R. P. McNitt, “Boundary Layer Theory for Micropolar Fluid,” Recent Adv. Eng. Sci. 5, 405–426 (1970).

689