This paper investigates the effects of thermal radiation on the magnetohydrodynamic (MHD) flow and heat transfer over a nonlinear shrinking porous she...

0 downloads
9 Views
429KB Size

No documents

Applied Mathematics and Mechanics (English Edition)

Eﬀects of thermal radiation on MHD viscous ﬂuid ﬂow and heat transfer over nonlinear shrinking porous sheet∗ G. C. SHIT,

R. HALDAR

(Department of Mathematics, Jadavpur University, Kolkata 700032, India)

Abstract This paper investigates the eﬀects of thermal radiation on the magnetohydrodynamic (MHD) ﬂow and heat transfer over a nonlinear shrinking porous sheet. The surface velocity of the shrinking sheet and the transverse magnetic ﬁeld are assumed to vary as a power function of the distance from the origin. The temperature dependent viscosity and the thermal conductivity are also assumed to vary as an inverse function and a linear function of the temperature, respectively. A generalized similarity transformation is used to reduce the governing partial diﬀerential equations to their nonlinear coupled ordinary diﬀerential equations, and is solved numerically by using a ﬁnite diﬀerence scheme. The numerical results concern with the velocity and temperature proﬁles as well as the local skin-friction coeﬃcient and the rate of the heat transfer at the porous sheet for diﬀerent values of several physical parameters of interest. Key words conductivity

thermal radiation, shrinking sheet, variable viscosity, variable thermal

Chinese Library Classification O361.3 2010 Mathematics Subject Classification

76E25

Nomenclature u, v, n, μ, μ∞ , T, Tw , T∞ , ρ∞ , B0 , σ, qr , σ∗, K∗,

velocity components along the x- and y-directions; power index; coeﬃcient of the viscosity; constant viscosity far away from the sheet; temperature of the ﬂuid; temperature at the sheet; free stream temperature; density of the ﬂuid; magnetic ﬁeld strength; electrical conductivity; radiative heat ﬂux; Stefan-Boltzman constant; mean absorption coeﬃcient;

βt , cp , g, k(T ), k∞ , v0 , f, η, θ, M, P r, θr , β, K,

coeﬃcient of the thermal expansion; speciﬁc heat at the constant pressure; acceleration due to the gravity; variable thermal conductivity; thermal conductivity of the ﬂuid far away from the sheet; suction velocity; dimensionless stream function; similarity space variable; non-dimensional temperature; magnetic parameter; Prandtl number; viscosity parameter; shrinking parameter; suction parameter;

∗ Received Feb. 9, 2011 / Revised Apr. 14, 2011 Project supported by the Department of Science and Technology, Government of India (DST-GOI) Funded Promotion of University Research and Scientiﬁc Excellence (PURSE) Programme of Jadavpur University (No. SR/S9/Z-23/2008/5) Corresponding author G. C. SHIT, Ph. D., E-mail: [email protected]

678

G. C. SHIT and R. HALDAR

Ec, Rex , Gr(x), Nr ,

1

Eckert number; local Reynolds number; local Grashof number; thermal radiation parameter;

λ, Cf , N u,

buoyancy parameter; skin-friction coeﬃcient; Nusselt number.

Introduction

The problem of ﬂow and heat transfer over a shrinking sheet is relatively a new consideration in the laminar boundary layer ﬂow. The surface velocity on the boundary towards a ﬁxed point is known as a shrinking phenomenon. The study of the boundary layer magnetohydrodynamic (MHD) ﬂow towards a shrinking sheet has gained considerable attention of many researchers because of its frequent occurrence in industrial technology, geothermal application, and high temperature plasmas applicable to nuclear fusion energy conversion and MHD power generation systems. Recently, several attempts[1–8] have been made on the study of shrinking phenomena. Muhaimin and Khamis[9] studied the eﬀects of heat and mass transfer on the nonlinear MHD viscous ﬂuid ﬂow over a shrinking sheet in the presence of suction. They explored the industrial application of the ﬂow over a shrinking sheet in detail, and observed that the shrinking of the sheet had a substantial eﬀect on the ﬂow ﬁelds. However, all the studies mentioned above are restricted in linear shrinking sheets with constant viscosities. The ﬂow over a nonlinear stretching sheet has been widely studied[10–13] . In many practical situations, the continuous stretching/shrinking surface is assumed to have a power-law velocity. It is well known that the physical properties of ﬂuid ﬂow may change with temperature, especially for the variable ﬂuid viscosity and the thermal conductivity. Prasad et al.[13] investigated the eﬀects of variable viscosity and variable thermal conductivity on the hydromagnetic ﬂow and heat transfer over a nonlinear stretching sheet. Nadeem and Hussain[5] examined the MHD ﬂow of a viscous ﬂuid on a nonlinear porous shrinking sheet with the homotopy analysis. They made an observation on the existence of the shrinking sheet solution and found that the solution might exist if either the magnetic ﬁeld or the stagnation point ﬂow is taken into account. However, the eﬀects of thermal radiation and viscous dissipation were not considered in their studies. Recently, a new idea has been added to the study of the viscous ﬂuid ﬂow, which is the consideration of the eﬀects of the thermal radiation and temperature-dependent viscosity. It is shown that the thermal radiation eﬀects may play a vital role in controlling the heat transfer process in polymer processing industry. In view of this, Raptis and Perdikis[14], Mohamed and Abo-Dahab[15] , and Seddeek and Aboeldahab[16] examined the eﬀects of the thermal radiation on the MHD ﬂow and heat transfer past a semi-inﬁnite porous plate with constant suction in the presence of heat generation. Moreover, an analytical solution was obtained by Fang and Zhang[17] for the ﬂow and heat transfer over a linear shrinking sheet with mass transfer, wherein the eﬀects of thermal radiation and variable thermal conductivity were not considered. In this paper, we study the eﬀects of thermal radiation and variable ﬂuid properties on the MHD ﬂuid ﬂow over a nonlinear porous shrinking sheet. The viscous dissipation and buoyancy eﬀects are taken into account in a situation when there is a temperature-dependent viscosity. The governing partial diﬀerential equations reduce to a system of ordinary diﬀerential equations by a suitable similarity transformation. The nonlinear coupled ordinary diﬀerential equations are solved numerically by a ﬁnite diﬀerence technique along with the Newton linearization method. The numerical results of the ﬂow characteristics are presented graphically.

2

Flow analysis

Let us consider the steady two-dimensional MHD ﬂow and heat transfer of an incompressible viscous ﬂuid over a nonlinear porous shrinking sheet. The x-axis is in the direction of the ﬂow and the y-axis is normal to it. Two equal forces with opposite directions are applied along

Eﬀects of thermal radiation on MHD viscous ﬂuid ﬂow and heat transfer

679

the x-axis so that the sheet shrinks towards the origin. The continuous shrinking surface is assumed to have a power-law velocity u = Uw = −bxn , where b is a constant, x denotes the distance from the slit, n is the power index, and Uw denotes the surface velocity of the sheet. We consider the ﬂuid to be electrically conducting under the inﬂuence of an applied magnetic ﬁeld B(x) normal to the shrinking sheet. The strength of the magnetic ﬁeld B(x) is B(x) = B0 x

n−1 2

,

in which B0 represents the constant magnetic ﬁeld strength applied perpendicular to the porous shrinking sheet. We assume that the induced magnetic ﬁeld produced by the motion of an electrically conducting ﬂuid is negligible since there is no electric ﬁeld because of the negligible polarization of charges. Following Lai and Kulacki[18], we assume that the ﬂuid viscosity varies as a reciprocal of a linear function of temperature given by 1 1 = (1 + γ(T − T∞ )) μ μ∞

(1)

1 = a(T − Tr ), μ

(2)

or

where ⎧ γ ⎪ a= , ⎪ ⎨ μ∞ ⎪ ⎪ ⎩ Tr = T∞ − 1 . γ In Eq. (2), both a and Tr are constants, and their values depend on the thermal property of the ﬂuid, i.e., γ. In general, a > 0 represents liquid, whereas a < 0 represents gas. By the Rosseland approximation for the radiation, the radiative heat ﬂux qr is qr = −

4σ ∗ ∂T 4 . 3K ∗ ∂y

(3)

The temperature diﬀerences within the ﬂow are so small under the consideration that T 4 may be expressed as a linear function of the temperature as shown in Ref. [19]. Expanding T 4 in a Taylor series about T∞ and neglecting the higher-order terms, we obtain 3 4 T4 ∼ T − 3T∞ . = 4T∞

(4)

Substituting Eq. (4) in Eq. (3) yields 3 ∂qr ∂2T 16σ ∗ T∞ =− . ∂y 3K ∗ ∂y 2

(5)

680

G. C. SHIT and R. HALDAR

Owing to the above mentioned assumptions, the boundary layer ﬂow over a porous shrinking sheet is governed by the following system of equations: ∂u ∂v + = 0, ∂x ∂y

(6)

∂u 1 ∂ ∂u σB 2 (x) ∂u +v = μ ± gβt (T − T∞ ) − u, ∂x ∂y ρ∞ ∂y ∂y ρ∞ ∂u 2 ∂q ∂T ∂T ∂ ∂T r +v = k(T ) +μ . ρ ∞ cp u − ∂x ∂y ∂y ∂y ∂y ∂y u

(7) (8)

The second term on the right-hand side of Eq. (7) represents the inﬂuence of the thermal buoyancy force, and the signs ± indicate the assisting ﬂow and the opposing ﬂow, respectively. Now, the temperature-dependent thermal conductivity is given by the following relation[20] : ε (T − T∞ ) , k(T ) = k∞ 1 + ΔT where ΔT = Tw − T∞ , Tw is the temperature at the sheet which is greater than T∞ , ε is a small parameter, and k∞ is the thermal conductivity of the ﬂuid far away from the sheet. The boundary conditions corresponding to the nonlinear porous shrinking sheet on the velocity and temperature ﬁelds are u(x, 0) = −bxn ,

v(x, 0) = −v0 x

n−1 2

,

T = Tw

at y = 0

(9)

and u → 0,

T → T∞

as y → ∞,

(10)

where b (> 0) is the constant shrinking rate, and v0 is the suction velocity. It is worth mentioning that the positive values of n indicate that the surface velocity is accelerated towards the extruded slit while the negative values of n indicate that the surface velocity is decelerated towards the extruded slit. To examine the ﬂow regime adjacent to the sheet, the following transformations are invoked: ⎧ u = bxn f (η), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b(n + 1) n−1 ⎪ ⎪ ⎪ x 2 y, η= ⎪ ⎪ 2ν ⎨ T − T∞ ⎪ ⎪ , θ(η) = ⎪ ⎪ Tw − T∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n−1 ⎪ ⎩ v = − bν(n + 1) x n−1 2 ηf (η) , f (η) + 2 n+1

(11)

where f is a dimensionless stream function, η is a similarity space variable, ν is the dynamic viscosity, and θ is the dimensionless temperature. Clearly, the continuity equation (6) is satisﬁed

Eﬀects of thermal radiation on MHD viscous ﬂuid ﬂow and heat transfer

681

automatically by u and v deﬁned in Eq. (11). Substituting Eq. (11) in Eqs. (7)–(10) yields θr − θ θ θr − θ 2 f − β ff + f θr θr − θ θr θr − θ θr − θ − (2 − β)M f + (2 − β)λ θ = 0, θr θr

f +

(12)

(3Nr (1 + εθ) + 4)θ + 3Nr P rf θ + 3Nr εθ2 − 3Nr P rEc

θr − θ 2 f = 0. θr

(13)

The transformed boundary conditions are given by f (η) = −1, f (η) → 0,

f (η) = K,

θ(η) = 1

at η = 0,

θ(η) → 0 as η → ∞,

(14) (15)

where primes denote diﬀerentiation with respect to η only. The dimensionless parameters in Eqs. (12) and (13) are deﬁned by ⎧ Tr − T∞ 1 ⎪ θr = , =− ⎪ ⎪ ⎪ T − T γ(T w ∞ w − T∞ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ cp μ 2σB02 ⎪ ⎪ ⎪ , Pr = M = , ⎪ ⎪ ρ∞ b(n + 1) k∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k∞ K ∗ 2n ⎪ ⎪ ⎪ , Nr = , β= ⎪ 3 ∗ ⎪ 4T∞ σ n+1 ⎨ v0 Gr(x) ⎪ ⎪ ⎪ K= , λ= , ⎪ ⎪ Re2x bν(n+1) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Uw2 bxn+1 ⎪ ⎪ ⎪ , Ec = , Rex = ⎪ ⎪ ν cp (Tw − T∞ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Gr(x) = gβt (Tw − T∞ ) . b2 x2n−1 It is worthwhile to mention here that λ > 0.0 indicates assisting of the ﬂow, λ < 0.0 indicates opposing of the ﬂow, while λ = 0.0 represents the case when the buoyancy forces are absent. Moreover, if λ is of a signiﬁcantly greater order of magnitude than one, the buoyancy forces will be predominant and the ﬂow will essentially be free convective. In our study, K > 0.0 means suction, while K < 0.0 indicates injection. Sometimes, K is also called the wall mass transfer parameter. Another important characteristics of the present investigation are the local 1 1 skin-friction Cf (Rex ) 2 and the local Nusselt number N u(Rex )− 2 , which are deﬁned by 1 θr f (0), Cf (Rex ) = − 1 − θr 2 − β 1 − 12 θ (0), =− N u(Rex ) 2−β 1 2

(16)

(17)

682

G. C. SHIT and R. HALDAR

where ⎧ τw (x) ⎪ Cf = , ⎪ ⎪ ⎪ ρUw2 ⎪ ⎪ ⎪ ⎪ ⎪ xqw ⎪ ⎪ ⎪ , Nu = ⎪ ⎪ k∞ (Tw − T∞ ) ⎨

θ 3n−1 b(n + 1) ∂u ⎪ ⎪ r ⎪ ⎪ f (0), ⎪ τw = μ ∂y y=0 = −μ∞ 1 − θ bx 2 ⎪ 2ν r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b(n + 1) n−1 ⎪ ⎩ qw = −k∞ ∂T x 2 (Tw − T∞ )θ (0). = −k∞ ∂y y=0 2ν

3

Numerical results and discussion

The inﬂuence of the temperature-dependent ﬂuid properties on the laminar boundary layer MHD ﬂow and heat transfer of an electrically conducting ﬂuid over a nonlinear shrinking sheet is investigated numerically. The system of the couple nonlinear ordinary diﬀerential equations (12) and (13) subject to the boundary conditions (14) and (15) are solved numerically by a ﬁnite diﬀerence scheme. Since Eqs. (12) and (13) are highly nonlinear ordinary diﬀerential equations, we used the Newton linearization method (see Ref. [21]) to linearize the discretized equation. The numerical procedure have been presented by Misra and Shit[22] and Misra et al.[23–24] in detail. The essential feature of this technique is that it is based on an iterative procedure with good stability and simple and tri-diagonal manipulation. The computational work is carried out by taking Δη = 0.012 5 as the calculation step size and 10−6 as the calculation convergency, respectively. Moreover, various values of the physical parameters M , θr , K, λ, Nr , and n have been chosen over a range, which are listed in each legend of the ﬁgures. To test the accuracy of our numerical solution, the comparison of f (η) with the analytical solution obtained by Nadeem and Hussain[5] is shown in Fig. 1 with M = 1.0, β = 1.0, K = 2.0, and λ = 0.0.

Fig. 1

f (η) with M , β =1.0, K = 2.0, Ec, λ, ε, P r, Nr = 0, and θr → −∞

The results of the axial velocity f (η) and the temperature distribution θ(η) with diﬀerent values of the magnetic parameter M are plotted in Figs. 2 and 4. It is obvious that with the increase in M , the axial velocity across the boundary layer decreases, while the dimensionless temperature increases. Because of the application of a transverse magnetic ﬁeld in an electrically conducting ﬂuid, a resistive force similar to a drag force is produced, which is known as the Lorentz force. This force has the tendency to slow down the ﬂuid motion and the resistance

Eﬀects of thermal radiation on MHD viscous ﬂuid ﬂow and heat transfer

683

oﬀered to the ﬂow. Therefore, it is responsible for the increase in the temperature. It is also noticed that the thermal boundary layer thickness increases in the presence of a magnetic ﬁeld. The variation of the shrinking parameter β on the axial velocity and temperature proﬁles are shown in Figs. 3 and 5 in the presence of a magnetic ﬁeld. We observe from these two ﬁgures that, as the shrinking parameter β increases, the axial velocity in the boundary layer decreases, whereas the temperature increases. It is interesting to note from Fig. 3 that, in the vicinity of the sheet, the axial velocity decreases, while the trend is reversed in the free stream. Therefore, in both cases, the shrinking parameter β has a reducing eﬀect on the momentum boundary layer thickness and an enhanced eﬀect on the thermal boundary thickness.

Fig. 2

Fig. 4

f (η) with diﬀerent M

θ(η) with diﬀerent M

Fig. 3

Fig. 5

f (η) with diﬀerent β

θ(η) with diﬀerent β

Figures 6–9 illustrate the axial velocity f (η) and the dimensionless temperature θ(η) for diﬀerent λ and diﬀerent K. The variation of the buoyancy parameter λ on the axial velocity is shown in Fig. 6. It is seem that the axial velocity increases as the buoyancy parameter increases in the ﬂow assisting region (λ > 0.0) while decreases as the buoyancy parameter increases in the ﬂow opposing region (λ < 0.0). Physically, λ > 0.0 means heating the ﬂuid or cooling the shrinking sheet, λ < 0.0 indicates cooling the ﬂuid or heating the sheet, and λ = 0.0 corresponds to the free convection parameter. From Fig. 6, it can be seen that heating the ﬂuid increases the momentum boundary layer thickness while cooling the ﬂuid or heating the surface decreases the momentum boundary layer thickness. From Fig. 8, it can be seen that in both cases, i.e., λ > 0.0 and λ < 0.0, the temperature decreases with the increase in the buoyancy parameter λ. Figures 7 and 9 depict the variations of the suction parameter K on the axial velocity and

684

G. C. SHIT and R. HALDAR

the temperature in the presence of a magnetic ﬁeld, respectively. From Fig. 7, it is known that the axial velocity in the boundary layer of the shrinking sheet increases with the increase in the suction parameter K. However, in the temperature distribution case, it decreases signiﬁcantly with the increase in the suction parameter K.

Fig. 6

f (η) with diﬀerent λ

Fig. 7

f (η) with diﬀerent K

Fig. 8

θ(η) with diﬀerent λ

Fig. 9

θ(η) with diﬀerent K

The eﬀects of the thermal radiation parameter Nr on the axial velocity and temperature distribution are presented in Figs. 10 and 11, respectively. These two ﬁgures reveal that as the thermal radiation increases, both the axial velocity and the temperature decrease gradually. This may attribute to the fact that the increase in Nr causes less interactions with the momentum as well as the thermal boundary layers. Figures 12 and 13 deal with the variations of the viscosity parameter θr on the axial velocity and the temperature. It is observed that as the viscosity parameter θr increases, the axial velocity increases in the boundary-layer thickness while the temperature decreases. It is seen that when θr → −∞, the eﬀect of the variable viscosity becomes constant. The eﬀects of the Prandalt number P r on the axial velocity and the temperature are presented in Figs. 14 and 15, respectively. It shows that both the axial velocity and the temperature decrease gradually with the increase in the Prandalt number P r. Therefore, the Prandtl number P r has a reducing eﬀect on the momentum as well as the thermal boundary layer thickness. 1 The inﬂuence of the variable thermal conductivity ε on the local skin-friction Cf (Rex ) 2 and 1 the local Nusselt number N u(Rex)− 2 are shown in Figs. 16 and 17, respectively. It is observed 1 1 that Cf (Rex ) 2 and N u(Rex )− 2 both decrease with the increase in the variable thermal conductivity ε. Therefore, the eﬀects of the variable thermal conductivity may alter the momentum as

Eﬀects of thermal radiation on MHD viscous ﬂuid ﬂow and heat transfer

Fig. 10

f (η) with diﬀerent Nr

Fig. 11

θ(η) with diﬀerent Nr

Fig. 12

f (η) with diﬀerent θr

Fig. 13

θ(η) with diﬀerent θr

Fig. 14

f (η) with diﬀerent P r

Fig. 15

θ(η) with diﬀerent P r

685

well as the thermal boundary layer thickness. In contrast, both the local skin-friction coeﬃcient and the rate of the heat transfer increase with the increase in the shrinking parameter β. 1 Figures 18 and 19 illustrate the variations of the local skin-friction Cf (Rex ) 2 and the local 1 Nusselt number N u(Rex )− 2 with the magnetic parameter M . It is shown in Fig. 18 that the local skin-friction coeﬃcient increases with the increase in the magnetic parameter M and the suction parameter K. However, it is interesting to note from Fig. 19 that the rate of the heat transfer decreases with the magnetic parameter M , whereas the rate of heat transfer increases

686

G. C. SHIT and R. HALDAR

signiﬁcantly with the increase in the suction parameter K.

4

1

Fig. 16

Cf (Rex ) 2 with diﬀerent β

Fig. 18

Cf (Rex ) 2 with diﬀerent K

1

1

Fig. 17

N u(Rex )− 2 with diﬀerent β

Fig. 19

N u(Rex )− 2 with diﬀerent K

1

Conclusions

This paper concerns the problem of the ﬂow and heat transfer of an incompressible, viscous, and electrically conducting ﬂuid over a nonlinearly shrinking porous sheet in the presence of a temperature-dependent variable viscosity. The numerical solutions of the velocity and temperature ﬁelds are obtained by the Newton linearization method with a developed ﬁnite diﬀerence scheme. The main conclusions are as follows: (i) The axial velocity across the shrinking sheet decreases and the temperature increases with the increase in the magnetic parameter M . (ii) The shrinking parameter β has a reducing eﬀect on the axial velocity and an enhancing eﬀect on the temperature distribution. (ii) The the buoyancy parameter λ has a dual eﬀect on the axial velocity. The axial velocity increases in the assisting region (λ > 0.0) and decreases in the opposing region (λ < 0.0) with the increase in λ, while the temperature decreases with the increase in λ. (iii) The suction parameter K has signiﬁcant reducing eﬀects on both the axial velocity and the temperature proﬁles. (iv) The axial velocity and the temperature both decrease as the thermal radiation parameter Nr increases.

Eﬀects of thermal radiation on MHD viscous ﬂuid ﬂow and heat transfer

687

Therefore, the present study explores a variety of information for the applications in industrial technologies such as shrink packaging, shrink wrapping, shrink ﬁlm, and temperature controlling of ﬁnal products.

References [1] Fang, T., Liang, W., and Lee, C. F. A new solution branch for the Blasius equation — a shrinking sheet problem. Computers and Mathematics with Applications, 56, 3088–3095 (2008) [2] Hayat, T., Javad, T., and Sajid, M. Analytic solution for MHD rotating ﬂow of a second grade ﬂuid over a shrinking surface. Physics Letters A, 372, 3264–3273 (2008) [3] Wang, C. Y. Stagnation ﬂow towards a shrinking sheet. International Journal of Non-Linear Mechanics, 43, 377–382 (2008) [4] Nadeem, S. and Awais, M. Thin ﬁlm ﬂow of an unsteady shrinking sheet through porous medium with variable viscosity. Physics Letters A, 372, 4965–4972 (2008) [5] Nadeem, S. and Hussain, A. MHD ﬂow of a viscous ﬂuid on a non-linear porous shrinking sheet with homotopy analysis method. Applied Mathematics and Mechanics (English Edition), 30(12), 1569–1578 (2009) DOI 10.1007/s10483-009-1208-6 [6] Fang, T. Boundary layer ﬂow over a shrinking sheet with power-law velocity. International Journal of Heat and Mass Transfer, 51, 5838–5843 (2008) [7] Fang, T. and Zhang, J. Closed-form exact solutions of MHD viscous ﬂow over a shrinking sheet. Communications in Non-Linear Science and Numerical Simulation, 14, 2853–2857 (2009) [8] Nadeem, S., Hussain, A., Malik, M. Y., and Hayat, T. Series solutions for the stagnation ﬂow of a second-grade ﬂuid over a shrinking sheet. Applied Mathematics and Mechanics (English Edition), 30 (10), 1255–1262 (2009) DOI 10.1007/s10483-009-1005-6 [9] Muhaimin, R. K. and Khamis, A. B. Eﬀects of heat and mass transfer on non-linear MHD boundary layer ﬂow over a shrinking sheet in the presence of suction. Applied Mathematics and Mechanics (English Edition), 29(10), 1309–1317 (2008) DOI 10.1007/s10483-008-1006-z [10] Cortell, R. Viscous ﬂow and heat transfer over a non-linearly stretching sheet. Applied Mathematics and Computation, 184, 864–873 (2007) [11] Sajid, M., Hayat, T., Asghar, S., and Vajravelu, K. Analytical solution for axisymmetric ﬂow over a non-linear stretching sheet. Archive of Applied Mechanics, 78, 127–134 (2008) [12] Prasad, K. V., Vajravelu, K., and Datti, P. S. The eﬀect of variable ﬂuid properties on the hydromagnetic ﬂow and heat transfer over a non-linearly stretching sheet. International Journal of Thermal Science, 49, 603–610 (2010) [13] Prasad, K. V., Vajravelu, K., and Datti, P. S. Mixed convection heat transfer over a non-linear stretching surface with variable ﬂuid properties. International Journal of Non-Linear Mechanics, 45, 320–330 (2010) [14] Raptis, A. and Perdikis, C. Viscoelastic ﬂow by the presence of radiation. Journal of Applied Mathematics and Mechanics, 78, 277–279 (1998) [15] Mohamed, R. A. and Abo-Dahab, S. M. Inﬂuence of chemical reaction and thermal radiation on the heat and mass transfer in MHD micropolar ﬂow over a vertical moving porous plate in a porous medium with heat generation. International Journal of Thermal Science, 48, 1800–1813 (2009) [16] Seddeek, M. A. and Aboeldahab, E. M. Radiation eﬀects on unsteady MHD free convection with Hall current near an inﬁnite vertical porous plate. International Journal of Mathematics and Mathematical Sciences, 26, 249–255 (2001) [17] Fang, T. and Zhang, J. Thermal boundary layers over a shrinking sheet: an analytical solution. Acta Mechanica, 209, 325–343 (2010) [18] Lai, F. C. and Kulacki, F. A. The eﬀect of variable viscosity on convective heat transfer along a vertical surface in a saturated porous medium. International Journal of Heat and Mass Transfer, 33, 1028–1031 (1990)

688

G. C. SHIT and R. HALDAR

[19] Chamakha, A. J. Hydromagnetic natural convection from an isothermal inclined surface adjacent to a thermally stratiﬁed porous medium. International Journal of Engineering Science, 35, 975– 986 (1997) [20] Chiam, T. C. Heat transfer with variable thermal conductivity in a stagnation point towards a stretching sheet. International Communications in Heat and Mass Transfer, 23, 239–248 (1996) [21] Cebeci, T. and Cousteix, J. Modeling and Computation of Boundary-Layer Flows, Springer-Verlag, Berlin (1999) [22] Misra, J. C. and Shit, G. C. Flow of a biomagnetic viscoelastic ﬂuid in a channel with stretching walls. ASME Journal of Applied Mechanics, 76, 061006 (2009) [23] Misra, J. C., Shit, G. C., and Rath, H. J. Flow and heat transfer of an MHD viscoelastic ﬂuid in a channel with stretching wall: some applications to hemodynamics. Computers and Fluids, 37(1), 1–11 (2008) [24] Misra, J. C., Sinha, A., and Shit, G. C. Flow of a biomagnetic viscoelastic ﬂuid: application to estimation of blood ﬂow in arteries during electromagnetic hyperthermia, a therapeutic procedure for cancer treatment. Applied Mathematics and Mechanics (English Edition), 31(11), 1405–1420 (2010) DOI 10.1007/s10483-010-1371-6