Appl. Math. Mech. -Engl. Ed., 32(6), 677–688 (2011) DOI 10.1007/s10483-011-1448-6 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011
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]
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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
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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)
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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)
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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
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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
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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
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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.
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[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