Heat Mass Transfer (2011) 47:155–162 DOI 10.1007/s00231-010-0693-4
ORIGINAL
MHD boundary layer flow and heat transfer over a stretching sheet with induced magnetic field Fadzilah Md. Ali • Roslinda Nazar Norihan Md. Arifin • Ioan Pop
•
Received: 13 July 2009 / Accepted: 19 September 2010 / Published online: 7 October 2010 Ó Springer-Verlag 2010
Abstract In this paper, the problem of steady magnetohydrodynamic boundary layer flow and heat transfer of a viscous and electrically conducting fluid over a stretching sheet is studied. The effect of the induced magnetic field is taken into account. The transformed ordinary differential equations are solved numerically using the finite-difference scheme known as the Keller-box method. Numerical results are obtained for various values of the magnetic parameter, the reciprocal magnetic Prandtl number and the Prandtl number. The effects of these parameters on the flow and heat transfer characteristics are determined and discussed in detail. When the magnetic field is absent, the closed analytical results for the skin friction are compared with the exact numerical results. Also the numerical results for the heat flux from the stretching surface are compared with the results reported by other authors when the magnetic field is absent. It is found that very good agreement exists. List of symbols Cf Skin friction coefficient f Dimensionless stream function F. Md. Ali N. Md. Arifin Department of Mathematics & Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia R. Nazar (&) School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia e-mail:
[email protected] I. Pop Faculty of Mathematics, University of Cluj, CP 253, 3400 Cluj, Romania
H H0 H 1, H 2 He(x) k l M Nu p P Pr qw Rem Re T Tw T? u, v ue(x) V x, y
Induced magnetic field vector Uniform magnetic field at ? Magnetic components along the x and y directions, respectively x magnetic field at the edge Thermal conductivity Characteristic length Hartmann number Nusselt number Fluid pressure Magnetohydrodynamic pressure Prandtl number Wall heat flux Magnetic Reynolds number Reynolds number Fluid temperature Plate temperature Ambient temperature Velocity components along the x and y directions, respectively x velocity at the edge Fluid velocity vector Cartesian coordinates along the surface and normal to it, respectively
Greek symbols a Thermal diffusivity b Magnetic parameter g Similarity variable g0 Magnetic diffusivity k Reciprocal magnetic Prandtl number h Dimensionless temperature l Dynamic viscosity le Magnetic permeability m Kinematic viscosity
123
156
q r sw
Heat Mass Transfer (2011) 47:155–162
Fluid density Electric conductivity Wall shear stress
Subscripts w Condition at the surface ? Condition at infinity Superscript 0 Differentiation with respect to g
1 Introduction Interest in magnetohydrodynamic (MHD) flow began in 1918, when Hartmann (see Rossow [1]) invented the electromagnetic pump. MHD boundary layer flows of a viscous and electrically conducting fluid in the presence of a transverse magnetic field are observed in various technical systems employing liquid metal and plasma flow (Liron and Wilhelm [2]). The viscous MHD flow over an infinite and semi-infinite flat plate has received much attention in the past for the physical and technical importance of the problem and for its interesting mathematical aspects. It seems that the first papers treating the flow of an electrically conducting fluid were by Hartmann [3], and Hartmann and Lazarus [4]. Since then a large body of literature has been developed, in which references [5–13] are typical. With the exception of linear problems, there are very few exact problems solved on MHD boundary layer flow of an electrically conducting fluid past a semi-infinite flat plate in the presence of a transverse magnetic flow. Although Rossow [1] provided the first attempt to examine the steady boundary layer flow in such circumstances problem, the equations governing the unsteady MHD flow of a viscous, incompressible and electrically conducting fluid past an infinite flat plate in the presence of a uniform transverse magnetic field have not been considered. Wilks [10, 11], and de Socio and Renno [12] were concerned with the MHD counterpart of the classic Rayleigh [14] viscous flow problem for a perfectly and non-perfectly conducting flat plate. With the exception of linear problems, there are, however, very few exact problems solved in this literature. Ingham [8, 9] presented numerical solutions of the motion of a viscous electrically conducting fluid past a semi-infinite flat plate, which is started impulsively from rest. Rossow [1] has shown that one difficulty for the problem of steady boundary layer flow over a semiinfinite flat plate in the presence of a magnetic field is that a similar solution does not exist. However, Cobble [15] has determined a similarity solution for the Rossow’s problem [1], assuming that the applied magnetic field is proportional to x-1/2, where x is the coordinate measured along the flat plate.
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The steady two-dimensional boundary layer flow of a viscous and incompressible fluid induced by a surface moving with a constant velocity in an ambient fluid was first studied by Sakiadis [16]. Later, Crane [17] extended this idea for the two-dimensional problem, where the velocity is proportional to the stream-wise distance x from the leading edge of the stretching surface or stretching sheet and gave an exact similarity solution in closed analytical form. The study of the heat and mass transfer over a stretching sheet subject to suction or blowing was investigated by Gupta and Gupta [18], and Magyari and Keller [19]. The temperature distribution on the flow over a stretching sheet has been studied by Dutta et al. [20]. The case of a steady three-dimensional flow over a stretching surface has been considered by Wang [21]. The study of a stretching sheet or a moving wall is relevant to several important engineering applications in the field of metallurgy and chemical engineering processes (such as drawing, annealing and tinning of copper wire, etc.). These applications involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. By drawing these strips in an electrically conducting fluid under the influence of a magnetic field, the rate of cooling can be controlled. Moreover, the study of MHD flow has become important in engineering applications (such as in designing cooling system with liquid metals, MHD generator and other devices in the petroleum industry). Pavlov [22] gave an exact similarity solution of the MHD boundary layer equations for the steady two-dimensional flow of an electrically conducting fluid due to the stretching of a plane elastic surface in the presence of a uniform transverse magnetic field. Chakrabarti and Gupta [23] found the temperature distribution in this flow when a uniform suction is applied at the stretching surface. Later, an exact solution of temperature distribution on the steady flow over a stretching sheet has been obtained by Dutta [24] for MHD flow of a viscous and electrically conducting fluid in the presence of a constant magnetic field. This problem has been also considered by Andersson [25], and Pop and Na [26]. Devi and Thiyagarajan [27] solved the problem of steady nonlinear MHD flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a powerlaw velocity in the presence of variable transverse magnetic field. Using a similarity transformation, the governing nonlinear partial differential equations are transformed into nonlinear ordinary differential equations, which are solved numerically. The heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink has been very recently considered by Abel and Nandeppanavar [28]. Mahapatra and Gupta [29] analyzed the steady two-dimensional
Heat Mass Transfer (2011) 47:155–162
MHD stagnation-point flow towards a stretching surface in its own plane. The steady MHD flow and heat transfer of a viscous fluid on a stretching surface in a rotating fluid with a magnetic field has been studied by Takhar et al. [30]. The unsteady two- and three-dimensional MHD boundary layer flow due to the impulsive motion of an isothermal stretching surface has been considered by Takhar and Nath [31], Lakshmisha et al. [32], Takhar et al. [33], and Kumari and Nath [34]. The case of an unsteady flow over a nonisothermal stretched surface in the presence of a transverse magnetic field has been considered by Ezzat et al. [35]. Those studies are some of the examples in which the effect of the induced magnetic field is negligible. It is worth mentioning to this end the recent papers by Sajid et al. [36], Sajid and Hayat [37], Fang and Zhang [38], and Lok et al. [39] on MHD flow of a viscous and electrically conducting fluid in the presence of a uniform transverse magnetic field over a shrinking sheet. These last three papers extend the paper by Miklavcic and Wang [40] on the flow over a shrinking sheet in the absence of the magnetic field. For this flow configuration, the fluid is stretched towards a slot and the flow is quite different from the stretching case. It should be mentioned that in all of the above mentioned papers, the induced magnetic field has been neglected. It seems that Kumari et al. [41] are the ones who have studied the effect of the induced magnetic field on the steady and unsteady MHD flow and heat transfer over a stretching sheet with prescribed wall temperature or wall heat flux. The magnetic field is applied parallel to the stretching sheet. The results of this paper are, unfortunately, not accurate because Eq. (8) in this paper is wrongly derived. The governing nonlinear ordinary differential equations were solved numerically using the shooting method. Therefore, the aim of the present paper is to reinvestigate the MHD flow over a stretching sheet in the presence of induced magnetic field. The governing nonlinear ordinary differential equations are solved numerically using an efficient implicit finite-difference scheme known as the Keller-box method. The obtained results are compared with some results from the open literature and it is shown that they are in excellent agreement.
157
magnetic field is taken into account. The basic equations for the flow of a viscous, electrically conducting, incompressible fluid can be written in the following form (see Cowling [42]): r V ¼ 0;
rH¼0
ðV rÞV
ð1Þ
le 1 ðH rÞH ¼ rP þ mr2 V q 4pq
ð2Þ
r ðV HÞ þ g0 r2 H ¼ 0
ð3Þ
2
ðV rÞT ¼ ar T
ð4Þ
where V is the fluid velocity vector, H is the induced magnetic field vector, P = (p ? le|H|2/8p) is the magnetohydrodynamic pressure, T is the fluid temperature, p is 1 the fluid pressure, le, m, r, q, a and g0 ¼ 4prl denote e
the magnetic permeability, kinematic viscosity, electric conductivity, density, thermal diffusivity and magnetic diffusivity, respectively. According to the boundary layer approximations, Eqs. (1–4) for the problem under consideration can be reduced to (see Davies [43]) ou ov þ ¼ 0; ox oy
oH1 oH2 þ ¼0 ox oy ou ou l oH1 oH1 þ H2 u þ v e H1 ox oy 4pq ox oy due le He dHe o2 u ¼ ue þm 2 oy dx 4pq dx
ð5Þ
ð6Þ
u
oH1 oH1 ou ou o2 H1 þv H1 H2 ¼ g0 2 ox oy ox oy oy
ð7Þ
u
oT oT o2 T þv ¼a 2 ox oy oy
ð8Þ
where x and y are the Cartesian coordinates along the stretching surface and normal to it, respectively, u, v and H1, H2 are the velocity components and the magnetic components along the x and y directions, respectively, while ue(x) and He(x) are the x- velocity and the x- magnetic field at the edge of the boundary layer. The boundary layer conditions of Eqs. (5–8) are v ¼ 0; T ¼ Tw
u ¼ uw ðxÞ ¼ cx; at
oH1 ¼ H2 ¼ 0; oy
y¼0
u ¼ ue ðxÞ ¼ 0;
H1 ¼ He ðxÞ ¼ H0 ðx=lÞ;
ð9Þ
2 Basic equations
T ¼ T1
Consider the steady laminar flow of an incompressible electrically conducting fluid by a stretching surface in an ambient fluid which is at rest. The effect of the induced
where c is a positive constant, H0 is the value of the uniform magnetic field at infinity upstream and l is the characteristic length of the stretching surface comparable
as
y!1
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158
Heat Mass Transfer (2011) 47:155–162
with the dimensions of the field. Applying the following transformations: g ¼ ðc=mÞ1=2 y;
v ¼ ðcmÞ1=2 f ðgÞ; 1=2 H2 ¼ m=l2 c H0 gðgÞ;
Re1=2 Nu ¼ h0 ð0Þ x
ð19Þ
where Rex = uw x/m is the local Reynolds number.
u ¼ cxf 0 ðgÞ;
H1 ¼ H0 ð x=lÞg0 ðgÞ;
00 Re1=2 x Cf ¼ f ð0Þ;
ð10Þ
hðgÞ ¼ ðT T1 Þ=ðTw T1 Þ
3 Results and discussion
to Eqs. (5–8), we obtain the following ordinary differential equations:
Equations (11–13) subject to the boundary conditions (14) have been solved numerically via the Keller-box method as described by Cebeci and Bradshaw [44]. In this study, we found that the magnetic parameter b is only valid for smaller values compared to unity, i.e. b 1 (see also Wilks [11]), and large values of the reciprocal magnetic Prandtl number k(1). For example, given k = 100, the maximum value of b is 0.001. However, when k = 1,000, the maximum value of b has increased to 0.01. As k increases, the valid magnetic parameter b also increases. It is noticed from (19) that the skin friction coefficient Cf is directly related to the dimensionless velocity gradient at the wall or skin friction coefficient f00 (0). Since f00 (0) is negative, the computed variation of -f00 (0) with b, k and Pr will be considered. It is to be noted that for b = 0 (magnetic field is absent), Eq. (11) reduces to that of Crane [17]. This equation subject to the boundary conditions (14) admits the exact analytical solution f(g) = 1 - exp(-g). Therefore both the numerical and analytical solutions of the present study obtained for -f00 (0) when b = 0 is a unique value, namely 1.0000 and this is in excellent agreement with that of Crane [17] (1.000000), Wang [21] (1.000000), Lakshmisha et al. [32] (0.999974), Rajeswari et al. [45] (0.999818), Takhar and Nath [31] (1.000000) and Ishak et al. [46] (1.000000) when the buoyancy forces are absent (forced convection flow). On the other hand, the numerical solutions for the heat flux from the stretching surface -h0 (0) have been compared with those of Hassanien et al. [47], and Salleh and Nazar [48] when b = 0 (magnetic field is absent) and the Prandtl number Pr varying in the range 0.08 B Pr B 100. These results are given in Table 1 and they show that the agreement between the present results and those of Hassanien et al. [47], and Salleh and Nazar [48] are very good. We are, therefore, confident that the present results are accurate and that the numerical method used works very efficiently also for the present problem. Further, it is observed from Table 1 that the heat transfer coefficient -h0 (0) increases with Pr, because a higher Prandtl number fluid has a relatively lower thermal conductivity, which reduces conduction and thereby increases the variations. This results in the reduction of the thermal boundary layer thickness and increase in the heat transfer rate at the surface, -h0 (0). Table 2 presents the variation with b of -f00 (0) and -h0 (0) for the fixed value of k = 10,000 when Pr = 0.72 and 10. We observed that -f00 (0) decreases with the increase of the magnetic
f 000 þ ff 00 f 02 þ bðg02 gg00 1Þ ¼ 0
ð11Þ
kg000 þ fg00 f 00 g ¼ 0
ð12Þ
00
0
h þ Prf h ¼ 0
ð13Þ
and the boundary conditions (9) reduce to f ð0Þ ¼ 0; gð0Þ ¼ 0;
f 0 ð0Þ ¼ 1; 00
g ð0Þ ¼ 0;
f 0 ð1Þ ¼ 0;
hð0Þ ¼ 1;
g0 ð1Þ ¼ 1;
hð1Þ ¼ 0
ð14Þ
where primes denote differentiation with respect to g, k is the reciprocal magnetic Prandtl number, b is the magnetic parameter and Pr is the Prandtl number, which are defined as g0 1 le H0 2 m k¼ ¼ ð15Þ ; b¼ ; Pr ¼ 4prle m a m 4pq lc The magnetic parameter b, which gives the order of the ratio of the magnetic energy and the kinetic energy per unit volume, is related to the Hartmann [3] number M and the flow and magnetic Reynolds numbers Re and Rem in the following way: M2 ; M ¼ le H0 lðr=le Þ1=2 Re Rem ðclÞl ðclÞl ; Rem ¼ Re ¼ ¼ 4prle ðclÞl m g0 b¼
ð16Þ
For MHD boundary layers, b B 1 and k C 1, see Kumari et al. [41]. It may be noted that for b = 0 (without magnetic field), Eq. (11) reduces to that of Crane [17]. As b = 0 implies the absence of the magnetic field, Eq. (12) governing the induced magnetic field is no longer needed. The physical quantities of interest are the skin friction coefficient Cf and the local Nusselt number Nu, which are defined as sw xqw ð17Þ Cf ¼ 2 ; Nu ¼ quw kðTw T1 Þ where the wall shear stress sw and the wall heat flux qw are given by ou oT sw ¼ l ; qw ¼ k ð18Þ oy y¼0 oy y¼0 where k being the thermal conductivity of the fluid. Using variables (10), we obtain
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Heat Mass Transfer (2011) 47:155–162
159
Table 1 Variation of -h0 (0) with Pr when b = 0 (magnetic field is absent)
Table 3 Variation of -f00 (0) and -h0 (0) with k for fixed b = 0.001 when Pr = 0.72 and 10
Pr
k
-f00 (0)
Pr = 0.72 -h0 (0)
Pr = 10 -h0 (0)
Hassanien et al. [47]
Salleh and Nazar [48]
Present
0.08
0.07431
0.0743
0.1
0.09132
0.0913
100
0.9914
0.4713
2.30818
0.1692
1,000
0.9993
0.4639
2.30809
0.29864
0.2986
10,000 100,000
0.9999 1.0000
0.4632 0.4632
2.30808 2.30808
0.2
0.16915
0.4 0.6
0.40594
0.4059
0.72
0.46325
0.46317
0.4632
1 3
0.58198 1.16525
0.58198 1.16522
0.5820 1.1652
1.56806
1.5681
1
5
1.89548
1.8954
0.9
10
2.30801
2.30821
2.3081
0.8
100
7.74925
7.76249
7.7697
0.7
7
b
-f000 (0)
Pr = 0.72 -h0 (0)
Pr = 10 -h0 (0)
0.01
0.9993
0.4639
2.30809
0.05
0.9970
0.4663
2.30812
0.10 0.15
0.9945 0.9922
0.4687 0.4707
2.30817 2.30822
parameter b. This increases the boundary layer thickness which, in turn, decreases the surface skin friction coefficient -f00 (0). On the other hand, the surface heat flux -h0 (0) increases when b increases. The reason for such a behavior is that the thermal boundary layer thickness becomes thin as the magnetic parameter b increases, which results in higher temperature gradient at the surface, hence higher heat transfer at the surface. However, the effect of the reciprocal magnetic Prandtl number k on the skin friction coefficient -f00 (0) and the heat transfer coefficient -h0 (0) is the opposite, as shown in Table 3, i.e. -f00 (0) increases and -h0 (0) decreases with k. This is because the increase in k results in the decrease of the momentum boundary layer and growth of the thermal boundary layer thicknesses. However, the heat transfer is strongly affected by the Prandtl number Pr. On the other hand, it is also noticed from these tables that as Pr increases the value of -h0 (0) also increases. However, the variations in -h0 (0) with the variations of b (Table 2) or k (Table 3) remain small (the case of Pr = 10 is even smaller than the case of Pr = 0.72) for a maximum of less than 1%. Therefore, it can be concluded that the effect of the induced magnetic field is quite negligible, especially for larger Pr. Some of the computed dimensionless velocity profiles f0 (g), induced magnetic field profiles g0 (g) and temperature
0.5 0.4 0.3 β = 0.0, 0.05, 0.1, 0.15
0.2 0.1 0
0
5
10
15
20
25
η
30
35
40
45
Fig. 1 Dimensionless velocity profiles f0 (g) for different values of b when Pr = 0.72 and k = 10,000
1 0.9 0.8 0.7 0.6
f ’(η)
Table 2 Variation of -f00 (0) and -h0 (0) with b for fixed k = 10,000 when Pr = 0.72 and 10
f ’(η)
0.6
0.5 0.4 0.3 0.2
λ = 100, 1000, 10000, 100000
0.1 0
0
5
10
15
20
η
25
30
35
40
45
Fig. 2 Dimensionless velocity profiles f0 (g) for different values of k when Pr = 0.72 and b = 0.001
profiles h(g) are presented in Figs. 1, 2, 3, 4, 5, 6, 7 for different values of the parameters b, k and Pr. Figure 1 presents the velocity profiles for Pr = 0.72, k = 10,000 and different values of the magnetic parameter b. It is seen that f0 (g) increases when b increases because the skin friction coefficient -f00 (0) decreases. On the other hand, for
123
Heat Mass Transfer (2011) 47:155–162 1.1
1
1.09
0.9
1.08
0.8
1.07
0.7
1.06
0.6
θ(η)
g ’(η)
160
1.05 1.04
0.5 0.4
1.03
0.3
β = 0.001, 0.05, 0.1
1.02
0.2
1.01
0.1
1
0
10
20
η
30
40
0
50
Fig. 3 Dimensionless induced magnetic field profiles g0 (g) for different values of b when Pr = 0.72 and k = 10,000
λ = 100, 1000, 100000
0
2
4
6
η
8
10
Fig. 6 Dimensionless temperature profiles h(g) for different values of k when Pr = 0.72 and b = 0.001
1.8
1
1.7
0.9
1.6
0.8 0.7
1.5
θ(η)
g ’(η)
0.6 1.4 1.3
0.5 0.4
1.2
λ = 1000, 10000, 100000
1.1
0.2
1 0.9
0.1 0
10
20
η
30
40
50
Fig. 4 Dimensionless induced magnetic field profiles g0 (g) for different values of k when Pr = 0.72 and b = 0.001
1 0.9 0.8 0.7
θ(η)
0.6 0.5 0.4 0.3 β = 0.0, 0.05, 0.2, 0.4
0.2 0.1 0
Pr = 0.72, 1.0, 6.8, 10.0
0.3
0
2
4
η
6
8
10
Fig. 5 Dimensionless temperature profiles h(g) for different values of b when Pr = 0.72 and k = 10,000
fixed values of Pr = 0.72 and b = 0.001, the velocity profiles decrease when the parameter k increases, as can be observed from Fig. 2. This behavior is due to the increase
123
0
0
1
2
3
4
η
5
6
7
8
9
Fig. 7 Dimensionless temperature profiles h(g) for different values of Pr when k = 1,000 and b = 0.01
of -f00 (0) with k. The dimensionless induced magnetic field profiles g0 (g) for Pr = 0.72 and various values of b and k can be seen in Figs. 3 and 4, where these profiles increase with b and decrease with k. However, g0 (g) remains almost constant for very large values of k (Fig. 4). Finally, Figs. 5, 6, 7 display the dimensionless temperature profiles h(g) for different values of the parameters b, k and Pr. A feature of these profiles is the decrease of the thermal boundary layer thickness with b due to the increase of the wall heat flux -h0 (0) (Fig. 5) and an increase of the thermal boundary layer thickness with k due to the decrease of -h0 (0) with k (Fig. 6). The variation of the dimensionless temperature profiles h(g) with g is illustrated in Fig. 7 for k = 1,000, b = 0.01 and some values of the Prandtl number Pr. This graph demonstrates that the increase of Pr results in decrease of the temperature distribution, which tends to zero as the similarity variable g increases from the stretched surface and hence the thermal boundary layer thickness decreases as Pr increases. However, the dimensionless
Heat Mass Transfer (2011) 47:155–162
temperature profiles are substantially affected by the Prandtl number.
4 Conclusions A numerical study is performed for the problem of steady MHD boundary layer flow and heat transfer of an incompressible viscous fluid over a stretching sheet with an induced magnetic field taken into consideration. The results show that the velocity and the induced magnetic field profiles increase with the applied magnetic field or magnetic parameter b. The velocity and also the induced magnetic field profiles reduce as the reciprocal magnetic Prandtl number k increases but it shows opposite effect on the temperature profiles. Further, it is found that the induced magnetic field profiles are affected the most by the parameters b and k. On the other hand, it shows opposite effect of the Prandtl number Pr on the temperature profiles. The skin friction coefficient, the induced magnetic field at the wall and the heat transfer are found to increase with the magnetic parameter b. However they decrease when the reciprocal magnetic Prandtl number k increases. It is also found that as the reciprocal magnetic Prandtl number k increases, the valid magnetic parameter b also increases. Finally, it should be mentioned that the present analysis is more general than any previous investigations. Acknowledgments The authors gratefully acknowledge the financial supports received in the form of a fundamental research grant (FRGS) from the Ministry of Higher Education, Malaysia and the research university grant. The authors are also grateful to the reviewers for the valuable comments and suggestions.
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