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Radiation and melting effects on MHD boundary layer flow over a moving surface
Ain Shams Engineering Journal (2014) xxx, xxx–xxx
Ain Shams University
Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com
MECHANICAL ENGINEERING
Radiation and melting effects on MHD boundary layer flow over a moving surface Kalidas Das
*
Department of Mathematics, Kalyani Government Engineering College, Kalyani, Nadia, West Bengal 741235, India Received 28 December 2013; revised 16 March 2014; accepted 12 April 2014
KEYWORDS Melting; MHD; Thermal radiation; Moving surface
Abstract This paper presents a mathematical analysis of MHD flow and heat transfer from a warm, electrically conducting fluid to melting surface moving parallel to a constant free stream in the presence of thermal radiation. The similarity transformation technique is used to convert the governing partial differential equations into the self-similar ordinary differential equations and then solved numerically using MATLAB BVP solver bvp4c. Numerical results for the dimensionless velocity and temperature profiles as well as for the skin friction and the Nusselt number are elucidated for different values of the pertinent parameters. Comparison with previously published work is presented and it found to be in excellent agreement. 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.
1. Introduction In the recent years, the effects of electrically conducting fluids, such as liquid metals, water and others on the flow and heat transfer in the presence of external magnetic field were discussed by many researchers [1–5]. The flow and heat transfer characteristics induced by a continuously moving surface are important in industrial engineering processes such as in the lamination and melt-spinning process, in the polymer industry, continuous casting, and spinning of fibers. The steady boundary layer flow due to a continuously moving surface was first considered by Sakiadis [6]. Thereafter, this problem was extended by a very good number of researchers [7–12] * Mobile: +91 9748603199. E-mail address: [email protected]. Peer review under responsibility of Ain Shams University.
Production and hosting by Elsevier
in various aspects for both Newtonian and non-Newtonian fluids. Heat transfer accompanied by melting phenomenon has recently received considerable research attention. This is due to large number of applications, including latent heat storage, material processing, crystal growth, castings of metals, glass industry, purification of materials, and others. Melting heat transfer from a flat plate was investigated by Epstein and Cho [13]. On the other hand melting from a vertical flat plate embedded in a porous medium was studied by Kazmierczak et al. [14,15]. Cheng and Lin [16] discussed the melting effect on convective heat transfer from a vertical plate in a liquid saturated porous medium. Recently, Ishak et al. [17] studied the heat transfer characteristic from a warm liquid to a melting and moving surface. However, the effect of thermal radiation on the heat transfer processes is very important in high operating temperature and has many important applications such as nuclear reactor cooling system, gas turbines and various propulsion devices or aircraft, missiles, satellites and space vehicles. Cogley et al. [18] observed that the fluid does not absorb its own emitted radiation in the optically thin limit but absorb radiation
2090-4479 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2014.04.008 Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
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K. Das
emitted by the boundaries. The free convection flow through a porous medium bounded by a porous plate in the presence of thermal radiation was investigated by Raptis [19]. Makinde [20] examined the transient free convection along moving vertical porous plate in the presence of thermal radiation. Ibrahim et al. [21] discussed the effect of thermal radiation on mixed convection flow. On the other hand, the fluid can be treated as electrically conducting in the sense that it is ionized due to the high operating temperature. Accordingly, the influence of magnetic field on the fluid flow characteristics is important in the presence of thermal radiation. Das [22] investigated the effect of thermal radiation on fluid flow over a flat plate in the presence of magnetic field. Hayat et al. [23] studied mixed convection MHD boundary layer flow through a porous medium in the presence of thermal radiation. The effects of thermal radiation on boundary layer flow and heat transfer toward a shrinking sheet were examined by Bhattacharyya and Layek [24]. Hayat et al. [25] discussed stretched flow of Jeffrey fluid in the presence of thermal radiation. The main purpose of the present investigation is to illustrate the effects of magnetic field and thermal radiation on steady laminar boundary layer flow and heat transfer from a warm electrically conducting fluid to a melting surface moving parallel to a constant free stream. The organization of the paper is as follows. Section 2 discusses the mathematical formulation of the problem. Numerical method is provided in Section 3. Section 4 consists of results and discussion. The concluding remarks are given in Section 5. 2. Mathematical formulation of the problem Consider a steady laminar flow of an incompressible electrically conducting fluid toward a moving surface melting at a steady rate into a warm liquid of the same material with constant property under the influence of a transverse magnetic ! field B . The external electric field is assumed to be zero and the magnetic Reynolds number is assumed to be small. Hence, the induced magnetic field is negligible compared with the ! externally applied magnetic field so that B = [0, B(x)] where B(x) is the applied magnetic field in the positive direction of y-axis and varies in strength as a function of x. The flow is assumed to be in the x-direction which is taken along the moving surface and y-axis is normal to it. It is assumed that the surface is moving with a constant velocity Uw. It is also assumed that the temperature of the melting surface is Tm. The liquid phase far from the plate is maintained at constant temperature T1 where T1 > Tm. In addition, the temperature of the solid medium far from the interface is constant and is denoted by TS where TS < Tm. Under the foregoing assumptions, the equations of motion and the equation representing temperature distribution in the liquid flow must obey the usual boundary layer equations @u @v þ ¼0 @x @y
ð1Þ
u
@u @u @ 2 u rB2 ðxÞ þv ¼m 2 ðu U1 Þ @x @x @y q
ð2Þ
u
@T @T @2T 1 @qr þv ¼a 2 @x @x @y qcp @y
ð3Þ
where u, v are velocity components along x, y-axis respectively, m is the kinematic viscosity of the fluid, r is the electrical conductivity of the fluid, T is the temperature of the fluid within the boundary layer, a is the thermal conductivity of the fluid, cp is the specific heat at constant pressure p, q is the density of the fluid and qr is the radiative heat flux. The boundary conditions necessary to complete the problem formulations are [13,17] u ¼ Uw ðxÞ; T ¼ Tm for y ¼ 0 ð4Þ u ! U1 ; T ! T1 as y ! 1 and @T j ¼ q½k þ cS ðTm TS Þvðx; 0Þ @y y¼0
ð5Þ
where U1 is the free stream velocity, k is the thermal conductivity, k is the latent heat of the fluid and cS is the heat capacity of the solid surface. Eq. (5) states that the heat conducted to the melting surface is equal to the heat of melting plus the sensible heat required raising the solid surface temperature TS to its melting temperature Tm [13]. Using the Rosseland approximation, the radiative heat flux term qr is given by qr ¼
4r @T4 3k @y
ð6Þ
where r* is the Stefan–Boltzmann constant and k* is the mean absorption coefficient. Assuming that the differences in temperature within the flow are such that T4 can be expressed as a linear combination of the temperature, we expand T4 in Taylor’s series about T1 and neglecting higher order terms, we get T4 ¼ 4T31 T 3T41
ð7Þ
Thus, we have @qr 16T31 r @ 2 T ¼ @y 3k @y2
ð8Þ
The momentum and energy Eqs. (1)–(3) can be transformed into the corresponding ordinary differential equations by using the following similarity transformations: rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi U1 T Tm ; wðx; yÞ ¼ mxU1 fðgÞ; hðgÞ ¼ g¼y ð9Þ mx T1 Tm where g is the similarity variable, f(g) is the dimensionless stream function and h(g) is the dimensionless temperature. Further, w is the stream function defined as u ¼ @w and @y v ¼ @w , which identically satisfies Eq. (1). @x In terms of new variables (9), Eqs. (2) and (3) can be rewritten as: f 000 þ 0:5ff 00 Ha2 ð f 0 1Þ ¼ 0
ð10Þ
ð1 þ Nr Þh00 þ Prfh0 ¼ 0
ð11Þ
where primes denote differentiation with respect to the similarqffiffiffiffiffiffiffi rx ity variable g, Ha ¼ BðxÞ qU is the magnetic field parameter, 1 3
1r Nr ¼ 16T is the thermal radiation parameter, Pr ¼ la is the 3k j Prandtl number. The momentum Eq. (10) and energy Eq. (11) to have a similarity solution, the magnetic field parameter Ha must be a constant. Therefore, if we assume the applied 1 magnetic field B(x) is proportional to x2 Helmy [26], then Ha
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
Radiation and melting effects on MHD boundary layer flow over a moving surface
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will be independent of x. We therefore assume that 1 BðxÞ ¼ B0 x2 where B0 is constant. The corresponding boundary conditions (4) and (5) become 0 f ð0Þ ¼ e; hð0Þ ¼ 0; Pr fð0Þ þ Mh0 ð0Þ ¼ 0; ð12Þ f 0 ð1Þ ¼ 1; hð1Þ ¼ 1
present results with the results put forward by Ishak et al. [17], the temperature profiles versus boundary layer coordinate g have been plotted in Fig. 1 for various values of melting parameter M in the absence of magnetic field and thermal radiation and taking Pr = 1 and e = 0.2. It is observed that the results agree very well with those of Ishak et al. [17].
where e ¼ UU1w is the moving parameter and M is the dimensionless melting parameter which is defined as
4. Numerical results and discussion
M¼
cf ðT1 Tm Þ k þ cS ðTm TS Þ
ð13Þ
It should be noted that the melting parameter M is a comc ðT T Þ bination of the Stefan number f 1k m and cS ðTmkTS Þ for the liquid and solid phases respectively. The physical quantities of interest are the skin friction coefficient (rate of shear stress) Cf and the local Nusselt number (rate of heat transfer) Nu which are important for this kind of flow. They are defined as follows: The equation defining the wall shear stress is @u sw ¼ l ð14Þ @y y¼0 So the local skin friction coefficient on the surface can be expressed as sw 1 00 2 Cf ¼ 2 ¼ Re ð15Þ x f ð0Þ que The quantity of heat transfer through the unit area of the surface is given by @T 4r @T4 qw ¼ j ð16Þ @y y¼0 3k @y y¼0 In practical applications, the rate of heat transfer is usually expressed as the local Nusselt number 1
Nu ¼ Re2x ð1 þ NrÞh0 ð0Þ
ð17Þ
U1 x m
where Rex ¼ is the local Reynolds number. It is worth mentioning that in the absence of magnetic field parameter Ha and thermal radiation parameter Nr, the problem reduces to those considered by Ishak et al. [17]. 3. Method of solution The set of non-linear differential Eqs. (10) and (11) is highly nonlinear and coupled and cannot be solved analytically. The numerical solutions of Eqs. (10) and (11) subject to the boundary conditions (12) are obtained using MATLAB BVP solver bvp4c. The asymptotic boundary conditions in (12) at g fi 1 are replaced by those at a large but finite value of g where no considerable variation in velocity, temperature etc. occur as is usually the standard practice in the boundary layer analysis. The procedure is repeated until we get the results up to the desired degree of accuracy, 106.
The effect of melting phenomenon, magnetic field and thermal radiation on the velocity and temperature distribution is depicted in Fig. 2–13 whereas the values of skin friction coefficient and local Nusselt number are presented in Table 1. In the present study, numerical computations are carried out for Pr = 1 while M, Ha, e and Nr are varied over a range which are listed in the figure legends. 4.1. Effect of melting parameter M The effect of the melting parameter M can be understood from the variation in the streamwise velocity component f 0 (g) with the similarity independent variable g as illustrated in Figs. 2 and 3. Since the present problem code has been solved at a specific location on the wall, a large value of g is relevant to a location far from the surface at a specified x-coordinate. It is observed from Fig. 2 that the fluid velocity enhances with the increase in the melting parameter M within the boundary layer region, when the moving parameter e(>0), and satisfies the far field boundary conditions (12) asymptotically, thus supporting the validity of the numerical results obtained. This increases the boundary layer thickness and as a result, the local velocity also accelerates. This happens due to the movement of the solid surface and the free stream in the same directions. But the reverse effect occurs for M when the moving parameter e(<0). Figs. 4 and 5 depict the effect of the melting parameter M on temperature distribution for moving surface. The dimensionless temperature profiles h(g) increase monotonically from Tm (i.e. h = 0) at the surface to T1 (i.e. h = 1) at infinity. With increase in the melting parameter M the temperature h(g) decreases across the boundary layer for both values of the moving parameter e > 0 and e < 0. Consequently, more intense melting tends to thicken the thermal boundary layer. Table 1 depicts the effect of the melting parameter M on the skin friction coefficient and the local Nusselt number. It is
3.1. Verification of the results In the absence of magnetic field and thermal radiation, the non-dimensional governing Eqs. (10) and (11) with the corresponding boundary conditions (12) exactly coincide with those (Eqs. (7)–(9)) of Ishak et al. [17]. In order to compare the
Figure 1 Temperature profiles in the absence of magnetic field and thermal radiation.
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
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K. Das
Figure 2 Velocity profiles for various values of M when e = 2.0, Nr = 0.5, Ha = 0.2.
Figure 5 Temperature profiles for various values of M when e = 2.0, Nr = 0.5, Ha = 0.2.
Figure 3 Velocity profiles for various values of M when e = 2.0, Nr = 0.5, Ha = 0.2.
Figure 6 Velocity profiles for various values of Ha when e = 2.0, Nr = 0.5, M = 1.0.
Figure 4 Temperature profiles for various values of M when e = 2.0, Nr = 0.5, Ha = 0.2.
Figure 7 Velocity profiles for various values of Ha when e = 2.0, Nr = 0.5, M = 1.0.
observed from the table that as M increases, the skin friction coefficient (in absolute sense) decreases. On the other hand, the local Nusselt number is negative for all values of M, as expected. Negative value of Nu indicates heat flow from the fluid to the solid surface. This fact is obvious since the fluid is hotter than the solid surface. Further, it is worth noticing from table that the local Nusselt number (in absolute sense) is higher in the absence of melting than in the presence of melting effect. Thus, increasing the melting parameter M decreases the heat transfer rate at the solid–fluid interface. These results are found to be identical to the results of Ishak et al. [17].
4.2. Effect of magnetic field parameter Ha The influence of magnetic field parameter Ha on the velocity distribution of a conducting fluid is presented in Figs. 6 and 7. As the parameter value of Ha increases in the presence of thermal radiation and moving surface, the flow rate retards and thereby leads to a decrease in the velocity profiles as shown in figures. The reason behind this phenomenon is that application of magnetic field to an electrically conducting fluid gives rise to a resistive type force called the Lorentz force. This
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
Radiation and melting effects on MHD boundary layer flow over a moving surface
Figure 8 Temperature profiles for various values of Ha when e = 2.0, Nr = 0.5, M = 1.0.
Figure 9 Temperature profiles for various values of Ha when e = 2.0, Nr = 0.5, M = 1.0.
Figure 10 Velocity profiles for various values of Nr when e = 2.0, Ha = 0.2, M = 1.0.
force has the tendency to slow down the motion of the conducting fluid in the boundary layer. Figs. 8 and 9 illustrate the temperature distribution for various values of the magnetic field parameter Ha. It is observed that the fluid temperature is minimum near the boundary layer region and it increases on increasing boundary layer coordinate g to approach the free stream value. For the electrically conducting fluid, the temperature decreases uniformly on increasing Ha in the presence of thermal radiation and for e > 0 and, as a consequence, thickness of the thermal boundary layer decreases. On the other hand, when the solid surface and free stream move in opposite
5
Figure 11 Velocity profiles for various values of Nr when e = 2.0, Ha = 0.2, M = 1.0.
Figure 12 Temperature profiles for various values of Nr when e = 2.0, Ha = 0.2, M = 1.0.
Figure 13 Temperature profiles for various values of Nr when e = 2.0, Ha = 0.2, M = 1.0.
directions, i.e., when e < 0, the fluid temperature decreases drastically in the presence of the magnetic field, as shown in Fig. 9. Table 1 predicts the influence of magnetic field parameter Ha on the skin friction coefficient and heat transfer rate in the presence of the thermal radiation. It may be remarked from the table that the skin friction coefficient increases with the increase in magnetic field parameter Ha in the presence of melting when the solid surface and free stream move in the same direction, i.e., when e > 0, whereas an opposite effect can be observed for skin friction coefficient when e < 0. An increase in the values of the magnetic field parameter Ha
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
6
K. Das Table 1
Effects of e, M, Nr and Ha on Cf and Nu.
e
M
Nr
Ha
Cf
Nu
2.0
0.0 1.0 2.0 1.0
0.5
0.2
1.434290 1.056146 0.859014 1.018122 1.087240 1.133311 0.684993 1.056141 1.383031 0.198220 0.119974 0.094834 0.125195 0.116827 0.113107 0.228420 0.169974 0.125747
1.165641 0.845626 0.679126 0.626344 1.027126 1.322133 0.889336 0.845626 0.802658 0.317664 0.190109 0.140986 0.116403 0.266503 0.423814 0.295695 0.190109 0.164744
0.2
0.0 1.0 2.0 1.0
0.0 0.5 1.0 0.5
0.5
0.0 0.5 1.0 0.5
0.0 0.2 0.4 0.2
0.0 0.2 0.4
results in the reduction of heat transfer rate (in absolute sense) on the moving surface for both e > 0 and e < 0. It is worth mentioning that the presence of magnetic field produces smaller values of the Nusselt number (in absolute sense) compared with the values when the magnetic field is absent.
increasing thermal radiation parameter in the presence of magnetic field and so the thickness of the thermal boundary layer decreases. Fig. 13 shows the pattern of the temperature distribution for different values of the thermal radiation parameter Nr. It can be observed that the temperature of the fluid decreases as Nr increases far away from the plate but the effect is not significant adjacent to the surface of the plate. The influence of thermal radiation parameter Nr on the skin friction coefficient and the Nusselt number is presented in tabular form in Table 1. An increase in the values of the thermal radiation parameter Nr results in the reduction of the skin friction coefficient but the effect is negligible. It is seen that the absolute value of the temperature gradient at the surface is higher for higher values of the thermal radiation parameter Nr and heat transfer rate from the surface enhances due to the presence of thermal radiation. 4.4. Effect of moving parameter e Figs. 14 and 15 depict the effect of e on velocity profiles of an electrically conducting fluid in the presence of transverse magnetic field. The results show that the momentum boundary layer thickness increases with increasing the values of moving parameter. Figs. 16 and 17 highlights the influence of moving parameter on temperature profiles in the boundary layer region in the presence of thermal radiation. It is found from these plots that the fluid temperature enhances for both
4.3. Effect of thermal radiation parameter Nr The impact of the thermal radiation parameter Nr in the presence of the magnetic field on the velocity profiles of an electrically conducting fluid for a moving surface is presented in Figs. 10 and 11. It can be easily seen from figure that the fluid velocity decreases as the boundary layer coordinate g increases for a fixed value of Nr and e > 0 but the effect is opposite for e < 0. For a fixed non-zero value of g, the velocity distribution across the boundary layer decreases with increase in values of Nr. The physics behind this is that the increased radiation decreases the thickness of the momentum boundary layer, which ultimately reduces the velocity. The effect of thermal radiation parameter Nr on the fluid temperature is illustrated in Figs. 12 and 13. From Fig. 12 it can be seen that the temperature distribution decreases uniformly with
Figure 15 Velocity profiles for various values of e(<0) when Ha = 0.2, M = 1.0, Nr = 0.5.
Figure 14 Velocity profiles for various values of e(>0) when Ha = 0.2, M = 1.0, Nr = 0.5.
Figure 16 Temperature profiles for various values of e(>0) when Ha = 0.2, M = 1.0, Nr = 0.5.
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
Radiation and melting effects on MHD boundary layer flow over a moving surface
Figure 17 Temperature profiles for various values of e(<0) when Ha = 0.2, M = 1.0, Nr = 0.5.
e > 0 and e < 0 and, as a consequence, thickness of the thermal boundary layer increases. 5. Conclusions In this work, the melting effect on MHD boundary layer flow of an incompressible electrically conducting fluid with thermal radiation near the moving surface is investigated numerically. Following conclusion can be drawn from the present investigation: Increasing the magnetic field and the thermal radiation parameter leads to deceleration of the fluid velocity but the effect is reverse for the melting parameter when the solid surface and the free stream move in the same direction. The fluid temperature and the thermal boundary layer thickness decrease for increasing thermal radiation, melting parameter and magnetic field whereas reverse effect occurs for moving parameter. It is seen that the skin friction coefficient decreases with the increase in the melting parameter, whereas the effect is opposite for magnetic field and the thermal radiation parameter when the solid surface and the free stream move in the same direction. Moreover, it is observed that with increase in the melting strength and magnetic field, rate of heat transfer at the solid–fluid interface decreases but the reverse effect occurs for the thermal radiation.
Acknowledgements The author wish to express his very sincere thanks to the reviewers for their valuable suggestions and comments to improve the presentation of this article. References [1] Vives V, Perry C. Effects of magnetically damped convection during the controlled solidification of metals and alloys. Int J Heat Mass Transf 1987;30:479–96. [2] Series RW, Hurle DTJ. The use of magnetic fields in semiconductor crystal growth. J Cryst Growth 1991;113:305–28. [3] Tagawa T, Ozoe H. Enhancement of heat transfer rate by application of a static magnetic field during natural convection of liquid Metal in a cube. ASME J Heat Transf 1997;119:265–71.
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[4] Davidson PA. Magnetohydrodynamics in materials processing. Annu Rev Fluid Mech 1999;31:273–300. [5] Hayat T, Naz R, Alsaedi A, Rashidi MM. Hydromagnetic rotating flow of third grade fluid. Appl Math Mech (English Edition) 2013;34(12):1481–94. [6] Sakiadis BC. Boundary layer behavior on continuous solid surfaces. II: The boundary layer on a continuous flat surface. AIChE J 1961;7:221–5. [7] Tsou FK, Sparrow EM, Goldstein RJ. Flow and heat transfer in the boundary layer on a continuous moving surface. Int J Heat Mass Transf 1967;10:219–35. [8] Afzal N, Varshney IS. The cooling of a low heat resistance stretching sheet moving through a fluid. Wa rme-und Stoffu Bertragung 1980;14:289–93. [9] Hussaini MY, Lakin WD, Nachman A. On similarity solutions of a boundary layer problem with an upstream moving wall. SIAM J Appl Math 1987;47:699–709. [10] Afzal N, Badaruddin A, Elgarvi AA. Momentum and transport on a continuous flat surface moving in a parallel stream. Int J Heat Mass Transf 1993;36:3399–403. [11] Ishak A, Nazar R, Pop I. Boundary-layer flow of a micropolar fluid on a continuous moving or fixed surface. Can J Phys 2006;84:399–410. [12] Ishak A, Nazar R, Pop I. Boundary layer on a moving wall with suction and injection. Chin Phys Lett 2007;24:2274–6. [13] Epstein M, Cho DH. Melting heat transfer in steady laminar flow over a flat plate. J Heat Transfer 1976;98:531–3. [14] Kazmierczak M, Poulikakos D, Sadowski D. Melting of a vertical plate in porous medium controlled by forced convection of a dissimilar fluid. Int Commun Heat Mass Transfer 1987;14:507–17. [15] Kazmierczak M, Poulikakos D, Pop I. Melting from a flat plate in a porous medium in the presence of steady convection. Numer Heat Transfer 1986;10:571–81. [16] Cheng WT, Lin CH. Transient mixed convection heat transfer with melting effect from the vertical plate in a liquid saturated porous medium. Int J Eng Sci 2006;44:1023–36. [17] Ishak A, Nazar R, Bachok N, Pop I. Melting heat transfer in steady laminar flow over a moving surface. Heat Mass Transfer 2010;46:463–8. [18] Cogley AC, Vincenty WE, Gilles SE. Differential approximation for radiation in a non-gray gas near equilibrium. AIAA J 1968;6:551–3. [19] Raptis A. Radiation and free convection flow through a porous medium. Int Commun Heat Mass Transfer 1998;25:289–95. [20] Makinde OD. Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate. Int Commun Heat Mass Transfer 2005;32:1411–9. [21] Ibrahim FS, Elaiw AM, Bakr AA. Influence of viscous dissipation and radiation on unsteady MHD mixed convection flow of micropolar fluids. Appl Math Inform Sci 2008;2:143–62. [22] Das K. Impact of thermal radiation on MHD slip flow over a flate plate with variable fluid properties. Heat Mass Transfer 2011;48(5):767–78. [23] Hayat T, Abbas Z, Pop I, Asghar S. 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 Transf 2010;53:466–74. [24] Bhattacharyya K, Layek GC. 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 Transf 2011;54(1–3):302–7. [25] Hayat T, Shehzad SA, Alsaedi A. Three-dimensional stretched flow of Jeffrey fluid with variable thermal conductivity and thermal radiation. Appl Math Mech (English Edition) 2013;34(12):1481–94. [26] Helmy KA. MHD boundary layer equations for power law fluids with variable electric conductivity. Meccanica 1995;30:187–200.
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
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K. Das Dr. Kalidas Das has completed his Ph.D. degree in Fluid Mechanics (MHD) from the University of Kalyani in 1997. He is now in the position of Assistant Professor (senior) of Mathematics and Head, Dept. of Mathematics, Kalyani Government Engineering college, West Bengal, India. So far he had 65 research papers published in National and International journals to his credit in the fields of fluid mechanics (MHD, CFD and nanofluids) and bio-mechanics. Three students have
completed their research work and two students are still being guided by him at the moment. He is the author and also co-author of many books on graduate (Hons.) and under graduate level.
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
Ain Shams University
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MECHANICAL ENGINEERING
Radiation and melting effects on MHD boundary layer flow over a moving surface Kalidas Das
*
Department of Mathematics, Kalyani Government Engineering College, Kalyani, Nadia, West Bengal 741235, India Received 28 December 2013; revised 16 March 2014; accepted 12 April 2014
KEYWORDS Melting; MHD; Thermal radiation; Moving surface
Abstract This paper presents a mathematical analysis of MHD flow and heat transfer from a warm, electrically conducting fluid to melting surface moving parallel to a constant free stream in the presence of thermal radiation. The similarity transformation technique is used to convert the governing partial differential equations into the self-similar ordinary differential equations and then solved numerically using MATLAB BVP solver bvp4c. Numerical results for the dimensionless velocity and temperature profiles as well as for the skin friction and the Nusselt number are elucidated for different values of the pertinent parameters. Comparison with previously published work is presented and it found to be in excellent agreement. 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.
1. Introduction In the recent years, the effects of electrically conducting fluids, such as liquid metals, water and others on the flow and heat transfer in the presence of external magnetic field were discussed by many researchers [1–5]. The flow and heat transfer characteristics induced by a continuously moving surface are important in industrial engineering processes such as in the lamination and melt-spinning process, in the polymer industry, continuous casting, and spinning of fibers. The steady boundary layer flow due to a continuously moving surface was first considered by Sakiadis [6]. Thereafter, this problem was extended by a very good number of researchers [7–12] * Mobile: +91 9748603199. E-mail address: [email protected]. Peer review under responsibility of Ain Shams University.
Production and hosting by Elsevier
in various aspects for both Newtonian and non-Newtonian fluids. Heat transfer accompanied by melting phenomenon has recently received considerable research attention. This is due to large number of applications, including latent heat storage, material processing, crystal growth, castings of metals, glass industry, purification of materials, and others. Melting heat transfer from a flat plate was investigated by Epstein and Cho [13]. On the other hand melting from a vertical flat plate embedded in a porous medium was studied by Kazmierczak et al. [14,15]. Cheng and Lin [16] discussed the melting effect on convective heat transfer from a vertical plate in a liquid saturated porous medium. Recently, Ishak et al. [17] studied the heat transfer characteristic from a warm liquid to a melting and moving surface. However, the effect of thermal radiation on the heat transfer processes is very important in high operating temperature and has many important applications such as nuclear reactor cooling system, gas turbines and various propulsion devices or aircraft, missiles, satellites and space vehicles. Cogley et al. [18] observed that the fluid does not absorb its own emitted radiation in the optically thin limit but absorb radiation
2090-4479 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2014.04.008 Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
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K. Das
emitted by the boundaries. The free convection flow through a porous medium bounded by a porous plate in the presence of thermal radiation was investigated by Raptis [19]. Makinde [20] examined the transient free convection along moving vertical porous plate in the presence of thermal radiation. Ibrahim et al. [21] discussed the effect of thermal radiation on mixed convection flow. On the other hand, the fluid can be treated as electrically conducting in the sense that it is ionized due to the high operating temperature. Accordingly, the influence of magnetic field on the fluid flow characteristics is important in the presence of thermal radiation. Das [22] investigated the effect of thermal radiation on fluid flow over a flat plate in the presence of magnetic field. Hayat et al. [23] studied mixed convection MHD boundary layer flow through a porous medium in the presence of thermal radiation. The effects of thermal radiation on boundary layer flow and heat transfer toward a shrinking sheet were examined by Bhattacharyya and Layek [24]. Hayat et al. [25] discussed stretched flow of Jeffrey fluid in the presence of thermal radiation. The main purpose of the present investigation is to illustrate the effects of magnetic field and thermal radiation on steady laminar boundary layer flow and heat transfer from a warm electrically conducting fluid to a melting surface moving parallel to a constant free stream. The organization of the paper is as follows. Section 2 discusses the mathematical formulation of the problem. Numerical method is provided in Section 3. Section 4 consists of results and discussion. The concluding remarks are given in Section 5. 2. Mathematical formulation of the problem Consider a steady laminar flow of an incompressible electrically conducting fluid toward a moving surface melting at a steady rate into a warm liquid of the same material with constant property under the influence of a transverse magnetic ! field B . The external electric field is assumed to be zero and the magnetic Reynolds number is assumed to be small. Hence, the induced magnetic field is negligible compared with the ! externally applied magnetic field so that B = [0, B(x)] where B(x) is the applied magnetic field in the positive direction of y-axis and varies in strength as a function of x. The flow is assumed to be in the x-direction which is taken along the moving surface and y-axis is normal to it. It is assumed that the surface is moving with a constant velocity Uw. It is also assumed that the temperature of the melting surface is Tm. The liquid phase far from the plate is maintained at constant temperature T1 where T1 > Tm. In addition, the temperature of the solid medium far from the interface is constant and is denoted by TS where TS < Tm. Under the foregoing assumptions, the equations of motion and the equation representing temperature distribution in the liquid flow must obey the usual boundary layer equations @u @v þ ¼0 @x @y
ð1Þ
u
@u @u @ 2 u rB2 ðxÞ þv ¼m 2 ðu U1 Þ @x @x @y q
ð2Þ
u
@T @T @2T 1 @qr þv ¼a 2 @x @x @y qcp @y
ð3Þ
where u, v are velocity components along x, y-axis respectively, m is the kinematic viscosity of the fluid, r is the electrical conductivity of the fluid, T is the temperature of the fluid within the boundary layer, a is the thermal conductivity of the fluid, cp is the specific heat at constant pressure p, q is the density of the fluid and qr is the radiative heat flux. The boundary conditions necessary to complete the problem formulations are [13,17] u ¼ Uw ðxÞ; T ¼ Tm for y ¼ 0 ð4Þ u ! U1 ; T ! T1 as y ! 1 and @T j ¼ q½k þ cS ðTm TS Þvðx; 0Þ @y y¼0
ð5Þ
where U1 is the free stream velocity, k is the thermal conductivity, k is the latent heat of the fluid and cS is the heat capacity of the solid surface. Eq. (5) states that the heat conducted to the melting surface is equal to the heat of melting plus the sensible heat required raising the solid surface temperature TS to its melting temperature Tm [13]. Using the Rosseland approximation, the radiative heat flux term qr is given by qr ¼
4r @T4 3k @y
ð6Þ
where r* is the Stefan–Boltzmann constant and k* is the mean absorption coefficient. Assuming that the differences in temperature within the flow are such that T4 can be expressed as a linear combination of the temperature, we expand T4 in Taylor’s series about T1 and neglecting higher order terms, we get T4 ¼ 4T31 T 3T41
ð7Þ
Thus, we have @qr 16T31 r @ 2 T ¼ @y 3k @y2
ð8Þ
The momentum and energy Eqs. (1)–(3) can be transformed into the corresponding ordinary differential equations by using the following similarity transformations: rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi U1 T Tm ; wðx; yÞ ¼ mxU1 fðgÞ; hðgÞ ¼ g¼y ð9Þ mx T1 Tm where g is the similarity variable, f(g) is the dimensionless stream function and h(g) is the dimensionless temperature. Further, w is the stream function defined as u ¼ @w and @y v ¼ @w , which identically satisfies Eq. (1). @x In terms of new variables (9), Eqs. (2) and (3) can be rewritten as: f 000 þ 0:5ff 00 Ha2 ð f 0 1Þ ¼ 0
ð10Þ
ð1 þ Nr Þh00 þ Prfh0 ¼ 0
ð11Þ
where primes denote differentiation with respect to the similarqffiffiffiffiffiffiffi rx ity variable g, Ha ¼ BðxÞ qU is the magnetic field parameter, 1 3
1r Nr ¼ 16T is the thermal radiation parameter, Pr ¼ la is the 3k j Prandtl number. The momentum Eq. (10) and energy Eq. (11) to have a similarity solution, the magnetic field parameter Ha must be a constant. Therefore, if we assume the applied 1 magnetic field B(x) is proportional to x2 Helmy [26], then Ha
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
Radiation and melting effects on MHD boundary layer flow over a moving surface
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will be independent of x. We therefore assume that 1 BðxÞ ¼ B0 x2 where B0 is constant. The corresponding boundary conditions (4) and (5) become 0 f ð0Þ ¼ e; hð0Þ ¼ 0; Pr fð0Þ þ Mh0 ð0Þ ¼ 0; ð12Þ f 0 ð1Þ ¼ 1; hð1Þ ¼ 1
present results with the results put forward by Ishak et al. [17], the temperature profiles versus boundary layer coordinate g have been plotted in Fig. 1 for various values of melting parameter M in the absence of magnetic field and thermal radiation and taking Pr = 1 and e = 0.2. It is observed that the results agree very well with those of Ishak et al. [17].
where e ¼ UU1w is the moving parameter and M is the dimensionless melting parameter which is defined as
4. Numerical results and discussion
M¼
cf ðT1 Tm Þ k þ cS ðTm TS Þ
ð13Þ
It should be noted that the melting parameter M is a comc ðT T Þ bination of the Stefan number f 1k m and cS ðTmkTS Þ for the liquid and solid phases respectively. The physical quantities of interest are the skin friction coefficient (rate of shear stress) Cf and the local Nusselt number (rate of heat transfer) Nu which are important for this kind of flow. They are defined as follows: The equation defining the wall shear stress is @u sw ¼ l ð14Þ @y y¼0 So the local skin friction coefficient on the surface can be expressed as sw 1 00 2 Cf ¼ 2 ¼ Re ð15Þ x f ð0Þ que The quantity of heat transfer through the unit area of the surface is given by @T 4r @T4 qw ¼ j ð16Þ @y y¼0 3k @y y¼0 In practical applications, the rate of heat transfer is usually expressed as the local Nusselt number 1
Nu ¼ Re2x ð1 þ NrÞh0 ð0Þ
ð17Þ
U1 x m
where Rex ¼ is the local Reynolds number. It is worth mentioning that in the absence of magnetic field parameter Ha and thermal radiation parameter Nr, the problem reduces to those considered by Ishak et al. [17]. 3. Method of solution The set of non-linear differential Eqs. (10) and (11) is highly nonlinear and coupled and cannot be solved analytically. The numerical solutions of Eqs. (10) and (11) subject to the boundary conditions (12) are obtained using MATLAB BVP solver bvp4c. The asymptotic boundary conditions in (12) at g fi 1 are replaced by those at a large but finite value of g where no considerable variation in velocity, temperature etc. occur as is usually the standard practice in the boundary layer analysis. The procedure is repeated until we get the results up to the desired degree of accuracy, 106.
The effect of melting phenomenon, magnetic field and thermal radiation on the velocity and temperature distribution is depicted in Fig. 2–13 whereas the values of skin friction coefficient and local Nusselt number are presented in Table 1. In the present study, numerical computations are carried out for Pr = 1 while M, Ha, e and Nr are varied over a range which are listed in the figure legends. 4.1. Effect of melting parameter M The effect of the melting parameter M can be understood from the variation in the streamwise velocity component f 0 (g) with the similarity independent variable g as illustrated in Figs. 2 and 3. Since the present problem code has been solved at a specific location on the wall, a large value of g is relevant to a location far from the surface at a specified x-coordinate. It is observed from Fig. 2 that the fluid velocity enhances with the increase in the melting parameter M within the boundary layer region, when the moving parameter e(>0), and satisfies the far field boundary conditions (12) asymptotically, thus supporting the validity of the numerical results obtained. This increases the boundary layer thickness and as a result, the local velocity also accelerates. This happens due to the movement of the solid surface and the free stream in the same directions. But the reverse effect occurs for M when the moving parameter e(<0). Figs. 4 and 5 depict the effect of the melting parameter M on temperature distribution for moving surface. The dimensionless temperature profiles h(g) increase monotonically from Tm (i.e. h = 0) at the surface to T1 (i.e. h = 1) at infinity. With increase in the melting parameter M the temperature h(g) decreases across the boundary layer for both values of the moving parameter e > 0 and e < 0. Consequently, more intense melting tends to thicken the thermal boundary layer. Table 1 depicts the effect of the melting parameter M on the skin friction coefficient and the local Nusselt number. It is
3.1. Verification of the results In the absence of magnetic field and thermal radiation, the non-dimensional governing Eqs. (10) and (11) with the corresponding boundary conditions (12) exactly coincide with those (Eqs. (7)–(9)) of Ishak et al. [17]. In order to compare the
Figure 1 Temperature profiles in the absence of magnetic field and thermal radiation.
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
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K. Das
Figure 2 Velocity profiles for various values of M when e = 2.0, Nr = 0.5, Ha = 0.2.
Figure 5 Temperature profiles for various values of M when e = 2.0, Nr = 0.5, Ha = 0.2.
Figure 3 Velocity profiles for various values of M when e = 2.0, Nr = 0.5, Ha = 0.2.
Figure 6 Velocity profiles for various values of Ha when e = 2.0, Nr = 0.5, M = 1.0.
Figure 4 Temperature profiles for various values of M when e = 2.0, Nr = 0.5, Ha = 0.2.
Figure 7 Velocity profiles for various values of Ha when e = 2.0, Nr = 0.5, M = 1.0.
observed from the table that as M increases, the skin friction coefficient (in absolute sense) decreases. On the other hand, the local Nusselt number is negative for all values of M, as expected. Negative value of Nu indicates heat flow from the fluid to the solid surface. This fact is obvious since the fluid is hotter than the solid surface. Further, it is worth noticing from table that the local Nusselt number (in absolute sense) is higher in the absence of melting than in the presence of melting effect. Thus, increasing the melting parameter M decreases the heat transfer rate at the solid–fluid interface. These results are found to be identical to the results of Ishak et al. [17].
4.2. Effect of magnetic field parameter Ha The influence of magnetic field parameter Ha on the velocity distribution of a conducting fluid is presented in Figs. 6 and 7. As the parameter value of Ha increases in the presence of thermal radiation and moving surface, the flow rate retards and thereby leads to a decrease in the velocity profiles as shown in figures. The reason behind this phenomenon is that application of magnetic field to an electrically conducting fluid gives rise to a resistive type force called the Lorentz force. This
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
Radiation and melting effects on MHD boundary layer flow over a moving surface
Figure 8 Temperature profiles for various values of Ha when e = 2.0, Nr = 0.5, M = 1.0.
Figure 9 Temperature profiles for various values of Ha when e = 2.0, Nr = 0.5, M = 1.0.
Figure 10 Velocity profiles for various values of Nr when e = 2.0, Ha = 0.2, M = 1.0.
force has the tendency to slow down the motion of the conducting fluid in the boundary layer. Figs. 8 and 9 illustrate the temperature distribution for various values of the magnetic field parameter Ha. It is observed that the fluid temperature is minimum near the boundary layer region and it increases on increasing boundary layer coordinate g to approach the free stream value. For the electrically conducting fluid, the temperature decreases uniformly on increasing Ha in the presence of thermal radiation and for e > 0 and, as a consequence, thickness of the thermal boundary layer decreases. On the other hand, when the solid surface and free stream move in opposite
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Figure 11 Velocity profiles for various values of Nr when e = 2.0, Ha = 0.2, M = 1.0.
Figure 12 Temperature profiles for various values of Nr when e = 2.0, Ha = 0.2, M = 1.0.
Figure 13 Temperature profiles for various values of Nr when e = 2.0, Ha = 0.2, M = 1.0.
directions, i.e., when e < 0, the fluid temperature decreases drastically in the presence of the magnetic field, as shown in Fig. 9. Table 1 predicts the influence of magnetic field parameter Ha on the skin friction coefficient and heat transfer rate in the presence of the thermal radiation. It may be remarked from the table that the skin friction coefficient increases with the increase in magnetic field parameter Ha in the presence of melting when the solid surface and free stream move in the same direction, i.e., when e > 0, whereas an opposite effect can be observed for skin friction coefficient when e < 0. An increase in the values of the magnetic field parameter Ha
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K. Das Table 1
Effects of e, M, Nr and Ha on Cf and Nu.
e
M
Nr
Ha
Cf
Nu
2.0
0.0 1.0 2.0 1.0
0.5
0.2
1.434290 1.056146 0.859014 1.018122 1.087240 1.133311 0.684993 1.056141 1.383031 0.198220 0.119974 0.094834 0.125195 0.116827 0.113107 0.228420 0.169974 0.125747
1.165641 0.845626 0.679126 0.626344 1.027126 1.322133 0.889336 0.845626 0.802658 0.317664 0.190109 0.140986 0.116403 0.266503 0.423814 0.295695 0.190109 0.164744
0.2
0.0 1.0 2.0 1.0
0.0 0.5 1.0 0.5
0.5
0.0 0.5 1.0 0.5
0.0 0.2 0.4 0.2
0.0 0.2 0.4
results in the reduction of heat transfer rate (in absolute sense) on the moving surface for both e > 0 and e < 0. It is worth mentioning that the presence of magnetic field produces smaller values of the Nusselt number (in absolute sense) compared with the values when the magnetic field is absent.
increasing thermal radiation parameter in the presence of magnetic field and so the thickness of the thermal boundary layer decreases. Fig. 13 shows the pattern of the temperature distribution for different values of the thermal radiation parameter Nr. It can be observed that the temperature of the fluid decreases as Nr increases far away from the plate but the effect is not significant adjacent to the surface of the plate. The influence of thermal radiation parameter Nr on the skin friction coefficient and the Nusselt number is presented in tabular form in Table 1. An increase in the values of the thermal radiation parameter Nr results in the reduction of the skin friction coefficient but the effect is negligible. It is seen that the absolute value of the temperature gradient at the surface is higher for higher values of the thermal radiation parameter Nr and heat transfer rate from the surface enhances due to the presence of thermal radiation. 4.4. Effect of moving parameter e Figs. 14 and 15 depict the effect of e on velocity profiles of an electrically conducting fluid in the presence of transverse magnetic field. The results show that the momentum boundary layer thickness increases with increasing the values of moving parameter. Figs. 16 and 17 highlights the influence of moving parameter on temperature profiles in the boundary layer region in the presence of thermal radiation. It is found from these plots that the fluid temperature enhances for both
4.3. Effect of thermal radiation parameter Nr The impact of the thermal radiation parameter Nr in the presence of the magnetic field on the velocity profiles of an electrically conducting fluid for a moving surface is presented in Figs. 10 and 11. It can be easily seen from figure that the fluid velocity decreases as the boundary layer coordinate g increases for a fixed value of Nr and e > 0 but the effect is opposite for e < 0. For a fixed non-zero value of g, the velocity distribution across the boundary layer decreases with increase in values of Nr. The physics behind this is that the increased radiation decreases the thickness of the momentum boundary layer, which ultimately reduces the velocity. The effect of thermal radiation parameter Nr on the fluid temperature is illustrated in Figs. 12 and 13. From Fig. 12 it can be seen that the temperature distribution decreases uniformly with
Figure 15 Velocity profiles for various values of e(<0) when Ha = 0.2, M = 1.0, Nr = 0.5.
Figure 14 Velocity profiles for various values of e(>0) when Ha = 0.2, M = 1.0, Nr = 0.5.
Figure 16 Temperature profiles for various values of e(>0) when Ha = 0.2, M = 1.0, Nr = 0.5.
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
Radiation and melting effects on MHD boundary layer flow over a moving surface
Figure 17 Temperature profiles for various values of e(<0) when Ha = 0.2, M = 1.0, Nr = 0.5.
e > 0 and e < 0 and, as a consequence, thickness of the thermal boundary layer increases. 5. Conclusions In this work, the melting effect on MHD boundary layer flow of an incompressible electrically conducting fluid with thermal radiation near the moving surface is investigated numerically. Following conclusion can be drawn from the present investigation: Increasing the magnetic field and the thermal radiation parameter leads to deceleration of the fluid velocity but the effect is reverse for the melting parameter when the solid surface and the free stream move in the same direction. The fluid temperature and the thermal boundary layer thickness decrease for increasing thermal radiation, melting parameter and magnetic field whereas reverse effect occurs for moving parameter. It is seen that the skin friction coefficient decreases with the increase in the melting parameter, whereas the effect is opposite for magnetic field and the thermal radiation parameter when the solid surface and the free stream move in the same direction. Moreover, it is observed that with increase in the melting strength and magnetic field, rate of heat transfer at the solid–fluid interface decreases but the reverse effect occurs for the thermal radiation.
Acknowledgements The author wish to express his very sincere thanks to the reviewers for their valuable suggestions and comments to improve the presentation of this article. References [1] Vives V, Perry C. Effects of magnetically damped convection during the controlled solidification of metals and alloys. Int J Heat Mass Transf 1987;30:479–96. [2] Series RW, Hurle DTJ. The use of magnetic fields in semiconductor crystal growth. J Cryst Growth 1991;113:305–28. [3] Tagawa T, Ozoe H. Enhancement of heat transfer rate by application of a static magnetic field during natural convection of liquid Metal in a cube. ASME J Heat Transf 1997;119:265–71.
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[4] Davidson PA. Magnetohydrodynamics in materials processing. Annu Rev Fluid Mech 1999;31:273–300. [5] Hayat T, Naz R, Alsaedi A, Rashidi MM. Hydromagnetic rotating flow of third grade fluid. Appl Math Mech (English Edition) 2013;34(12):1481–94. [6] Sakiadis BC. Boundary layer behavior on continuous solid surfaces. II: The boundary layer on a continuous flat surface. AIChE J 1961;7:221–5. [7] Tsou FK, Sparrow EM, Goldstein RJ. Flow and heat transfer in the boundary layer on a continuous moving surface. Int J Heat Mass Transf 1967;10:219–35. [8] Afzal N, Varshney IS. The cooling of a low heat resistance stretching sheet moving through a fluid. Wa rme-und Stoffu Bertragung 1980;14:289–93. [9] Hussaini MY, Lakin WD, Nachman A. On similarity solutions of a boundary layer problem with an upstream moving wall. SIAM J Appl Math 1987;47:699–709. [10] Afzal N, Badaruddin A, Elgarvi AA. Momentum and transport on a continuous flat surface moving in a parallel stream. Int J Heat Mass Transf 1993;36:3399–403. [11] Ishak A, Nazar R, Pop I. Boundary-layer flow of a micropolar fluid on a continuous moving or fixed surface. Can J Phys 2006;84:399–410. [12] Ishak A, Nazar R, Pop I. Boundary layer on a moving wall with suction and injection. Chin Phys Lett 2007;24:2274–6. [13] Epstein M, Cho DH. Melting heat transfer in steady laminar flow over a flat plate. J Heat Transfer 1976;98:531–3. [14] Kazmierczak M, Poulikakos D, Sadowski D. Melting of a vertical plate in porous medium controlled by forced convection of a dissimilar fluid. Int Commun Heat Mass Transfer 1987;14:507–17. [15] Kazmierczak M, Poulikakos D, Pop I. Melting from a flat plate in a porous medium in the presence of steady convection. Numer Heat Transfer 1986;10:571–81. [16] Cheng WT, Lin CH. Transient mixed convection heat transfer with melting effect from the vertical plate in a liquid saturated porous medium. Int J Eng Sci 2006;44:1023–36. [17] Ishak A, Nazar R, Bachok N, Pop I. Melting heat transfer in steady laminar flow over a moving surface. Heat Mass Transfer 2010;46:463–8. [18] Cogley AC, Vincenty WE, Gilles SE. Differential approximation for radiation in a non-gray gas near equilibrium. AIAA J 1968;6:551–3. [19] Raptis A. Radiation and free convection flow through a porous medium. Int Commun Heat Mass Transfer 1998;25:289–95. [20] Makinde OD. Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate. Int Commun Heat Mass Transfer 2005;32:1411–9. [21] Ibrahim FS, Elaiw AM, Bakr AA. Influence of viscous dissipation and radiation on unsteady MHD mixed convection flow of micropolar fluids. Appl Math Inform Sci 2008;2:143–62. [22] Das K. Impact of thermal radiation on MHD slip flow over a flate plate with variable fluid properties. Heat Mass Transfer 2011;48(5):767–78. [23] Hayat T, Abbas Z, Pop I, Asghar S. 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 Transf 2010;53:466–74. [24] Bhattacharyya K, Layek GC. 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 Transf 2011;54(1–3):302–7. [25] Hayat T, Shehzad SA, Alsaedi A. Three-dimensional stretched flow of Jeffrey fluid with variable thermal conductivity and thermal radiation. Appl Math Mech (English Edition) 2013;34(12):1481–94. [26] Helmy KA. MHD boundary layer equations for power law fluids with variable electric conductivity. Meccanica 1995;30:187–200.
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008
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K. Das Dr. Kalidas Das has completed his Ph.D. degree in Fluid Mechanics (MHD) from the University of Kalyani in 1997. He is now in the position of Assistant Professor (senior) of Mathematics and Head, Dept. of Mathematics, Kalyani Government Engineering college, West Bengal, India. So far he had 65 research papers published in National and International journals to his credit in the fields of fluid mechanics (MHD, CFD and nanofluids) and bio-mechanics. Three students have
completed their research work and two students are still being guided by him at the moment. He is the author and also co-author of many books on graduate (Hons.) and under graduate level.
Please cite this article in press as: Das K, Radiation and melting effects on MHD boundary layer flow over a moving surface, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.008