- Email: [email protected]
Hall effects on natural convection of participating MHD with thermal radiation in a cavity
International Journal of Heat and Mass Transfer 66 (2013) 838–843
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Hall effects on natural convection of participating MHD with thermal radiation in a cavity Jing-Kui Zhang a,b, Ben-Wen Li a,⇑, Yuan-Yuan Chen a,b a b
Key Laboratory for Ferrous Metallurgy and Resources Utilization of Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China Key Laboratory of Electromagnetic Processing of Materials of Ministry of Education, Northeastern University, Shenyang 110819, China
a r t i c l e
i n f o
Article history: Received 19 January 2013 Received in revised form 28 July 2013 Accepted 28 July 2013 Available online 23 August 2013 Keywords: Thermal radiation Participating MHD Hall effects Square cavity
a b s t r a c t In this work, Hall effects on natural convection of participating MHD with thermal radiation are investigated numerically. An external uniform magnetic field is applied on a square cavity which is filled with participating magnetic fluid. The full filled fluid has characteristics of gray, absorbing, emitting, scattering and electrically conducting. All walls of the cavity are opaque and diffusively reflection. The finite volume method (FVM) is employed to solve the momentum equation and energy equation. The radiative flux which is treated as a heat source term is obtained by solving radiative transfer equation (RTE) instead of the widely used Rosseland approximation. Graphical results and detailed analysis show the Hall effects on fluid flow and heat transfer with thermal radiation under different Pl number and Ha number. Results also show that the Hall effects on participating MHD cannot be neglected in certain range of parameters. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The study of MHD flow and heat transfer with thermal radiation has gained considerable attention during recent years. This is in view of its wide application in engineering and scientific research such as extrusion of molten polymers, MHD generator, space weather forecasting, thermonuclear reaction device, aeronautics, astrophysics and so on. Many excellent works have been presented for thermal radiation effects on natural convection of MHD. Hayat and his partners [1–6] investigated the radiation effects on MHD flow and heat transfer by homotopy analysis method (HAM). They studied the free convection of Maxwell fluid and incompressible second grade fluid over a stretching sheet, in a channel, in a porous space and past a semi-infinite fixed plate, respectively. Some other relevant works [7,8] are also performed by them about the radiative features on different fluids flow. Turkyilmazoglu [9] investigated thermal radiation effects on the time-dependent MHD permeable flow with variable viscosity numerically by a technique based on the spectral Chebyshev collocation in the direction normal to the disk and forward marching in time. Seddeek et al. [10], Mahmoud [11], Cookey et al. [12] and Ghaly [13] found that the rate of heat transfer of magnetic fluid on a plate increased with the increasing of radiation parameter. The interaction of free convection with thermal radiation of a viscous incompressible unsteady flow past a ⇑ Corresponding author. Tel.: +86 136 64102 228. E-mail addresses: [email protected], [email protected] (B.-W. Li). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.07.090
moving vertical cylinder was analyzed by Ganesan and Loganathan [14]. Seddeek [15], Mahmoud [8] and Turkyilmazoglu [9] conducted works on thermal radiation on MHD flow and heat transfer with variable viscosity or thermal conductivity. As we known, just for low values of electrical conductivity, magnetic flux density and velocity, the Hall effects can be neglected. However, in all works mentioned above, it is assumed that the Hall effects were neglected no matter the magnetic flied was strong or weak in their works. In fact, the magnetic field and flow intensity are moderate or even more in some of above works. From [16–21], we can find that the Hall effects have notable impacts on the natural convection of electrically conductive fluid without radiation in certain range of parameters. Thus, investigation of Hall effects on participating MHD flow and heat transfer with thermal radiation is necessary. Furthermore, the radiative heat flux was obtained on one dimension by a same simplified method (Rosseland approximation) in all works above, so, the coupling of radiation and free convection was performed at the expense of accuracy. Accurate calculation has been achieved for coupled free convection and radiative heat transfer in our very recent work [22]. The optical parameters effects on fluid flow and heat transfer of participating MHD was investigated with the assumption that the Hall effects could be neglected in that work, same as in works [1–6,9– 15]. That assumption may result in some possible changes of phenomena on flow field and temperature field. Thereafter, based on our former work [22], investigation is carried out in our present work for different Ha and Pl number to understand the Hall effects on participating MHD with thermal radiation.
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843
839
Fig. 1. Streamlines for different Pl and m with Ha = 50.
Fig. 3. Local Nusselt number on the hot wall for different Pl and m with Ha = 50.
Fig. 2. Isotherms for different Pl and m with Ha = 50.
2. Mathematical formulation and numerical techniques The physical model together with the coordinate system and boundary conditions can be referred to Fig. 1 of [22]. Also the
properties of the media, and the situation considered are similar to those in [22]. We assume the fluid density is constant, except for the density in the buoyancy term which follows the Boussinesq approximation. The induced magnetic field is neglected by assuming a very small magnetic Reynolds number [23]. Through dimensionless exchange continued from [22], the governing equations can be written as
840
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843
Fig. 4. Streamlines for different Ha and m with Pl = 0.02.
@U @V þ ¼0 @X @Y
ð1Þ
walls are hL = 0.5 and hR = +0.5, respectively. On the adiabatic T ref 1 e q c þ T T q ¼ 0. walls, the boundary condition is e 4Pl r c
h
@ 2 @ @P 1 Ha2 U þ ðUVÞ ¼ þ r2 U U @X @Y @X Re Reð1 þ m2 Þ @ @ 2 @P 1 ðUVÞ þ þ r2 V þ Rih V ¼ @X @Y @Y Re T ref @ @ 1 ðUhÞ þ ðVhÞ ¼ r2 h @X @Y RePr Th Tc Z sL ð1 xÞ 4 1 eX Id H RePlPr 4p 4p J 2 U þ ð1 þ m2 Þ
x reI ¼ eI þ ð1 xÞH4 þ sL 4p 1
ð2Þ
eI ðrw ; Xn Þ ¼ ew H4 þ 1 ew
p
ð3Þ
ð4Þ
4p
eI 0 UdX0
ð5Þ
where the definitions of dimensionless variables and numbers have already been given in [22], except for the Hall parameter m and the rB20 U0 L
.
h T c Þ
All the walls are considered to be no-slip for velocity, thus, U = 0 and V = 0 on all walls. The values of temperature on isothermal
N X 0 0 0 I rw ; Xn n Xn wn
0
n Xn < 0
n0 ¼1
ð6Þ where r is the spatial coordinates vector. n is the angular direction of radiation. n is the angular direction vector, and N is the number of discrete-ordinates directions. ew is the emissivity of wall. When the temperature field and radiation intensity are obtained, the conductive and radiative heat flux in non-dimensional forms on the walls can be obtained by
e qc ¼
Z
Joule heating parameter J ¼ qcp ðT
Due to the opaque and diffusively reflection walls, the boundary condition of RTE is given by
dh ; dY C
Z 1 eIdX e q r ¼ ew H4
p
ð7Þ
C
Local conductive, radiative and total Nusselt number Nuc, Nur and Nut are defined as
qc ; Nuc ¼ e
Nur ¼
T ref 1 e qr ; T h T c 4Pl
Nut ¼ Nuc þ Nur
ð8Þ
This work is based on our former work. The physical problem, governing equations, boundary conditions and numerical methods in
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843
841
Fig. 5. Isotherms for different Ha and m with Pl = 0.02.
this work are same as the former, and the grid test and codes validation have been performed in that work which are available in Ref. [22]. 3. Results and discussion Different Ha number, Pl number and Hall parameter m are selected to investigate the Hall effects on fluid flow and heat transfer of participating magnetic fluid. In all the following calculations, Re, Ri, Pr and J are taken as Re = 300, Ri = 1, Pr = 0.733 and J = 1. The temperature of the right vertical wall is set to be Th = 2Tc, and the dimensionless temperature for RTE is obtained by H = 2h/3 + 1. Emissivity of all walls is set to be a uniform value of ew = 0.8. The scattering of the fluid is isotropic and x = 0.5, and the optical thickness is set as sL = 1. 3.1. Different Pl numbers The Pl number means the rate of conduction to radiation as its expression. The results illustrated in Figs. 1–3 demonstrate the Hall effects on fluid flow and heat transfer with different Pl. Ha number is set as constant Ha = 50. It can be clearly seen in Fig. 1 that the flow intensity increases as m increases for 0–3 under a fixed Pl. On the one hand, this is because the Lorentz force term in the NS equations decreases as m increases which can be seen in Eq.
(2), which leads to the decrease of the suppression of magnetic field on MHD flow. On the other hand, with the increasing of m, the Joule heating term is decreased, so, the loss of kinetic energy of the fluid decreases as shown in Eq. (4). For a fixed m, the flow intensity decreases obviously as Pl increases due to the significant decrease of thermal radiation as shown in Fig. 3. The corresponding temperature field is shown in Fig. 2. With the increasing of m, isotherms near the bottom wall move to the right while isotherms near the top wall move to the left when Pl is kept at fixed due to the increasing of flow intensity and the anticlockwise flow direction. When Pl increases, the distribution of isotherms becomes more center symmetric for a fixed m. This is because the convection plays more important role on the heat transfer with the decrease of thermal radiation. In order to reveal the influence of Hall effects on heat transfer on the boundary, the local Nusselt number on the hot wall is selected as shown in Fig. 3. When Pl increases from 0.02 to 2, the values of Nut decrease obviously for a fixed m due to the quick drop of thermal radiation, while, Nuc has different changes with different m. With the increasing of Pl, Nuc increases near the bottom of the hot wall and decreases near the top of the hot wall for a fixed value of m. For a fixed Pl, the average values of Nut, Nur and Nuc are all increases with the increasing of m because of the increase of flow intensity. That means the heat transfer is enhanced with the increasing of Hall parameter. It also can be observed in Fig. 3 that the average values of Nut changes just a little as m increases from 0
842
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843
Hall effects on natural convection of participating MHD under different Ha, where Pl = 0.02 is kept at fixed. Fig. 4 shows the streamlines for different Ha and m. It can be seen that the flow intensity is increased as m increases from 0 to 3 under Ha = 10, 50and 100, respectively. This is also because the Lorentz force term and the Joule heating term decrease as m increases. With the increasing of m, it also can be observed that streamlines distribution and flow intensity change slightly for Ha = 10, and with the increasing of Ha, these changes become more. So, the Hall parameter m plays more important role on fluid flow as Ha increases. When m is kept at fixed, the flow intensity decreases obviously with the increasing of Ha due to increase of the suppression of magnetic field on MHD flow. The influence of Hall effects on the temperature distribution is shown in Fig. 5 under different Ha. For a fixed Ha, isotherms move to the right near the bottom wall and move to the left near the top wall. When m is kept at fixed, isotherms become more paralleled to the vertical walls as Ha increases because of the increase of the suppression of the magnetic field imposed on the fluid as shown in Fig. 4. The rate of heat transfer on the hot wall is shown in Fig. 6 for different Ha and m. It can be seen that the average values of Nuc, Nut and Nur increase as m increases for a fixed Ha. The difference of the average values of Nusselt number is significant as m increases from 0 to 3 for Ha = 10, and the difference becomes less for Ha = 50. When Ha increases up to 100, the difference of the average values of Nusselt number is minimum as m increases from 0 to 3. Thus, m plays more important impacts on the heat transfer as Ha decreases. For a fixed m, the average values of Nuc, Nut and Nur decreases obviously with the increasing of Ha. That means the heat transfer is also suppressed by the magnetic flied.
4. Conclusions For the first time, the Hall effects on fluid flow and heat transfer of participating MHD with different Pl number and Ha number are investigated in this work. Thermal radiative heat flux is obtained accurately by solving RTE instead of simplified method (Rosseland approximation). In view of the obtained results, we can find that the fluid flow and heat transfer of the participating MHD are all enhanced as the Hall parameter m increases. Under different levels of thermal radiation, the Hall effects have obvious impacts on the flow field and temperature flied, and the Hall effects play more important role on heat transfer with the increasing of Pl. For weak magnetic field, the Hall effects on the flow field of the participating MHD are slight. With the increasing of Hall parameter m, Hall effects play more important role on the fluid flow while the Hall effects have less impacts on the heat transfer. Acknowledgement This work was supported by the Natural Science Foundation of China (NSFC) with granted contract (No. 51176026).
Fig. 6. Local Nusselt number on the hot wall for different Ha and m with Pl = 0.02.
to 3 for Pl = 0.02. With the increasing of Pl, the average values of Nut changes more as m increases. So, the Hall parameter plays more important role on heat transfer with the increasing of Pl.
3.2. Different Ha numbers As a synthesized dimensionless parameter of electrical conductivity and magnetic flux density Ha number, Figs. 4–6 show the
References [1] T. Hayat, Z. Abbas, M. Sajid, S. Asghar, The influence of thermal radiation on MHD flow of a second grade fluid, International Journal of Heat and Mass Transfer 50 (2007) 931–941. [2] T. Hayat, M. Nawaza, M. Sajid, S. Asghar, The effect of thermal radiation on the flow of a second grade fluid, Computers and Mathematics with Applications 58 (2009) 369–379. [3] T. Hayat, M. Qasim, Influence of thermal radiation and Joule heating on MHD flow of a Maxwell fluid in the presence of thermophoresis, International Journal of Heat and Mass Transfer 53 (2010) 4780–4788. [4] T. Hayat, R. Sajjad, Z. Abbas, M. Sajid, A.A. Hendi, Radiation effects on MHD flow of Maxwell fluid in a channel with porous medium, International Journal of Heat and Mass Transfer 54 (2011) 854–862.
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843 [5] T. Hayat, M. Awais, S. Asghar, Radiative effects in a three-dimensional flow of MHD Eyring–Powell fluid, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.02.009 (in press). [6] Z. Abbas, T. Hayat, Radiation effects on MHD flow in a porous space, International Journal of Heat and Mass Transfer 51 (2008) 1024–1033. [7] M. Sajid, T. Hayat, Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet, International Communications in Heat and Mass Transfer 35 (2008) 347–356. [8] T. Hayat, S.A. Shehzad, M. Qasim, S. Obaidat, Radiative flow of Jeffery fluid in a porous medium with power law heat flux and heat source, Nuclear Engineering and Design 243 (2012) 15–19. [9] M. Turkyilmazoglu, Thermal radiation effects on the time-dependent MHD permeable flow having variable viscosity, International Journal of Thermal Sciences 50 (2011) 88–96. [10] M.A. Seddeek, S.N. Odda, M.Y. Akl, M.S. Abdelmeguid, Analytical solution for the effect of radiation on flow of a magneto-micropolar fluid past a continuously moving plate with suction and blowing, Computational Materials Science 45 (2009) 423–428. [11] M.A.A. Mahmoud, Thermal radiation effects on MHD flow of a micropolar fluid over a stretching surface with variable thermal conductivity, Physica A 375 (2007) 401–410. [12] C.I. Cookey, A. Ogulu, V.B.O. Pepple, Influence of viscous dissipation and radiation on unsteady MHD free-convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction, International Journal of Heat and Mass Transfer 46 (2003) 2305–2311. [13] A.Y. Ghaly, Radiation effects on a certain MHD free-convection flow, Chaos, Solitons & Fractals 13 (2002) 1843–1850. [14] P. Ganesan, P. Loganathan, Radiation and mass transfer effects on flow of an incompressible viscous fluid past a moving vertical cylinder, International Journal of Heat and Mass Transfer 45 (2002) 4281–4288.
843
[15] M.A. Seddeek, Effects of radiation and variable viscosity on a MHD free convection flow past a semi-infinite flat plate with an aligned magnetic field in the case of unsteady flow, International Journal of Heat and Mass Transfer 45 (2002) 931–935. [16] E. Osalusi, J. Side, R. Harris, The effects of Ohmic heating and viscous dissipation on unsteady MHD and slip flow over a porous rotating disk with variable properties in the presence of Hall and ion-slip currents, International Communications in Heat and Mass Transfer 34 (2007) 1017–1029. [17] L. Chacón, D.A. Knoll, J.M. Finn, Hall MHD effects on the 2D Kelvin–Helmholtz/ tearing instability, Physics Letters A 308 (2003) 187–197. [18] H.S. Takhar, A.J. Chamkha, G. Nath, MHD flow over a moving plate in a rotating fluid with magnetic field, Hall currents and free stream velocity, International Journal of Engineering Science 40 (2002) 1511–1527. [19] M.A. Rana, A.M. Siddiqui, N. Ahmed, Hall effect on Hartmann flow and heat transfer of a Burgers’ fluid, Physics Letters A 372 (2008) 562–568. [20] J. Zueco, P. Eguía, E. Granada, J.L. Míguez, O.A. Bég, An electrical network for the numerical solution of transient mhd couette flow of a dusty fluid: effects of variable properties and hall current, International Communications in Heat and Mass Transfer 37 (2010) 1432–1439. [21] L.K. Saha, M.A. Hossain, R.S.R. Gorla, Effect of Hall current on the MHD laminar natural convection flow from a vertical permeable flat plate with uniform surface temperature, International Journal of Thermal Sciences 46 (2007) 790– 801. [22] J.K. Zhang, B.W. Li, Z.M. Hu, Effects of optical parameters on fluid flow and heat transfer of participating magnetic fluid, International Journal of Heat and Mass Transfer 59 (2013) 126–136. [23] G.W. Sutton, A. Sherman, Engineering Magnetohydrodynamics, McGraw-Hill, New York, 1965.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Hall effects on natural convection of participating MHD with thermal radiation in a cavity Jing-Kui Zhang a,b, Ben-Wen Li a,⇑, Yuan-Yuan Chen a,b a b
Key Laboratory for Ferrous Metallurgy and Resources Utilization of Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China Key Laboratory of Electromagnetic Processing of Materials of Ministry of Education, Northeastern University, Shenyang 110819, China
a r t i c l e
i n f o
Article history: Received 19 January 2013 Received in revised form 28 July 2013 Accepted 28 July 2013 Available online 23 August 2013 Keywords: Thermal radiation Participating MHD Hall effects Square cavity
a b s t r a c t In this work, Hall effects on natural convection of participating MHD with thermal radiation are investigated numerically. An external uniform magnetic field is applied on a square cavity which is filled with participating magnetic fluid. The full filled fluid has characteristics of gray, absorbing, emitting, scattering and electrically conducting. All walls of the cavity are opaque and diffusively reflection. The finite volume method (FVM) is employed to solve the momentum equation and energy equation. The radiative flux which is treated as a heat source term is obtained by solving radiative transfer equation (RTE) instead of the widely used Rosseland approximation. Graphical results and detailed analysis show the Hall effects on fluid flow and heat transfer with thermal radiation under different Pl number and Ha number. Results also show that the Hall effects on participating MHD cannot be neglected in certain range of parameters. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The study of MHD flow and heat transfer with thermal radiation has gained considerable attention during recent years. This is in view of its wide application in engineering and scientific research such as extrusion of molten polymers, MHD generator, space weather forecasting, thermonuclear reaction device, aeronautics, astrophysics and so on. Many excellent works have been presented for thermal radiation effects on natural convection of MHD. Hayat and his partners [1–6] investigated the radiation effects on MHD flow and heat transfer by homotopy analysis method (HAM). They studied the free convection of Maxwell fluid and incompressible second grade fluid over a stretching sheet, in a channel, in a porous space and past a semi-infinite fixed plate, respectively. Some other relevant works [7,8] are also performed by them about the radiative features on different fluids flow. Turkyilmazoglu [9] investigated thermal radiation effects on the time-dependent MHD permeable flow with variable viscosity numerically by a technique based on the spectral Chebyshev collocation in the direction normal to the disk and forward marching in time. Seddeek et al. [10], Mahmoud [11], Cookey et al. [12] and Ghaly [13] found that the rate of heat transfer of magnetic fluid on a plate increased with the increasing of radiation parameter. The interaction of free convection with thermal radiation of a viscous incompressible unsteady flow past a ⇑ Corresponding author. Tel.: +86 136 64102 228. E-mail addresses: [email protected], [email protected] (B.-W. Li). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.07.090
moving vertical cylinder was analyzed by Ganesan and Loganathan [14]. Seddeek [15], Mahmoud [8] and Turkyilmazoglu [9] conducted works on thermal radiation on MHD flow and heat transfer with variable viscosity or thermal conductivity. As we known, just for low values of electrical conductivity, magnetic flux density and velocity, the Hall effects can be neglected. However, in all works mentioned above, it is assumed that the Hall effects were neglected no matter the magnetic flied was strong or weak in their works. In fact, the magnetic field and flow intensity are moderate or even more in some of above works. From [16–21], we can find that the Hall effects have notable impacts on the natural convection of electrically conductive fluid without radiation in certain range of parameters. Thus, investigation of Hall effects on participating MHD flow and heat transfer with thermal radiation is necessary. Furthermore, the radiative heat flux was obtained on one dimension by a same simplified method (Rosseland approximation) in all works above, so, the coupling of radiation and free convection was performed at the expense of accuracy. Accurate calculation has been achieved for coupled free convection and radiative heat transfer in our very recent work [22]. The optical parameters effects on fluid flow and heat transfer of participating MHD was investigated with the assumption that the Hall effects could be neglected in that work, same as in works [1–6,9– 15]. That assumption may result in some possible changes of phenomena on flow field and temperature field. Thereafter, based on our former work [22], investigation is carried out in our present work for different Ha and Pl number to understand the Hall effects on participating MHD with thermal radiation.
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843
839
Fig. 1. Streamlines for different Pl and m with Ha = 50.
Fig. 3. Local Nusselt number on the hot wall for different Pl and m with Ha = 50.
Fig. 2. Isotherms for different Pl and m with Ha = 50.
2. Mathematical formulation and numerical techniques The physical model together with the coordinate system and boundary conditions can be referred to Fig. 1 of [22]. Also the
properties of the media, and the situation considered are similar to those in [22]. We assume the fluid density is constant, except for the density in the buoyancy term which follows the Boussinesq approximation. The induced magnetic field is neglected by assuming a very small magnetic Reynolds number [23]. Through dimensionless exchange continued from [22], the governing equations can be written as
840
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843
Fig. 4. Streamlines for different Ha and m with Pl = 0.02.
@U @V þ ¼0 @X @Y
ð1Þ
walls are hL = 0.5 and hR = +0.5, respectively. On the adiabatic T ref 1 e q c þ T T q ¼ 0. walls, the boundary condition is e 4Pl r c
h
@ 2 @ @P 1 Ha2 U þ ðUVÞ ¼ þ r2 U U @X @Y @X Re Reð1 þ m2 Þ @ @ 2 @P 1 ðUVÞ þ þ r2 V þ Rih V ¼ @X @Y @Y Re T ref @ @ 1 ðUhÞ þ ðVhÞ ¼ r2 h @X @Y RePr Th Tc Z sL ð1 xÞ 4 1 eX Id H RePlPr 4p 4p J 2 U þ ð1 þ m2 Þ
x reI ¼ eI þ ð1 xÞH4 þ sL 4p 1
ð2Þ
eI ðrw ; Xn Þ ¼ ew H4 þ 1 ew
p
ð3Þ
ð4Þ
4p
eI 0 UdX0
ð5Þ
where the definitions of dimensionless variables and numbers have already been given in [22], except for the Hall parameter m and the rB20 U0 L
.
h T c Þ
All the walls are considered to be no-slip for velocity, thus, U = 0 and V = 0 on all walls. The values of temperature on isothermal
N X 0 0 0 I rw ; Xn n Xn wn
0
n Xn < 0
n0 ¼1
ð6Þ where r is the spatial coordinates vector. n is the angular direction of radiation. n is the angular direction vector, and N is the number of discrete-ordinates directions. ew is the emissivity of wall. When the temperature field and radiation intensity are obtained, the conductive and radiative heat flux in non-dimensional forms on the walls can be obtained by
e qc ¼
Z
Joule heating parameter J ¼ qcp ðT
Due to the opaque and diffusively reflection walls, the boundary condition of RTE is given by
dh ; dY C
Z 1 eIdX e q r ¼ ew H4
p
ð7Þ
C
Local conductive, radiative and total Nusselt number Nuc, Nur and Nut are defined as
qc ; Nuc ¼ e
Nur ¼
T ref 1 e qr ; T h T c 4Pl
Nut ¼ Nuc þ Nur
ð8Þ
This work is based on our former work. The physical problem, governing equations, boundary conditions and numerical methods in
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843
841
Fig. 5. Isotherms for different Ha and m with Pl = 0.02.
this work are same as the former, and the grid test and codes validation have been performed in that work which are available in Ref. [22]. 3. Results and discussion Different Ha number, Pl number and Hall parameter m are selected to investigate the Hall effects on fluid flow and heat transfer of participating magnetic fluid. In all the following calculations, Re, Ri, Pr and J are taken as Re = 300, Ri = 1, Pr = 0.733 and J = 1. The temperature of the right vertical wall is set to be Th = 2Tc, and the dimensionless temperature for RTE is obtained by H = 2h/3 + 1. Emissivity of all walls is set to be a uniform value of ew = 0.8. The scattering of the fluid is isotropic and x = 0.5, and the optical thickness is set as sL = 1. 3.1. Different Pl numbers The Pl number means the rate of conduction to radiation as its expression. The results illustrated in Figs. 1–3 demonstrate the Hall effects on fluid flow and heat transfer with different Pl. Ha number is set as constant Ha = 50. It can be clearly seen in Fig. 1 that the flow intensity increases as m increases for 0–3 under a fixed Pl. On the one hand, this is because the Lorentz force term in the NS equations decreases as m increases which can be seen in Eq.
(2), which leads to the decrease of the suppression of magnetic field on MHD flow. On the other hand, with the increasing of m, the Joule heating term is decreased, so, the loss of kinetic energy of the fluid decreases as shown in Eq. (4). For a fixed m, the flow intensity decreases obviously as Pl increases due to the significant decrease of thermal radiation as shown in Fig. 3. The corresponding temperature field is shown in Fig. 2. With the increasing of m, isotherms near the bottom wall move to the right while isotherms near the top wall move to the left when Pl is kept at fixed due to the increasing of flow intensity and the anticlockwise flow direction. When Pl increases, the distribution of isotherms becomes more center symmetric for a fixed m. This is because the convection plays more important role on the heat transfer with the decrease of thermal radiation. In order to reveal the influence of Hall effects on heat transfer on the boundary, the local Nusselt number on the hot wall is selected as shown in Fig. 3. When Pl increases from 0.02 to 2, the values of Nut decrease obviously for a fixed m due to the quick drop of thermal radiation, while, Nuc has different changes with different m. With the increasing of Pl, Nuc increases near the bottom of the hot wall and decreases near the top of the hot wall for a fixed value of m. For a fixed Pl, the average values of Nut, Nur and Nuc are all increases with the increasing of m because of the increase of flow intensity. That means the heat transfer is enhanced with the increasing of Hall parameter. It also can be observed in Fig. 3 that the average values of Nut changes just a little as m increases from 0
842
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843
Hall effects on natural convection of participating MHD under different Ha, where Pl = 0.02 is kept at fixed. Fig. 4 shows the streamlines for different Ha and m. It can be seen that the flow intensity is increased as m increases from 0 to 3 under Ha = 10, 50and 100, respectively. This is also because the Lorentz force term and the Joule heating term decrease as m increases. With the increasing of m, it also can be observed that streamlines distribution and flow intensity change slightly for Ha = 10, and with the increasing of Ha, these changes become more. So, the Hall parameter m plays more important role on fluid flow as Ha increases. When m is kept at fixed, the flow intensity decreases obviously with the increasing of Ha due to increase of the suppression of magnetic field on MHD flow. The influence of Hall effects on the temperature distribution is shown in Fig. 5 under different Ha. For a fixed Ha, isotherms move to the right near the bottom wall and move to the left near the top wall. When m is kept at fixed, isotherms become more paralleled to the vertical walls as Ha increases because of the increase of the suppression of the magnetic field imposed on the fluid as shown in Fig. 4. The rate of heat transfer on the hot wall is shown in Fig. 6 for different Ha and m. It can be seen that the average values of Nuc, Nut and Nur increase as m increases for a fixed Ha. The difference of the average values of Nusselt number is significant as m increases from 0 to 3 for Ha = 10, and the difference becomes less for Ha = 50. When Ha increases up to 100, the difference of the average values of Nusselt number is minimum as m increases from 0 to 3. Thus, m plays more important impacts on the heat transfer as Ha decreases. For a fixed m, the average values of Nuc, Nut and Nur decreases obviously with the increasing of Ha. That means the heat transfer is also suppressed by the magnetic flied.
4. Conclusions For the first time, the Hall effects on fluid flow and heat transfer of participating MHD with different Pl number and Ha number are investigated in this work. Thermal radiative heat flux is obtained accurately by solving RTE instead of simplified method (Rosseland approximation). In view of the obtained results, we can find that the fluid flow and heat transfer of the participating MHD are all enhanced as the Hall parameter m increases. Under different levels of thermal radiation, the Hall effects have obvious impacts on the flow field and temperature flied, and the Hall effects play more important role on heat transfer with the increasing of Pl. For weak magnetic field, the Hall effects on the flow field of the participating MHD are slight. With the increasing of Hall parameter m, Hall effects play more important role on the fluid flow while the Hall effects have less impacts on the heat transfer. Acknowledgement This work was supported by the Natural Science Foundation of China (NSFC) with granted contract (No. 51176026).
Fig. 6. Local Nusselt number on the hot wall for different Ha and m with Pl = 0.02.
to 3 for Pl = 0.02. With the increasing of Pl, the average values of Nut changes more as m increases. So, the Hall parameter plays more important role on heat transfer with the increasing of Pl.
3.2. Different Ha numbers As a synthesized dimensionless parameter of electrical conductivity and magnetic flux density Ha number, Figs. 4–6 show the
References [1] T. Hayat, Z. Abbas, M. Sajid, S. Asghar, The influence of thermal radiation on MHD flow of a second grade fluid, International Journal of Heat and Mass Transfer 50 (2007) 931–941. [2] T. Hayat, M. Nawaza, M. Sajid, S. Asghar, The effect of thermal radiation on the flow of a second grade fluid, Computers and Mathematics with Applications 58 (2009) 369–379. [3] T. Hayat, M. Qasim, Influence of thermal radiation and Joule heating on MHD flow of a Maxwell fluid in the presence of thermophoresis, International Journal of Heat and Mass Transfer 53 (2010) 4780–4788. [4] T. Hayat, R. Sajjad, Z. Abbas, M. Sajid, A.A. Hendi, Radiation effects on MHD flow of Maxwell fluid in a channel with porous medium, International Journal of Heat and Mass Transfer 54 (2011) 854–862.
J.-K. Zhang et al. / International Journal of Heat and Mass Transfer 66 (2013) 838–843 [5] T. Hayat, M. Awais, S. Asghar, Radiative effects in a three-dimensional flow of MHD Eyring–Powell fluid, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.02.009 (in press). [6] Z. Abbas, T. Hayat, Radiation effects on MHD flow in a porous space, International Journal of Heat and Mass Transfer 51 (2008) 1024–1033. [7] M. Sajid, T. Hayat, Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet, International Communications in Heat and Mass Transfer 35 (2008) 347–356. [8] T. Hayat, S.A. Shehzad, M. Qasim, S. Obaidat, Radiative flow of Jeffery fluid in a porous medium with power law heat flux and heat source, Nuclear Engineering and Design 243 (2012) 15–19. [9] M. Turkyilmazoglu, Thermal radiation effects on the time-dependent MHD permeable flow having variable viscosity, International Journal of Thermal Sciences 50 (2011) 88–96. [10] M.A. Seddeek, S.N. Odda, M.Y. Akl, M.S. Abdelmeguid, Analytical solution for the effect of radiation on flow of a magneto-micropolar fluid past a continuously moving plate with suction and blowing, Computational Materials Science 45 (2009) 423–428. [11] M.A.A. Mahmoud, Thermal radiation effects on MHD flow of a micropolar fluid over a stretching surface with variable thermal conductivity, Physica A 375 (2007) 401–410. [12] C.I. Cookey, A. Ogulu, V.B.O. Pepple, Influence of viscous dissipation and radiation on unsteady MHD free-convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction, International Journal of Heat and Mass Transfer 46 (2003) 2305–2311. [13] A.Y. Ghaly, Radiation effects on a certain MHD free-convection flow, Chaos, Solitons & Fractals 13 (2002) 1843–1850. [14] P. Ganesan, P. Loganathan, Radiation and mass transfer effects on flow of an incompressible viscous fluid past a moving vertical cylinder, International Journal of Heat and Mass Transfer 45 (2002) 4281–4288.
843
[15] M.A. Seddeek, Effects of radiation and variable viscosity on a MHD free convection flow past a semi-infinite flat plate with an aligned magnetic field in the case of unsteady flow, International Journal of Heat and Mass Transfer 45 (2002) 931–935. [16] E. Osalusi, J. Side, R. Harris, The effects of Ohmic heating and viscous dissipation on unsteady MHD and slip flow over a porous rotating disk with variable properties in the presence of Hall and ion-slip currents, International Communications in Heat and Mass Transfer 34 (2007) 1017–1029. [17] L. Chacón, D.A. Knoll, J.M. Finn, Hall MHD effects on the 2D Kelvin–Helmholtz/ tearing instability, Physics Letters A 308 (2003) 187–197. [18] H.S. Takhar, A.J. Chamkha, G. Nath, MHD flow over a moving plate in a rotating fluid with magnetic field, Hall currents and free stream velocity, International Journal of Engineering Science 40 (2002) 1511–1527. [19] M.A. Rana, A.M. Siddiqui, N. Ahmed, Hall effect on Hartmann flow and heat transfer of a Burgers’ fluid, Physics Letters A 372 (2008) 562–568. [20] J. Zueco, P. Eguía, E. Granada, J.L. Míguez, O.A. Bég, An electrical network for the numerical solution of transient mhd couette flow of a dusty fluid: effects of variable properties and hall current, International Communications in Heat and Mass Transfer 37 (2010) 1432–1439. [21] L.K. Saha, M.A. Hossain, R.S.R. Gorla, Effect of Hall current on the MHD laminar natural convection flow from a vertical permeable flat plate with uniform surface temperature, International Journal of Thermal Sciences 46 (2007) 790– 801. [22] J.K. Zhang, B.W. Li, Z.M. Hu, Effects of optical parameters on fluid flow and heat transfer of participating magnetic fluid, International Journal of Heat and Mass Transfer 59 (2013) 126–136. [23] G.W. Sutton, A. Sherman, Engineering Magnetohydrodynamics, McGraw-Hill, New York, 1965.