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Effect of Magnetic Field on the Buoyancy and Thermocapillary View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ePrints@Bangalore University International Journal of Heat and Fluid Flow 32 (2011) 402–412 Contents lists available at ScienceDirect International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff Effect of magnetic field on the buoyancy and thermocapillary driven convection of an electrically conducting fluid in an annular enclosure ⇑ M. Sankar a,b, , M. Venkatachalappa c, Younghae Do a a Department of Mathematics, Kyungpook National University, 1370 Sangyeok-Dong, Buk-Gu, Daegu 702-701, Republic of Korea b Department of Mathematics, East Point College of Engineering and Technology, Bangalore 560 049, India c UGC Centre for Advanced Studies in Fluid Mechanics, Department of Mathematics, Bangalore University, Bangalore 560 001, India article info abstract Article history: The main objective of this article is to study the effect of magnetic field on the combined buoyancy and Received 25 December 2009 surface tension driven convection in a cylindrical annular enclosure. In this study, the top surface of the Received in revised form 30 November 2010 annulus is assumed to be free, and the bottom wall is insulated, whereas the inner and the outer cylin- Accepted 13 December 2010 drical walls are kept at hot and cold temperatures respectively. The governing equations of the flow sys- Available online 8 January 2011 tem are numerically solved using an implicit finite difference technique. The numerical results for various governing parameters of the problem are discussed in terms of the streamlines, isotherms, Nusselt num- Keywords: ber and velocity profiles in the annuli. Our results reveal that, in tall cavities, the axial magnetic field sup- Magnetic field presses the surface tension flow more effectively than the radial magnetic field, whereas, the radial Cylindrical annulus Finite difference magnetic field is found to be better for suppressing the buoyancy driven flow compared to axial magnetic Thermocapillary convection field. However, the axial magnetic field is found to be effective in suppressing both the flows in shallow Surface tension cavities. From the results, we also found that the surface tension effect is predominant in shallow cavities Radii ratio compared to the square and tall annulus. Further, the heat transfer rate increases with radii ratio, but decreases with the Hartmann number. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction orientation with respect to the convective flow direction. Substan- tial theoretical and numerical work thus far has appeared on the Thermocapillary convection is a fluid motion driven by the sur- magnetic damping of natural convection. face tension variation at a free surface resulting from the thermal Langlois (1985) has made a detailed review on the effect of an gradients along the surface. The combination of buoyancy and free external magnetic field on Czochralski crystal growth. He con- surface shear frequently exhibits a complex flow field. These two cluded that when the magnetic Reynolds number is very small, it forces may assist or compete with each other to dominate the flow is not necessary to consider the complete MHD theory to investi- field. In Czochralski crystal growth process, the convection may be gate the Czochralski flow. A theoretical analysis of the effect of driven either by a buoyancy due to density gradient caused by lat- magnetic field on the source distribution in Czochralski grown eral heating, or a surface tension force due to temperature gradient crystals has been given by Kobayashi (1986). A review on the sur- along the free surface. The ever-growing demand for a high quality face tension driven convection in the crystal growth melt flow has crystal material has resulted in a substantial research focused on been carried out by Schwabe (1988). Ozoe and Okada (1989) have the understanding, and eventually controlling the convection in numerically investigated the effect of magnetic damping in a cubic material processing systems. Since the convective flow is the major enclosure. They found that the strongest damping effect is cause of some macroscopic defects in the crystal produced, it is achieved when the magnetic field is applied perpendicular to the desirable to remove this deleterious flow. One of the effective hot wall. Venkatachalappa and Subbaraya (1993) investigated nat- methods to control a thermally induced flow is the magnetic ural convection in a rectangular enclosure in the presence of a damping, in which the applied magnetic field will generate a vertical magnetic field. Their numerical results revealed that Lorentz force to damp the convective flow. The damping effect the temperature and velocity fields are significantly modified by depends on the strength of the applied magnetic field and its the applied magnetic field. BenHadid and Henry (1997) studied the damping effect of a steady convection in a 4  1  1 cavity with a magnetic field applied in different directions. Their results were ⇑ Corresponding author. Tel.: +82 53 950 7951; fax: +82 53 950 7954. in good agreement with the analytical predictions of Garandet E-mail address: [email protected] (M. Sankar). et al. (1992) and Alboussière et al. (1996). 0142-727X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2010.12.001 M. Sankar et al. / International Journal of Heat and Fluid Flow 32 (2011) 402–412 403 Nomenclature A aspect ratio c coefficient of surface tension à B0 magnetic field strength g dimensional vorticity (B0r, B0x) magnetic field strength in (r, x) direction g dimensionless vorticity D annulus gap h dimensional temperature (Fr, Fx) electromagnetic force components in (r, x) direction j thermal diffusivity g acceleration due to gravity k radii ratio Har Hartmann number due to radial magnetic field m kinematic viscosity Hax Hartmann number due to axial magnetic field q fluid density L height of the annulus r surface tension Ma Marangoni number re electrical conductivity Nu Nusselt number s dimensionless time P pressure w dimensional stream function Pr Prandtl number W dimensionless stream function Ra Rayleigh number Wmax,s maximum value of thermocapillary driven stream func- T dimensionless temperature tion t dimensional time (r, x) dimensional radial and axial co-ordinates Subscripts (R, X) dimensionless radial and axial co-ordinates h condition at hot wall (ri, r0) radii of inner and outer cylinders c condition at cold wall. (u, v) dimensional velocity components in (r, x) direction 0 reference state (U, V) dimensionless velocity components in (R, X) direction max maximum value Greek letters b coefficient of thermal expansion Carpenter and Homsy (1989) have studied the problem of com- hydrothermal waves. Sim et al. (2003) carried out numerical sim- bined buoyancy and thermocapillary convection in a differentially ulations of oscillatory thermocapillary convection in a cylindrical heated square cavity with a free surface. Numerical simulations of annulus with a free liquid surface and found a steady and axisym- combined buoyancy and surface tension driven convection in a metric convection at sufficiently low Reynolds numbers. Bauer and rectangular cavity have been performed by Rudraiah et al. (1995) Eidel (2007) analytically investigated the effect of aspect and diam- in the presence of a magnetic field. They found that the average eter ratios on the thermocapillary convection in an annular cylin- Nusselt number increases with Marangoni number, but decreases drical container. Recently, Kim and Kim (2008) performed a with Hartmann number. Some notable studies on the numerical numerical investigation of two-dimensional axisymmetric thermo- simulation of combined buoyancy and/or thermocapillary convec- capillary convection during laser melting processes with deform- tion in a differentially heated rectangular enclosures are due to able free surfaces. More recently, Kakarantzas et al. (2009) Strani et al. (1983), Srinivasan and Basu (1986), Bergman and demonstrated the effect of sinusoidal temperature on the magne- Ramadhyani (1986), Bergman and Keller (1988), Juel et al. tohydrodynamic natural convection in a cylindrical cavity. (1999), Moessner and Mueller (1999) and Gelfgat and Bar-Yoseph Most of the previous studies found in the literature on the com- (2001). The effect of magnetic field on the combined buoyancy and bined buoyancy and surface tension driven convection were con- thermocapillary driven convection in a rectangular enclosure con- centrated either on a rectangular cavity, with or without a taining a heat generating substance has been numerically investi- magnetic field, or on a cylindrical geometry without a magnetic gated by Hossain et al. (2005). Their study was mainly focused field. However, to the best of our knowledge, the problem of com- upon the magnetic field direction and the effect of a heat genera- bined buoyancy and surface tension driven convection in a vertical tion parameter on the flow and heat transfer. Ha and Kim (2005) cylindrical annulus with a free surface has not yet been studied in investigated the transient molten pool geometry during a laser the presence of a magnetic field. This problem attracts by its sheer melting process with a deformable flat surface. The recent studies existence and intellectual curiosity, but its treatment and investi- on the natural convection in rectangular enclosure under the influ- gation would be quite complex and expensive for the experiments. ence of magnetic field are by Lee and Hyun (2005) and Sarris et al. Thus the objective of the present numerical study is to investigate (2005). the effect of direction of magnetic field on the combined buoyancy Vrentas et al. (1981) have studied the surface tension and buoy- and surface tension driven convection in a vertical cylindrical ancy driven convection in a vertical circular cylinder for a wide annulus, which is filled with an electrically conducting fluid. range of aspect ratios. They determined the critical Marangoni and Rayleigh numbers, structure of convective motion, and nature of bifurcation branching. An experimental study of oscillatory ther- 2.
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