The skin effect appears when a rotating magnetic field (RMF) penetrates the melt with electrical conductivity, and the skin effect becomes more obvious with the increased dimensionless shielding parameter K. In this paper, an analytic solution, based on the Bessel functions for the distribution of the rotating magnetic field with the skin effect in melt was derived. In order to model the interaction of the rotating magnetic field and melt convection, the polynomial of the analytic solution was truncated in different terms. In addition, the applicative range of K for different truncation was estimated. Then the rotating magnetic field with truncation was introduced to the rotating magnetic field Φ1-Φ2 model. Through the simulation results of the finite volume method, the estimated range of K for the approximate magnetic field with different truncation was verified.
AfrandM., FarahatS. and NezhadA.H., 3-D numerical investigation of natural convection in a tilted cylindrical annulus containing molten potassium and controlling it using various magnetic fields, International Journal of Applied Electromagnetics and Mechanics46 (2014), 809-821.
2.
MahmoodiM., EsfeM.H., AkbariM., KarimipourA. and AfrandM., Magneto-natural convection in square cavities with a source-sink pair on different walls, International Journal of Applied Electromagnetics and Mechanics47 (2015), 21-32.
3.
PalJ., CramerA. and GerbethG., Experimental investigation on the electromagnetically controlled buoyancy-induced flow in a model of a Czochralski puller, International Journal of Applied Electromagnetics and Mechanics (2014), 163-170.
4.
AfrandM., RostamiS., AkbariM., WongwisesS., EsfeM.H. and KarimipourA., Effect of induced electric field on magneto-natural convection in a vertical cylindrical annulus filled with liquid potassium, International Journal of Heat and Mass Transfer90 (2015), 418-426.
5.
TeimouriH., AfrandM., SinaN., KarimipourA. and IsfahaniA.H.M., Natural convection of liquid metal in a horizontal cylindrical annulus under radial magnetic field, International Journal of Applied Electromagnetics and Mechanics49 (2015), 453-461.
6.
AfrandM., FarahatS., NezhadA.H., SheikhzadehG.A., SarhaddiF. and WongwisesS., Multi-objective optimization of natural convection in a cylindrical annulus mold under magnetic field using particle swarm algorithm, International Communications in Heat and Mass Transfer60 (2015), 13-20.
7.
AfrandM., FarahatS., NezhadA.H., SheikhzadehG.A. and SarhaddiF., Numerical simulation of electrically conducting fluid flow and free convection heat transfer in an annulus on applying magnetic field, Heat Transfer Research45(8) (2014), 749-766.
8.
AfrandM., SinaN., TeimouriH., MazaheriA., SafaeiM.R., EsfeM.H., KamaliJ. and ToghraieD., Effect of magnetic field on free convection in inclined cylindrical annulus containing molten potassium, Int J Appl Mech7 (2015), DOI: 10.1142/S1758825115500520.
9.
DostS., liuY., LentB. and ReddenR.F., A numerical simulation study for the effect of applied magnetic field in growth of CdTe single crystals by the traveling heater method, International Journal of Applied Electromagnetics and Mechanics17 (2003), 271-188.
10.
YaoL.P., ZengZ. and ZhangY.X., Effects of transverse rotating magnetic field on thermocapillary flow under microgravity, Journal of Chongqing University35 (2012), 115-120.
11.
BrückerF.-U. and SchwerdtfegerK., Single crystal growth with the czochralski method involving rotational electromagnetic stirring of the melt, Journal of Crystal Growth139 (1994), 351-356.
12.
DoldP., CröllA., KaiserTh., SalkM. and BenzK.W., Semiconductor crystals grown under microgravity and in magnetic fields, 33rd Aerospace Sciences Meeting and Exhibit, AIAA, (1995), 95-0264.
13.
DoldP. and BenzK.W., Modification of fluid flow and heat transport in vertical bridgman configurations by rotating magnetic fields, Crystal Research and Technology32 (1997), 51-60.
14.
YaoL., ZengZ., MizusekiH. et al., Effects of rotating magnetic fields on thermocapillary flow in a floating half-zone, Journal of Crystal Growth316 (2011), 177-184.
15.
DavidsonP.A. and HuntJ.C., Swirling recirculating flow in a liquid-metal column generated by a rotating magnetic field, Journal of Fluid Mechanics185 (1987), 67-106.
16.
KaiserT. and BenzK., Taylor vortex instabilities induced by a rotating magnetic field: A numerical approach, Physics of Fluids10 (1998), 1104.
17.
JaberT., SaghirM. and SourceF., The effect of rotating magnetic fields on the growth of SiGe using the traveling solvent method, FDMP: Fluid Dynamics and Materials Processing2 (2006), 175-190.
18.
JaberT.J., SaghirM.Z. and VivianiA., Three-dimensional modelling of GeSi growth in presence of axial and rotating magnetic fields, European Journal of Mechanics B-Fluids28 (2009), 214-223.
19.
MartyP.H., WitkowskiL.M. and TrombettaP., On the stability of rotating MHD flows, Transfer Phenomena in Magnetohydrodynamic and Electroconducting Flows (1999), 327-343.
20.
WalkerJ.S., WitkowskiL.M. and HouchensB.C., Effects of a rotating magnetic field on the thermocapillary instability in the floating zone process, Journal of Crystal Growth252 (2003), 413-423.
21.
WitkowskiL.M. and WalkerJ.S., Flow driven by Marangoni convection and rotating magnetic field, Magnetohydrodynamics37 (2001), 112-118.
22.
DoldP. and BenzK., Modification of fluid flow and heat transport in vertical bridgman configurations by rotating magnetic fields, Crystal Research and Technology32 (1997), 47-56.
23.
DoldP. and BenzK., Rotating magnetic fields: Fluid flow and crystal growth applications, Progress in Crystal Growth and Characterization of Materials38 (1999), 39-58.
24.
PriedeJ., Theoretical study of a flow in an axisymmetric cavity of finite length driven by a rotating magnetic field, University of Salaspils, Latvia, Ph D Thesis, 1993.
25.
BarzR.U., GerbethG. and WunderwaldU., Modelling of the isothermal melt flow due to rotating magnetic fields in crystal growth, Journal of Crystal Growth180 (1997), 410-421.
26.
MöβnerR. and GerbethG., Buoyant melt flows under the influence of steady and rotating magnetic fields, Journal of Crystal Growth197 (1999), 341-354.
27.
YildizE., DostS. and YildizM., A numerical simulation study for the effect of magnetic fields in liquid phase diffusion growth of SiGe single crystals, Journal of Crystal Growth291 (2006), 497-511.
28.
YaoL., ZengZ. and MizusekiH. et al., Effects of rotating magnetic fields on thermocapillary flow: Comparison of the infinite and the Φ1-Φ2 models, Int J ThermSci49 (2010), 2413-2418.
29.
YaoL., ZengZ., ZhangY. et al., Three-dimensional unsteady thermocapillary flow under rotating magnetic field, Cryst Res Technol47 (2012), 816-823.
30.
YaoL., ZengZ., ZhangY. et al., Influence of rotating magnetic field strength on three-dimensional thermocapillary flow in a floating half-zone model, Heat Mass Transfer48 (2012), 2103-2111.
31.
YaoL., ZengZ., ChenJ. et al., Investigation of convection control under the non-uniform RMF in a liquid bridge, Procedia Engineering31 (2012), 659-664.
32.
DahlbergE., On the action of a rotating magnetic field on a conducting liquid, AB Atomenergi Report AE-477, Sweden, 1972, 1-59.
33.
BarzR.U., GerbethG. and WunderwaldU., Modelling of the isothermal melt flow due to rotating magnetic fields in crystal growth, Journal of Crystal Growth180 (1997), 410-421.
34.
VolzM.P. and MazurukK., Thermoconvective instability in a rotating magnetic field, International Journal of Heat and Mass Transfer42 (1999), 1037-1045.
35.
SpitzerK.H., DubkeM. and SchwerdtfegerK., Rotational electromagnetic stirring in continous casting of round stands, Metallurgical Transactions17 (1986), 119-131.
36.
SpitzerK.H., Application of rotating magnetic fields in czochralski crystal growth, Progress in Grystal Growth and Characterization of Materials (1999), 39-58.
