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Rotating magnetic field effect on convection and its stability in a horizontal cylinder subjected to a longitudinal temperature gradient

Published online by Cambridge University Press:  15 October 2010

D. V. LYUBIMOV ,
A. V. BURNYSHEVA ,
H. BENHADID ,
T. P. LYUBIMOVA  and
D. HENRY
D. V. LYUBIMOV
Affiliation:
Perm State University, 15 Bukirev Street, 614990 Perm, Russia
A. V. BURNYSHEVA
Affiliation:
Institute of Continuous Media Mechanics UB RAS, 1 Koroleva Street, 614013 Perm, Russia
H. BENHADID
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, Université de Lyon, École Centrale de Lyon, Université Lyon 1, INSA de Lyon, UMR-CNRS 5509, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne CEDEX, France
T. P. LYUBIMOVA
Affiliation:
Institute of Continuous Media Mechanics UB RAS, 1 Koroleva Street, 614013 Perm, Russia
D. HENRY*
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, Université de Lyon, École Centrale de Lyon, Université Lyon 1, INSA de Lyon, UMR-CNRS 5509, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne CEDEX, France
*
†Email address for correspondence: daniel.henry@ec-lyon.fr
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Abstract

A rotating magnetic field (RMF) is used in crystal growth applications during the solidification process in order to improve the crystal quality. Its influence on the convective flows in molten metals and on their stability is studied here in the case of a horizontal infinite cylindrical channel subjected to a longitudinal temperature gradient. The steady convective flows, which correspond to the usual longitudinal counterflow structure, with four vortices in the cross-section for non-zero Prandtl number, Pr, are modified by the RMF (parametrized by the magnetic Taylor number Tam). For zero Prandtl number, the flow in the cross-section corresponds to circular streamlines and the longitudinal flow structure is moved in the direction of the magnetic field rotation, with a decrease in its intensity and an asymptotic variation as 1/Tam. For non-zero Prandtl numbers, depending on the respective values of Tam on one side and Prandtl and Grashof numbers on the other side, different structures ranging from the circular streamlines with transport by rotation of the longitudinal velocity and the temperature field, to the more usual counterflow structure almost insensitive to the RMF with four cross-section vortices, can be obtained. The decrease in the flow intensity with increasing Tam is also delayed for non-zero Pr, but the same asymptotic limit is eventually reached. The stability analysis of these convective flows for Tam = 0 shows a steep increase of the thresholds around Pr = Prt,0 ≈ 3 × 10−4, corresponding to the transition between the usual counterflow shear mode and a new sidewall shear mode. This transition is still present with an RMF, but it occurs for smaller Pr values as Tam is increased. Strong stabilizing effects of the rotating magnetic field are found for Pr < Prt,0, particularly for Pr = 0 where an exponential increase of the threshold with Tam is found. For Pr > Prt,0 (i.e. in the domain where the sidewall instability is dominant), in contrast, the stabilization by the RMF is weak.

JFM classification

Convection: Convection in cavities Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 664 , 10 December 2010 , pp. 108 - 137
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Barz, R. U., Gerbeth, G., Wunderwald, U., Buhrig, E. & Gelfgat, Y. M. 1997 Modelling of the isothermal melt flow due to rotating magnetic fields in crystal growth. J. Cryst. Growth 180, 410–421.CrossRefGoogle Scholar
Bejan, A. & Tien, C. L. 1978 Fully developed natural counterflow in a long horizontal pipe with different end temperatures. Intl J. Heat Mass Transfer 21, 701–708.CrossRefGoogle Scholar
Ben Hadid, H., Vaux, S. & Kaddeche, S. 2001 Three-dimensional flow transitions under a rotating magnetic field. J. Cryst. Growth 230, 57–62.CrossRefGoogle Scholar
Bontoux, P., Smutek, C., Roux, B. & Lacroix, J. M. 1986 Three-dimensional buoyancy-driven flows in cylindrical cavities with differentially heated endwalls. Part 1. Horizontal cylinders. J. Fluid Mech. 169, 211–227.CrossRefGoogle Scholar
Braunsfurth, M. G. & Mullin, T. 1996 An experimental study of oscillatory convection in liquid gallium. J. Fluid Mech. 327, 199–219.CrossRefGoogle Scholar
Davoust, L., Cowley, M. D., Moreau, R. & Bolcato, R. 1999 Buoyancy-driven convection with a uniform magnetic field. Part 2. Experimental investigation. J. Fluid Mech. 400, 59–90.CrossRefGoogle Scholar
Dold, P. & Benz, K. W. 1995 Convective temperature fluctuations in liquid gallium in dependence on static and rotating magnetic fields. Cryst. Res. Technol. 30, 1135–1145.CrossRefGoogle Scholar
Dold, P. & Benz, K. W. 1997 Modification of fluid flow and heat transport in vertical Bridgman configurations by rotating magnetic fields. Cryst. Res. Technol. 32, 51–60.CrossRefGoogle Scholar
Dold, P. & Benz, K. W. 1999 Rotating magnetic field: fluid flow and crystal growth applications. Prog. Cryst. Growth Charact. Mater. 38, 7–38.CrossRefGoogle Scholar
Gelfgat, A. Y., Bar-Yoseph, P. Z. & Yarin, A. L. 1999 Stability of multiple steady states of convection in laterally heated cavities. J. Fluid Mech. 388, 315–334.CrossRefGoogle Scholar
Grants, I. & Gerbeth, G. 2001 Stability of axially symmetric flow driven by a rotating magnetic field in a cylindrical cavity. J. Fluid Mech. 431, 407–426.CrossRefGoogle Scholar
Grants, I. & Gerbeth, G. 2002 Linear three-dimensional instability of a magnetically driven rotating flow. J. Fluid Mech. 463, 229–239.CrossRefGoogle Scholar
Hart, J. E. 1972 Stability of thin non-rotating Hadley circulations. J. Atmos. Sci. 29, 687–697.2.0.CO;2>CrossRefGoogle Scholar
Henry, D. & Ben Hadid, H. 2007 Multiple flow transitions in a box heated from the side in low-Prandtl-number fluids. Phys. Rev. E 76, 016314.CrossRefGoogle Scholar
Henry, D. & Buffat, M. 1998 Two- and three-dimensional numerical simulations of the transition to oscillatory convection in low-Prandtl-number fluids. J. Fluid Mech. 374, 145–171.CrossRefGoogle Scholar
Hof, B., Juel, A., Zhao, L., Henry, D., Ben Hadid, H. & Mullin, T. 2004 On the onset of oscillatory convection in molten gallium. J. Fluid Mech. 515, 391–413.CrossRefGoogle Scholar
Hurle, D. T. J., Jakeman, E. & Johnson, C. P. 1974 Convective temperature oscillations in molten gallium. J. Fluid Mech. 64 (3), 565–576.CrossRefGoogle Scholar
Kaiser, T. & Benz, K. W. 1998 Taylor vortex instabilities induced by a rotating magnetic field: a numerical approach. Phys. Fluids 10, 1104–1110.CrossRefGoogle Scholar
Kamotani, Y. & Sahraoui, T. 1990 Oscillatory natural convection in rectangular enclosures filled with mercury. J. Heat Transfer 112, 235–255.CrossRefGoogle Scholar
Kuo, H. P. & Korpela, S. A. 1988 Stability and finite amplitude natural convection in a shallow cavity with insulated top and bottom and heated from a side. Phys. Fluids 31, 33–42.CrossRefGoogle Scholar
Laure, P. & Roux, B. 1987 Synthèse des résultats obtenus par l'étude de stabilité des mouvements de convection dans une cavité horizontale de grande extension. C. R. Acad. Sci. Paris 305, 1137–1143.Google Scholar
Lyubimov, D. V., Lyubimova, T. P. & Morozov, V. A. 2001 Software package for numerical investigation of linear stability of multidimensional flows. Bull. Perm Univ.: Inf. Syst. Technol. 5, 74–81.Google Scholar
Lyubimova, T. P., Lyubimov, D. V., Morozov, V. A., Scuridin, R. V., Ben Hadid, H. & Henry, D. 2009 Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 1. Effect of aspect ratio and Prandtl number. J. Fluid Mech. 635, 275–295.CrossRefGoogle Scholar
Lyubimov, D. V., Lyubimova, T. P., Perminov, A. B., Henry, D. & Ben Hadid, H. 2009 Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 2. Effect of a magnetic field. J. Fluid Mech. 635, 297–319.CrossRefGoogle Scholar
Martin Witkowski, L., Walker, J. S. & Marty, P. 1999 Nonaxisymmetric flow in a finite-length cylinder with a rotating magnetic field. Phys. Fluids 11, 1821–1826.CrossRefGoogle Scholar
Moreau, R. 1990 Magnetohydrodynamics. Kluwer Academic.CrossRefGoogle Scholar
Mössner, R. & Gerbeth, G. 1999 Buoyant melt flows under the influence of steady and rotating magnetic fields. J. Cryst. Growth 197, 341–354.CrossRefGoogle Scholar
Müller, A. & Wiehelm, M. 1964 Periodische temperaturschwankungen in flüssigem InSb als ursache schichtweisen einbaus von Te in kristallisierendes InSb. Z. Naturforsch. A 19, 254–263.CrossRefGoogle Scholar
Roux, B. (Ed.) 1990 Numerical simulation of oscillatory convection in low-Pr fluids. GAMM Workshop, Notes on Numerical Fluid Mechanics, vol. 27. Vieweg.Google Scholar
Skeldon, A. C., Riley, D. C. & Cliffe, K. A. 1996 Convection in a low-Prandtl-number fluid. J. Cryst. Growth 162, 95–106.CrossRefGoogle Scholar
Utech, H. P. & Flemings, M. C. 1966 Elimination of solute banding in Indium Antimonide crystals by growth in a magnetic field. J. Appl. Phys. 37 (5), 2021–2024.CrossRefGoogle Scholar
Vaux, S., Ben Hadid, H. & Henry, D. 2006 Study of the hydrodynamic instabilities in a differentially heated horizontal circular cylinder corresponding to a Bridgman growth configuration. J. Cryst. Growth 290, 674–682.CrossRefGoogle Scholar
Volz, M. P. & Mazuruk, K. 1999 Thermoconvective instability in a rotating magnetic field. Intl J. Heat Mass Transfer 42, 1037–1045.CrossRefGoogle Scholar
Volz, M. P. & Mazuruk, K. 2001 An experimental study of the influence of a rotating magnetic field on Rayleigh–Bénard convection. J. Fluid Mech. 444, 79–98.CrossRefGoogle Scholar
Wakitani, S. 2001 Numerical study of three-dimensional oscillatory natural convection at low Prandtl number in rectangular enclosures. J. Heat Transfer 123, 77–83.CrossRefGoogle Scholar
Winters, K. H. 1988 Oscillatory convection in liquid metals in a horizontal temperature gradient. Intl J. Numer. Meth. Engng 25, 401–414.CrossRefGoogle Scholar