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Electromagnetic control of thermal convection of a fluid with strongly temperature-dependent material properties

Published online by Cambridge University Press:  10 January 2009

CORNELIA GIESSLER  and
ANDRÉ THESS
CORNELIA GIESSLER
Affiliation:
Department of Mechanical Engineering, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany
ANDRÉ THESS*
Affiliation:
Department of Mechanical Engineering, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany
*
†Email address for correspondence: [email protected]
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Abstract

We study a one-dimensional model describing buoyancy-driven laminar steady flow of a glass melt in a closed loop under the influence of a localized electromagnetic (Lorentz) force. The loop is a highly simplified representation of a closed streamline in glass melt flow in a real furnace under the influence of an artificially produced Lorentz force. The model is based on the energy equation for the temperature and the Stokes equation for the velocity distribution inside the loop. We take into account the full nonlinear temperature dependence of the viscosity and the electrical conductivity of the melt. The three-dimensional problem is then reduced to a single nonlinear equation for the cross-section averaged velocity from which the one-dimensional temperature distribution along the loop can be readily obtained. We show that the two-way interaction between the velocity and temperature resulting from the temperature-dependent material properties and Lorentz force leads to the result that the mean velocity as a function of the control parameters is non-unique and involves bifurcations. For some parameters we even observe freezing, which refers to a regime in which the fluid is almost at rest. Our model reveals the role of temperature-dependent viscosity and conductivity in glass melt flows in a pure form that is not visible in full numerical simulations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 618 , 10 January 2009 , pp. 135 - 154
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Booker, J. R. 1976 Thermal convection with strongly temperature-dependent viscosity. J. Fluid Mech. 76 (4), 741–754.CrossRefGoogle Scholar
Creveling, H. F., De Paz, J. F., Baladi, J. Y. & Schoenhals, R. J. 1975 Stability characteristics of a single-phase free convection loop. J. Fluid Mech. 67 (1), 65–84.CrossRefGoogle Scholar
Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.CrossRefGoogle Scholar
Desrayaud, G., Fichera, A. & Marcoux, M. 2006 Numerical investigation of natural circulation in a 2d-annular closed-loop thermosyphon. Intl J. Heat Fluid Flow 27, 154–166.CrossRefGoogle Scholar
Ehrhard, P. & Müller, U. 1990 Dynamical behaviour of natural convection in a single-phase loop. J. Fluid Mech. 217, 487–518.CrossRefGoogle Scholar
Fowler, A. C. 1985 Fast thermoviscous convection. Stud. Appl. Math. 72 (3), 189–219.CrossRefGoogle Scholar
Ghaddar, N. 1998 a Analytical model of a side-heated free convection loop placed in a transverse magnetic field. J. Fluid Engng 120, 62–69.CrossRefGoogle Scholar
Ghaddar, N. 1998 b Analytical model of induced electric current from a free convection loop placed in a transverse magnetic field. Intl J. Heat Mass Transfer 41 (8–9), 1075–1086.CrossRefGoogle Scholar
Giessler, C. 2008 Theoretical investigations of electromagnetic control of glass melt flow. PhD thesis, TU Ilmenau.Google Scholar
Giessler, C., Lange, U. & Thess, A. 2007 Nonlinear laminar pipe flow of fluids with strongly temperature-dependent material properties. Phys. Fluids 19, 043601.CrossRefGoogle Scholar
Giessler, C., Schlegel, R. & Thess, A. 2008 Numerical investigation of the flow of a glass melt through along circular pipe. Intl J. Heat Fluid Flow 29 (5), 1462–1468.CrossRefGoogle Scholar
Giessler, C., Sievert, C., Krieger, U., Halbedel, B., Hülsenberg, D., Lüdke, U. & Thess, A. 2005 A model for electromagnetic control of buoyancy driven convection in glass melts. Fluid Dyn. Mat. Process. 1 (3), 247–266.Google Scholar
Giessler, C. & Thess, A. (in press) Numerical simulation of electromagnetically controlled thermal convection of glass melt in a crucible. Intl J. Heat Mass Transfer.Google Scholar
Greif, R. 1988 Natural circulation loops. J. Heat Transfer 110, 1243–1258.CrossRefGoogle Scholar
Helfrich, K. R. 1995 Thermo-viscous fingering of flow in a thin gap: a model of magma flow in dikes and fissures. J. Fluid Mech. 305, 219–238.CrossRefGoogle Scholar
Hofmann, O. R. & Thess, A. 2002 Elektromagnetische Beeinflussung der Glasbadströmung – Ein neues Anwendungsgebiet der Magnetohydrodynamik (in German). Glas-Ingenieur 1, 39–45.Google Scholar
Hülsenberg, D., Halbedel, B., Conrad, G., Thess, A., Kolesnikov, Y. & Lüdtke, U. 2004 Electromagnetic stirring of glass melts using Lorentz forces – experimental results. Glass Sci. Technol. 77, 186–193.Google Scholar
Hülsenberg, D., Halbedel, B., Krieger, U., Schröpfer, D., Thess, A. & Lüdtke, U. 2006 Elektromagnetische Modifizierung von Strömungen in Schmelzen (in German). Elektrowärme Intl 2, 107–110.Google Scholar
Keller, J. B. 1966 Periodic oscillations in a model of thermal convection. J. Fluid Mech. 26 (3), 599–606.CrossRefGoogle Scholar
Krieger, U. 2007 Einfluss elektromagnetisch generierter Kraftwirkungen auf die strömungen in Glasschmelzen (in German). PhD thesis, TU Ilmenau.Google Scholar
Kunert, C., Langsdorf, A., Lentes, F., Duch, K., Thess, A. & Kolesnikow, Y. 2004 Verfahren und Anordnung zur Zufuhr einer Glasschmelze zu einem Verarbeitungsprozess (in German). DE 10 2004 015 055.Google Scholar
Lange, U. & Loch, H. 2002 Instabilities and stabilization of glass pipe flow. In Mathematical Simulation in Glass Technology, Schott Series on Glass and Glass Ceramics (ed. Krause, D. & Loch, H.). Springer Verlag.Google Scholar
Moresi, L. & Solomatov, V. 1995 Numerical investigation of 2d convection with extremely large viscosity variations. Phys. Fluids 7 (9), 2154–2162.CrossRefGoogle Scholar
Morris, S. 1996 Stability of thermoviscous Hele–Shaw flow. J. Fluid Mech. 308, 111–128.CrossRefGoogle Scholar
Morris, S. & Canright, D. 1984 A boundary-layer analysis of Benard convection in a fluid of strongly temperature-dependent viscosity. Phys. Earth Planet. Inter. 36, 355–373.CrossRefGoogle Scholar
Ockendon, H. & Ockendon, J. R. 1977 Variable-viscosity flows in heated and cooled channels. J. Fluid Mech. 83 (1), 177–190.CrossRefGoogle Scholar
Ogawa, M., Schubert, G. & Zebib, A. 1991 Numerical simulations of three-dimensional thermal convection in a fluid with strongly temperature-dependent viscosity. J. Fluid Mech. 233, 299–328.CrossRefGoogle Scholar
Osmanis, A. D., Snijedze, A. K. & Aglitis, A. M. 1987 Influence of electromagnetic stirring of glass melts (in Russian). In Proceedings of the 12th Riga Symposium on Magnetohydrodynamics, Salaspils, USSR, pp. 179–183.Google Scholar
Poddubnaya, L. G. & Shaidurov, G. F. 1969 Convective stability of a conducting fluid in a closed circuit. Magnetohydrodynamics 5 (2), 63–66.Google Scholar
Richardson, S. M. 1986 Injection moulding of theroplastics: freezing of variable-viscosity fluids. iii. Fully-developed flows. Rheol. Acta 25, 372–379.CrossRefGoogle Scholar
Richter, F. M., Nataf, H.-C. & Daly, S. F. 1983 Heat transfer and horizontally averaged temperature of convection with large viscosity variations. J. Fluid Mech. 129, 173–192.CrossRefGoogle Scholar
Torrance, K. E. & Turcotte, D. L. 1971 Thermal convection with large viscosity variation. J. Fluid Mech. 47 (1), 113–125.CrossRefGoogle Scholar
Vogel, W. 1992 Glaschemie (in German). Springer.CrossRefGoogle Scholar
Welander, P. 1967 On the oscillatory instability of a differentially heated fluid loop. J. Fluid Mech. 29 (1), 17–30.CrossRefGoogle Scholar
Wylie, J. J. & Lister, J. R. 1995 The effects of temperature-dependent viscosity on flow in a cooled channel with application to basaltic fissure eruption. J. Fluid Mech. 305, 239–261.CrossRefGoogle Scholar
Zvirin, Y. 1981 A review on natural circulation loops in pressurized water reactors and other systems. Nucl. Engng Des. 67, 203–225.CrossRefGoogle Scholar