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Exact energy stability of Bénard–Marangoni convection at infinite Prandtl number

Published online by Cambridge University Press:  01 June 2017

Giovanni Fantuzzi Open the ORCID record for Giovanni Fantuzzi [Opens in a new window]  and
Andrew Wynn
Giovanni Fantuzzi*
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Andrew Wynn
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
†Email address for correspondence: gf910@ic.ac.uk
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Abstract

Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

JFM classification

Convection: Marangoni convection Instability: Nonlinear instability Mathematical Foundations: Variational methods
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , R1
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Copyright
© 2017 Cambridge University Press 

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References

Boeck, T. & Thess, A. 1998 Turbulent Bénard–Marangoni convection: results of two-dimensional simulations. Phys. Rev. Lett. 80 (6), 1216–1219.CrossRefGoogle Scholar
Boeck, T. & Thess, A. 2001 Bénard–Marangoni convection at large Prandtl numbers. Phys. Rev. E 64 (2), 027303.Google ScholarPubMed
de Bruyn, J. R., Bodenschatz, E., Morris, S. W., Trainoff, S. P., Hu, Y. & Cannell, D. S. 1996 Apparatus for the study of Rayleigh–Bénard convection in gases under pressure. Rev. Sci. Instrum. 67 (6), 2043–2067.Google Scholar
Chernyshenko, S. I., Huang, D., Goulart, P. J., Lasagna, D. & Tutty, O. R. 2013 Nonlinear stability analysis of fluid flow using sum of squares of polynomials. In AIP Conf. Proc., vol. 1558, pp. 265–268.Google Scholar
Constantin, P. & Doering, C. R. 1995a Variational bounds in dissipative systems. Physica D 82 (3), 221–228.Google Scholar
Constantin, P. & Doering, C. R. 1995b Variational bounds on energy dissipation in incompressible flows. II. Channel flow. Phys. Rev. E 51 (4), 3192–3198.Google Scholar
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics, 1st edn. vol. 1. Interscience Publisher Inc.Google Scholar
Davis, S. H. 1969 Buoyancy-surface tension instability by the method of energy. J. Fluid Mech. 39 (2), 347–359.Google Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19 (1), 403–435.Google Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69 (11), 1648–1651.Google Scholar
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49 (5), 4087–4099.Google Scholar
Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53 (6), 5957–5981.Google ScholarPubMed
Giaquinta, M. & Hildebrandt, S. 1996 Calculus of Variations I, Grundlehren der mathematischen Wissenschaften, vol. 310. Springer.Google Scholar
Goulart, P. J. & Chernyshenko, S. I. 2012 Global stability analysis of fluid flows using sum-of-squares. Physica D 241 (6), 692–704.CrossRefGoogle Scholar
Hagstrom, G. & Doering, C. R. 2010 Bounds on heat transport in Bénard–Marangoni convection. Phys. Rev. E 81 (4), 047301.Google Scholar
Jones, G. M. 1977 Thermal interaction of the core and the mantle and long-term behavior of the geomagnetic field. J. Geophys. Res. 82 (11), 1703–1709.CrossRefGoogle Scholar
Kumar, A. & Roy, S. 2009 Effect of three-dimensional melt pool convection on process characteristics during laser cladding. Comput. Mater. Sci. 46 (2), 495–506.CrossRefGoogle Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.Google Scholar
Schatz, M. F. & Neitzel, G. P. 2001 Experiments on thermocapillary instabilities. Annu. Rev. Fluid Mech. 33, 93–127.Google Scholar
Yiantsios, S. G., Serpetsi, S. K., Doumenc, F. & Guerrier, B. 2015 Surface deformation and film corrugation during drying of polymer solutions induced by Marangoni phenomena. Intl J. Heat Mass Transfer 89, 1083–1094.CrossRefGoogle Scholar