Comptes Rendus
Mécanique
Unstable spectrum of a Rayleigh–Bénard system with variable viscosity
Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1165-1178.

This Note studies a Rayleigh–Bénard system in an infinite layer, in the case of temperature-dependent viscosity, with rigid boundary conditions for the velocity at the bottom and free-slip at a top of the layer. It states the linearized problem in the relevant functional operator set-up and identifies, for each nonzero transverse frequency k and Rayleigh number R the (finite) number of modes which are unstable in time. This number is equal to the number of eigenvalues of a particular operator which are smaller than R.

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DOI : 10.5802/crmath.232
Classification : 34L05, 76E15

Olivier Lafitte 1, 2

1 IRL CRM, UMI3457, Centre de recherches Mathématiques, Université de Montréal, Montréal, Canada.
2 LAGA, UMR7539, Université Sorbonne Paris Nord, 93430 Villetaneuse, France.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Olivier Lafitte. Unstable spectrum of a Rayleigh–Bénard system with variable viscosity. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1165-1178. doi : 10.5802/crmath.232. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.232/

[1] Henri Bénard Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent. Thèse, Ann. de Chim. et Phys., Volume 23 (1901) no. 7, pp. 62-144 | Zbl

[2] Subrahmanyan Chandrasekhar Hydrodynamic and hydromagnetic stability, International Series of Monographs on Physics, Clarendon Press, 1961 | Zbl

[3] Philip G. Drazin; William H. Reid Hydrodynamic stability, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1982 | Zbl

[4] Jerome J. Erpenbeck Theory of detonation stability, Symposium (International) on Combustion, Volume 12 (1969) no. 1, pp. 711-721 | DOI

[5] Emmanuel Grenier; Yan Guo; Toan T. Nguyen Spectral instability of characteristic boundary layer flows, Duke Math. J., Volume 165 (2016) no. 16, pp. 3085-3146 | MR | Zbl

[6] Yan Guo; Hyung J. Hwang On the dynamical Rayleigh–Taylor instability, Arch. Ration. Mech. Anal., Volume 167 (2003) no. 3, pp. 235-253 | MR | Zbl

[7] Yan Guo; I. Tice Linear Rayleigh–Taylor instability for viscous, compressible fluids, SIAM J. Math. Anal., Volume 42 (2010) no. 4, pp. 1688-1720 | MR | Zbl

[8] D. L. Harris; William H. Reid Some further results on the Bénard problem, Phys. Fluids, Volume 1 (1958), pp. 102-110 | Zbl

[9] Bernard Helffer; Olivier Lafitte Asymptotic methods for the eigenvalues of the Rayleigh equation for the linearized Rayleigh– Taylor instability, Asymptotic Anal., Volume 33 (2003) no. 3-4, pp. 189-235 | MR | Zbl

[10] Olivier Lafitte; Mark William; Kevin R. Zumbrun The Erpenbeck high frequency instability Theorem for Zeldovich–Von Neumann–Döring detonations, Arch. Ration. Mech. Anal., Volume 204 (2012) no. 1, pp. 141-187 | DOI | Zbl

[11] Toan T. Nguyen; Olivier Lafitte Spectrum of the viscous Rayleigh–Taylor Instability (ongoing) (submitted in PhD Thesis: Université Sorbonne Paris Nord, Paris, France)

[12] Francisco Pla; Henar Herrero; Olivier Lafitte Theoretical and numerical study of a thermal convection problem with temperature dependent viscosity in a infinite layer, Physica D, Volume 239 (2010) no. 13, pp. 1108-1119 | MR | Zbl

[13] John W. Rayleigh (Strutt) On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Phil. Mag., Volume 32 (1916), pp. 529-546 | DOI | Zbl

[14] Sergey L. Skorokhodov Numerical analysis of the Spectrum of the Orr–Sommerfeld Problem, Comput. Math. Math. Phys., Volume 47 (2007), pp. 1603-1621 | DOI | MR

[15] Ying Tan; Weidong Su A trigonometric series expansion method for the Orr–Sommerfeld equation, AMM, Appl. Math. Mech., Engl. Ed., Volume 40 (2019) no. 6, pp. 877-888 | MR | Zbl

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