logo CRAS
Comptes Rendus. Mathématique
Mechanics
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.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: https://doi.org/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.
@article{CRMATH_2021__359_9_1165_0,
     author = {Olivier Lafitte},
     title = {Unstable spectrum of a {Rayleigh{\textendash}B\'enard} system with variable viscosity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1165--1178},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {9},
     year = {2021},
     doi = {10.5802/crmath.232},
     language = {en},
}
TY  - JOUR
AU  - Olivier Lafitte
TI  - Unstable spectrum of a Rayleigh–Bénard system with variable viscosity
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 1165
EP  - 1178
VL  - 359
IS  - 9
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.232
DO  - 10.5802/crmath.232
LA  - en
ID  - CRMATH_2021__359_9_1165_0
ER  - 
%0 Journal Article
%A Olivier Lafitte
%T Unstable spectrum of a Rayleigh–Bénard system with variable viscosity
%J Comptes Rendus. Mathématique
%D 2021
%P 1165-1178
%V 359
%N 9
%I Académie des sciences, Paris
%U https://doi.org/10.5802/crmath.232
%R 10.5802/crmath.232
%G en
%F CRMATH_2021__359_9_1165_0
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 32.0760.03

[2] Subrahmanyan Chandrasekhar Hydrodynamic and hydromagnetic stability, International Series of Monographs on Physics, Clarendon Press, 1961 | Zbl 0142.44103

[3] Philip G. Drazin; William H. Reid Hydrodynamic stability, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1982 | Zbl 0513.76031

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

[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 3566199 | Zbl 1359.35129

[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 1978583 | Zbl 1090.76527

[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 2679591 | Zbl 1429.76054

[8] D. L. Harris; William H. Reid Some further results on the Bénard problem, Phys. Fluids, Volume 1 (1958), pp. 102-110 | Zbl 0082.39701

[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 1981886 | Zbl 1065.34084

[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 | Article | Zbl 1427.76081

[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 2644677 | Zbl 1189.37094

[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 | Article | Zbl 46.1249.04

[14] Sergey L. Skorokhodov Numerical analysis of the Spectrum of the Orr–Sommerfeld Problem, Comput. Math. Math. Phys., Volume 47 (2007), pp. 1603-1621 | Article | MR 2388619

[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 3967590 | Zbl 1416.76042

Cited by Sources: