no. 9
Mécanique
Unstable spectrum of a Rayleigh–Bénard system with variable viscosity
Lafitte, Olivier1, 2 
1 IRL CRM, UMI3457, Centre de recherches Mathématiques, Université de Montréal, Montréal, Canada.
2 LAGA, UMR7539, Université Sorbonne Paris Nord, 93430 Villetaneuse, France.
Comptes Rendus. Mathématique, Tome 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.

Reçu le :
Accepté le :
Accepté après révision le :
Publié le :
DOI : 10.5802/crmath.232
Classification : 34L05, 76E15
Lafitte, Olivier 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 = {Lafitte, Olivier},
     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},
     url = {http://www.numdam.org/articles/10.5802/crmath.232/}
}
TY  - JOUR
AU  - Lafitte, Olivier
TI  - Unstable spectrum of a Rayleigh–Bénard system with variable viscosity
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 1165
EP  - 1178
VL  - 359
IS  - 9
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.232/
DO  - 10.5802/crmath.232
LA  - en
ID  - CRMATH_2021__359_9_1165_0
ER  - 
%0 Journal Article
%A Lafitte, Olivier
%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 http://www.numdam.org/articles/10.5802/crmath.232/
%R 10.5802/crmath.232
%G en
%F CRMATH_2021__359_9_1165_0
Lafitte, Olivier. Unstable spectrum of a Rayleigh–Bénard system with variable viscosity. Comptes Rendus. Mathématique, Tome 359 (2021) no. 9, pp. 1165-1178. doi : 10.5802/crmath.232. http://www.numdam.org/articles/10.5802/crmath.232/

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

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

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

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

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

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

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

[9] Helffer, Bernard; Lafitte, Olivier 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] Lafitte, Olivier; William, Mark; Zumbrun, Kevin R. 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] Nguyen, Toan T.; Lafitte, Olivier Spectrum of the viscous Rayleigh–Taylor Instability (ongoing) (submitted in PhD Thesis: Université Sorbonne Paris Nord, Paris, France)

[12] Pla, Francisco; Herrero, Henar; Lafitte, Olivier 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] Rayleigh (Strutt), John W. 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] Skorokhodov, Sergey L. Numerical analysis of the Spectrum of the Orr–Sommerfeld Problem, Comput. Math. Math. Phys., Volume 47 (2007), pp. 1603-1621 | DOI | MR

[15] Tan, Ying; Su, Weidong 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

Cité par Sources :