Comptes Rendus
no. 8
Joseph Boussinesq and his approximation: a contemporary view
[Joseph Boussinesq et son approximation : un aperçu actuel]
Radyadour Kh. Zeytounian1
1 12, rue Saint-Fiacre, 75002 Paris, France
Comptes Rendus. Mécanique, Volume 331 (2003) no. 8, pp. 575-586.

A hundred years ago, in his 1903 volume II of the monograph devoted to ‘Théorie Analytique de la Chaleur’, Joseph Valentin Boussinesq observes that: “The variations of density can be ignored except were they are multiplied by the acceleration of gravity in equation of motion for the vertical component of the velocity vector.” A spectacular consequence of this Boussinesq observation (called, in 1916, by Rayleigh, the ‘Boussinesq approximation’) is the possibility to work with a quasi-incompressible system of coupled dynamic, (Navier) and thermal (Fourier) equations where buoyancy is the main driving force. After a few words on the life of Boussinesq and on his observation, the applicability of this approximation is briefly discussed for various thermal, geophysical, astrophysical and magnetohydrodynamic problems in the framework of ‘Boussinesquian fluid dynamics’. An important part of our contemporary view is devoted to a logical (100 years later) justification of this Boussinesq approximation for a perfect gas and an ideal liquid in the framework of an asymptotic modelling of the full fluid dynamics (Euler and Navier–Stokes–Fourier) equations with especially careful attention given to the validity of this approximation.

En 1903, Gauthier-Villars éditait à Paris le tome II, du traité de Joseph Boussinesq intitulé : « Théorie Analytique de la Chaleur ». A la page VII de l'Avertissement à ce tome II Boussinesq écrit :

« .. il fallait encore observer que, dans la plupart des mouvements provoqués par la chaleur sur nos fluides pesants, les volumes ou les densités se conservent à très peu près, quoique la variation correspondante du poids¯ de l'unité de volume soit justement la cause des phénomènes qu'il s'agit d'analyser.

De là résulte la possibilité de négliger les variations de la densité, là où elles ne sont pas multipliées par la gravité g, tout en conservant, dans les calculs, leur produit par celle-ci ».

Cette observation est, ce que l'on appelle, aujourd'hui : « l'approximation de Boussinesq » (en accord avec l'appellation, en 1916, de Rayleigh), et une conséquence spectaculaire en est la possibilité de considérer un système d'équations quasi-incompressible couplé pour la dynamique (équation de Navier) et la température (équation de Fourier) pour lequel la poussée d'Archimède est la force active principale régissant le mouvement. Après un bref aperçu sur la vie de Boussinesq et sur son observation, l'application de l'approximation de Boussinesq (dans le cadre d'une « dynamique des fluides de Boussinesq ») pour les problèmes thermiques, géophysiques, astrophysiques et magnétohydrodynamiques fait l'objet de divers commentaires. Une part importante de notre aperçu actuel est consacrée à une justification logique de cette approximation de Boussinesq (100 ans après) pour un gaz parfait et un liquide ideal, dans le cadre d'une modélisation asymptotique des équations (d'Euler et de Navier–Stokes–Fourier) de la dynamique des fluides, avec une attention toute particulière pour ce qui concerne la validité de cette approximation.

Publié le :
DOI : 10.1016/S1631-0721(03)00120-7
Keywords: Fluid mechanics, Asymptotic modelling in fluid dynamics, Thermal convection, Geo-astro physical fluid dynamics, Magneto-hydrodynamics
Mots-clés : Mécanique des fluides, Modélisation asymptotique en dynamique des fluides, Convection thermique, Dynamique des fluides, Géo-astro physiques : Magnéto-hydrodynamique

Radyadour Kh. Zeytounian 1

1 12, rue Saint-Fiacre, 75002 Paris, France 
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Radyadour Kh. Zeytounian. Joseph Boussinesq and his approximation: a contemporary view. Comptes Rendus. Mécanique, Volume 331 (2003) no. 8, pp. 575-586. doi : 10.1016/S1631-0721(03)00120-7. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00120-7/

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  • Fei Yang; Xuejun Shao; Xudong Fu; Ehsan Kazemi Simulated Flow Velocity Structure in Meandering Channels: Stratification and Inertia Effects Caused by Suspended Sediment, Water, Volume 11 (2019) no. 6, p. 1254 | DOI:10.3390/w11061254
  • Yu Zhou; Mengying Wang; Manning Wang; Yi Wang Predictive accuracy of Boussinesq approximation in opposed mixed convection with a high-temperature heat source inside a building, Building and Environment, Volume 144 (2018), p. 349 | DOI:10.1016/j.buildenv.2018.08.043
  • Alejandro Allendes; Gabriel R. Barrenechea; César Naranjo A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem, Computer Methods in Applied Mechanics and Engineering, Volume 340 (2018), pp. 90-120 | DOI:10.1016/j.cma.2018.05.020 | Zbl:1440.65170
  • Eduard Feireisl Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2018), p. 2771 | DOI:10.1007/978-3-319-13344-7_70
  • Nabila Labsi; Youb Khaled Benkahla; Abdelkader Boutra; Meriem Titouah Convection heat transfer inside a lid-driven cavity filled with a shear-thinning Herschel–Bulkley fluid, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Volume 40 (2018) no. 3 | DOI:10.1007/s40430-018-1051-6
  • Bernardo Alan de Freitas Duarte; Rafael Romão da Silva Melo; Millena Martins Villar; Ricardo Serfaty; Aristeu da Silveira Neto An extension of Oberbeck–Boussinesq approximation for thermal convection problems, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Volume 40 (2018) no. 6 | DOI:10.1007/s40430-018-1181-x
  • Yu Zhou; Yelin Deng; Peng Wu; Shi-Jie Cao The effects of ventilation and floor heating systems on the dispersion and deposition of fine particles in an enclosed environment, Building and Environment, Volume 125 (2017), p. 192 | DOI:10.1016/j.buildenv.2017.08.049
  • Philipp W. Schroeder; Gert Lube Stabilised DG-FEM for incompressible natural convection flows with boundary and moving interior layers on non-adapted meshes, Journal of Computational Physics, Volume 335 (2017), pp. 760-779 | DOI:10.1016/j.jcp.2017.01.055 | Zbl:1380.65286
  • Aneta Wróblewska-Kamińska The asymptotic analysis of the complete fluid system on a varying domain: from the compressible to the incompressible flow, SIAM Journal on Mathematical Analysis, Volume 49 (2017) no. 5, pp. 3299-3334 | DOI:10.1137/15m1029655 | Zbl:1375.35407
  • Eduard Feireisl; Antonín Novotný Fluid Flow Modeling, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 1 | DOI:10.1007/978-3-319-63781-5_1
  • Eduard Feireisl; Antonín Novotný Asymptotic Analysis: An Introduction, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 145 | DOI:10.1007/978-3-319-63781-5_4
  • Eduard Feireisl; Antonín Novotný Singular Limits: Low Stratification, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 167 | DOI:10.1007/978-3-319-63781-5_5
  • Eduard Feireisl; Antonín Novotný Weak Solutions, A Priori Estimates, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 21 | DOI:10.1007/978-3-319-63781-5_2
  • Eduard Feireisl; Antonín Novotný Stratified Fluids, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 221 | DOI:10.1007/978-3-319-63781-5_6
  • Eduard Feireisl; Antonín Novotný Interaction of Acoustic Waves with Boundary, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 263 | DOI:10.1007/978-3-319-63781-5_7
  • Eduard Feireisl; Antonín Novotný Problems on Large Domains, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 313 | DOI:10.1007/978-3-319-63781-5_8
  • Eduard Feireisl; Antonín Novotný Vanishing Dissipation Limits, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 369 | DOI:10.1007/978-3-319-63781-5_9
  • Eduard Feireisl; Antonín Novotný Acoustic Analogies, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 409 | DOI:10.1007/978-3-319-63781-5_10
  • Eduard Feireisl; Antonín Novotný Appendix, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 429 | DOI:10.1007/978-3-319-63781-5_11
  • Eduard Feireisl; Antonín Novotný Existence Theory, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 49 | DOI:10.1007/978-3-319-63781-5_3
  • Eduard Feireisl; Antonín Novotný Bibliographical Remarks, Singular Limits in Thermodynamics of Viscous Fluids (2017), p. 501 | DOI:10.1007/978-3-319-63781-5_12
  • Eduard Feireisl Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2016), p. 1 | DOI:10.1007/978-3-319-10151-4_70-1
  • Aicha Belcaid; Georges Le Palec; Abdeslam Draoui Numerical and experimental study of Boussinesq wall horizontal turbulent jet of fresh water in a static homogeneous environment of salt water, Journal of Hydrodynamics, Volume 27 (2015) no. 4, p. 604 | DOI:10.1016/s1001-6058(15)60522-4
  • Christian Komo Influence of surface roughness to solutions of the Boussinesq equations with Robin boundary condition, Revista Matemática Complutense, Volume 28 (2015) no. 1, pp. 123-151 | DOI:10.1007/s13163-014-0152-8 | Zbl:1309.35084
  • Christian Komo Optimal initial value conditions for the existence of strong solutions of the Boussinesq equations, Annali dell'Università di Ferrara. Sezione VII. Scienze Matematiche, Volume 60 (2014) no. 2, pp. 377-396 | DOI:10.1007/s11565-014-0208-1 | Zbl:1307.35228
  • Eduard Feireisl; Antonín Novotný Inviscid incompressible limits under mild stratification: a rigorous derivation of the Euler-Boussinesq system, Applied Mathematics and Optimization, Volume 70 (2014) no. 2, pp. 279-307 | DOI:10.1007/s00245-014-9243-7 | Zbl:1308.76231
  • Anthony Randriamampianina; Emilia Crespo del Arco A High‐Resolution Method for Direct Numerical Simulation of Instabilities and Transitions in a Baroclinic Cavity, Modeling Atmospheric and Oceanic Flows (2014), p. 297 | DOI:10.1002/9781118856024.ch16
  • Franck Varenne Modélisation et interdisciplinarité, Modélisation et interdisciplinarité (2014), p. 299 | DOI:10.3917/quae.nicol.2014.01.0299
  • Elisabetta Felaco; Ingenuin Gasser Modelling, asymptotic analysis and simulation of the gas dynamics in a chimney, Journal of Mathematics in Industry, Volume 3 (2013), p. 20 (Id/No 3) | DOI:10.1186/2190-5983-3-3 | Zbl:1274.35282
  • Maria Bauer; Elisabetta Felaco; Ingenuin Gasser On One-Dimensional Low Mach Number Applications, Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Volume 120 (2013), p. 25 | DOI:10.1007/978-3-642-33221-0_2
  • Esmail M. A. Mokheimer Buoyancy Effects on Entropy Generation in the Entrance Region of Isothermal/Adiabatic Vertical Channel, Arabian Journal for Science and Engineering, Volume 37 (2012) no. 6, p. 1681 | DOI:10.1007/s13369-012-0271-9
  • Eduard Feireisl; Maria E. Schonbek On the Oberbeck–Boussinesq Approximation on Unbounded Domains, Nonlinear Partial Differential Equations, Volume 7 (2012), p. 131 | DOI:10.1007/978-3-642-25361-4_7
  • Angiolo Farina; Antonio Fasano; Andro Mikelić Non-Isothermal Flow of Molten Glass: Mathematical Challenges and Industrial Questions, Mathematical Models in the Manufacturing of Glass, Volume 2010 (2011), p. 173 | DOI:10.1007/978-3-642-15967-1_4
  • Alexander F. Shchepetkin; James C. McWilliams Accurate Boussinesq oceanic modeling with a practical, “Stiffened” Equation of State, Ocean Modelling, Volume 38 (2011) no. 1-2, p. 41 | DOI:10.1016/j.ocemod.2011.01.010
  • Gary B. Brassington System Design for Operational Ocean Forecasting, Operational Oceanography in the 21st Century (2011), p. 441 | DOI:10.1007/978-94-007-0332-2_18
  • Eduard Feireisl Mathematical models of incompressible fluids as singular limits of complete fluid systems, Milan Journal of Mathematics, Volume 78 (2010) no. 2, pp. 523-560 | DOI:10.1007/s00032-010-0128-1 | Zbl:1222.76030
  • N. A. Vinnichenko; A. V. Uvarov Improvement of the Oberbeck-Boussinesq approximation for solving unstationary convection problems in a closed cavity, Moscow University Physics Bulletin, Volume 65 (2010) no. 6, p. 454 | DOI:10.3103/s0027134910060056
  • The Simple Rayleigh (1916) Thermal Convection Problem, Convection in Fluids, Volume 90 (2009), p. 55 | DOI:10.1007/978-90-481-2433-6_3
  • A. Barletta Local energy balance, specific heats and the Oberbeck-Boussinesq approximation, International Journal of Heat and Mass Transfer, Volume 52 (2009) no. 21-22, pp. 5266-5270 | DOI:10.1016/j.ijheatmasstransfer.2009.06.006 | Zbl:1255.76128
  • Eduard Marušić-Paloka; Igor Pažanin Modelling of heat transfer in a laminar flow through a helical pipe, Mathematical and Computer Modelling, Volume 50 (2009) no. 11-12, pp. 1571-1582 | DOI:10.1016/j.mcm.2009.09.006 | Zbl:1185.76449
  • K. R. Rajagopal; G. Saccomandi; L. Vergori On the Oberbeck-Boussinesq approximation for fluids with pressure dependent viscosities, Nonlinear Analysis. Real World Applications, Volume 10 (2009) no. 2, pp. 1139-1150 | DOI:10.1016/j.nonrwa.2007.12.003 | Zbl:1167.76368
  • Dorin Bucur; Eduard Feireisl The incompressible limit of the full Navier-Stokes-Fourier system on domains with rough boundaries, Nonlinear Analysis. Real World Applications, Volume 10 (2009) no. 5, pp. 3203-3229 | DOI:10.1016/j.nonrwa.2008.10.024 | Zbl:1162.76048
  • E. Feireisl Chapter 2 Mathematical Methods in the Theory of Viscous Fluids, Volume 4 (2008), p. 57 | DOI:10.1016/s1874-5717(08)00002-9
  • E. Feireisl Asymptotic Properties of a Class of Weak Solutions to the Navier–Stokes–Fourier System, Hyperbolic Problems: Theory, Numerics, Applications (2008), p. 511 | DOI:10.1007/978-3-540-75712-2_49
  • Javier Principe; Ramon Codina A stabilized finite element approximation of low speed thermally coupled flows, International Journal of Numerical Methods for Heat Fluid Flow, Volume 18 (2008) no. 7-8, pp. 835-867 | DOI:10.1108/09615530810898980 | Zbl:1231.76162
  • ANGIOLO FARINA; ANTONIO FASANO; ANDRO MIKELIĆ ON THE EQUATIONS GOVERNING THE FLOW OF MECHANICALLY INCOMPRESSIBLE, BUT THERMALLY EXPANSIBLE, VISCOUS FLUIDS, Mathematical Models and Methods in Applied Sciences, Volume 18 (2008) no. 06, p. 813 | DOI:10.1142/s0218202508002875
  • Eduard Feireisl; Antonín Novotný; Hana Petzeltová Layered incompressible fluid flow equations in the limit of low Mach number and strong stratification, Physica D, Volume 237 (2008) no. 10-12, pp. 1466-1487 | DOI:10.1016/j.physd.2008.03.027 | Zbl:1143.76562
  • Chantal Staquet Internal gravity waves: parametric instability and deep ocean mixing, Comptes Rendus. Mécanique, Volume 335 (2007) no. 9-10, p. 665 | DOI:10.1016/j.crme.2007.08.009
  • Эдуард Файрайзл; Eduard Feireisl Асимптотический анализ полной системы Навье - Стокса - Фурье: от течений сжимаемой к течениям несжимаемой жидкости, Успехи математических наук, Volume 62 (2007) no. 3, p. 169 | DOI:10.4213/rm6803
  • Radyadour Kh. Zeytounian The Many Faces of the Asymptotics of Low-Mach-Number Flows, Topics in Hyposonic Flow Theory, Volume 672 (2005), p. 27 | DOI:10.1007/11414346_2

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