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Convection in Fluids pp 85–132Cite as

The Bénard (1900, 1901) Convection Problem, Heated from below

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Part of the book series: Fluid Mechanics and its Applications ((FMIA,volume 90))

In this chapter, we take into account the influence of a deformable free surface and, as a consequence, we revisit the mathematical formulation of the classical problem describing the Bénard instability of a horizontal layer of fluid, heated from below, and bounded by an upper deformable free surface. Because the deformation of the free surface, subject to a temperature-dependent surface tension, is taken into account in the full Bénard convection problem, heated from below, we have not specified this convection as being a ‘thermal convection’.

Indeed, in the full starting Bénard problem, heated from below, when we take into account the influence of a deformable free surface, subject to a temperature-dependent surface tension, the fluid being an expansible liquid, it is necessary to take into account, simultaneously, four main effects. Namely:

  1. (a)

    the conduction adverse temperature gradient (Bénard) effect in motionless steady-state conduction temperature

  2. (b)

    the temperature-dependent surface tension (Marangoni) effect

  3. (c)

    the heat flux across the upper, free surface (Biot) effect

  4. (d)

    the buoyancy (Archimedean—Boussinesq) effect arising from the volume (gravity) forces.

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(2009). The Bénard (1900, 1901) Convection Problem, Heated from below. In: Convection in Fluids. Fluid Mechanics and its Applications, vol 90. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2433-6_4

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  • DOI: https://doi.org/10.1007/978-90-481-2433-6_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-2432-9

  • Online ISBN: 978-90-481-2433-6

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