Benard Layers with Heat or Mass Transfer

  • M. G. Velarde
Part of the International Centre for Mechanical Sciences book series (CISM, volume 428)


I have taken the Benard liquid-layer set-up to succintly illustrate the variety of phenomena and mathematical problems that the Marangoni stress can originate already in the simplest case of (l+l)D geometry. This surface (tangential) stress may be due to heat or mass transfer creating a surface tension gradient that ultimately leads to flow motion and/or instability and subsequent convection (Marangoni effect). For the (1+1)D geometry I provide, first, under drastically simplifying approximations the conditions for the onset and nonlinear evolution of steady convection (as a caricature of Benard cells). The evolutionary problem is presented for two different effective gravity levels, one of them corresponding to microgravity (the kind of effective gravity existing in the free fall of an Earth‘s orbital space lab. or station). Then I present the theory when instability leads to oscillatory motions and waves. I give discussion of physical mechanisms leading to either capillary-gravity (transverse) waves or dilational (sound-like, longitudinal) waves, and even internal waves, and their possible resonance. For the transverse mode I discuss its nonlin­ear evolution by deriving a (“long wave”, shallow layer) generalized (dissipation-modified) Boussinesq-Korteweg-de Vries equation. I discuss how solitary waves and solitons are expected to appear in the Benard layer. Finally, I comment on a few experimental results about dissipative solitons.


Solitary Wave Rayleigh Number Internal Wave Liquid Layer Marangoni Number 
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© Springer-Verlag Wien 2002

Authors and Affiliations

  • M. G. Velarde
    • 1
  1. 1.Instituto PluridisciplinarUCMMadridSpain

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