Low Prandtl Number Fluids, a Paradigm for Dynamical System Studies

  • A. Libchaber
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 24)


In the recent development of experiments related to the road to chaos [1], fluids have played an important role. In this paper, we want to stress the usefulness of low Prandtl number fluids in Rayleigh-Benard experiments. In such studies, the three basic equations are related to the mass, momentum and energy conservation [2], the two last ones being called the Navier-Stokes and the Fourier equations. The Prandtl number is a dimensionless number which characterizes the fluid and depends on the energy and the momentum transport. A small Prandtl number fluid has a small kinematic viscosity or a large heat diffusivity. The richness of the fluid behavior is associated with the nonlinear terms in the equations. For the low Prandtl number case, the nonlinear term in the Navier-Stokes equation is the dominant one. When this term, \(\left( {\vec V\vec \nabla } \right)\vec V\), is larger than the diffusion one, \(\nu \Delta \vec V\), the Reynolds number for the velocity of convection is large. For a critical value of the Reynolds number, the fluid becomes unstable, through a Hopf bifurcation to a time-dependent state, called by Busse [2] the oscillatory instability (O.I. from now on). As the Prandtl number decreases, the relative importance of the nonlinear term increases, and the onset of the O.I. appears at lower Rayleigh numbers. One should notice that the O.I. is the only time-dependent transition occurring in the fluid, which can be derived from the three basic equations [2].


Prandtl Number Rayleigh Number Hopf Bifurcation Homoclinic Orbit Period Doubling 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • A. Libchaber
    • 1
  1. 1.Groupe de Physique des Solides de l’Ecole Normale SupérieureParis Cedex 05France

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