Some Further Results on the Bénard Problem

Abstract

The exact solution of the sixth-order differential equation which governs the stability of a viscous fluid contained between two rigid walls and heated from below is briefly reviewed and extended to include detailed results on the curve of neutral stability and the cell pattern at the onset of instability. Two approximate methods of solution are then discussed which employ a Fourier or Fourier-type expansion and which require the solution of only a fourth- or a second-order differential equation. A comparison of these approximate results with the exact solution gives some insight into the relative accuracy of these methods when applied to other more general problems for which an exact solution cannot be obtained.

A Lattice Boltzmann Method for a Binary Miscible Fluid Mixture and Its Application to a Heat-Transfer Problem

“…3, the solid straight line is drawn through the data points using the least-squares fitting and the intersection with the horizontal axis gives the critical Rayleigh number. The calculated critical Rayleigh number is 1708.48 and agrees well with the theoretical value of 1707.76 obtained by linear stability theory [25] within 0.042 %.…”

Large-scale instabilities in a nonrotating turbulent convection

“…Following the procedure described in [37,38,39], we adopted the even solution which has minimum critical effective Rayleigh number. Our numerical analysis showed that the anisotropy of turbulent The effective Rayleigh number versus the aspect ratio L z /L ⊥ of the perturbations for two rigid boundaries is plotted in Fig.…”

On the emergence of patterns of order

“…There is experimental evidence for such a process. 15 But the presence of such distortions does not explain why in fully developed turbulence the longitudinal vortices seem to play a more important part than transverse vortices. It is as if they completely take over the role of the latter in keeping up the mixing process between the horizontal layers, which is characteristic for turbulence.…”

“…Mathematical investigations have been carried out by Lord Rayleigh, H. Jeffreys, and A. R. Low; see for instance [14]. Some newer calculations have been given by W. H. Reid and D. L. Harris, see [15].…”

“…17 15 See F. R. Hama [24; further publications of Professor Hama and his group on this subject are forthcoming]; and J. R. Weske [25]. 18 Compare the discussion on the relation between the "momentum transfer theory" and the "vorticity transfer theory" in S. Goldstein [26] and for more details: A.…”