Quantitative comparison of molecular dynamics with hydrodynamics in Rayleigh-Bénard convection

Two-dimensional convection in a compressible hard-disk fluid confined between free boundaries is studied from both a microscopic and a macroscopic level, i.e., by molecular-dynamics (MD) simulation on the one hand and by (analytical and numerical) solution of the hydrodynamic equations on the other hand. For a fluid of 5000 hard disks in a rectangular box of an aspect ratio of 2, both approaches are in remarkable quantitative agreement, showing a stable double-roll convection pattern in the final state. From the hydrodynamical point of view the system is just beyond instability (R=1.07Rc) and involves the nonlinear coupling of two modes (codimension-2 bifurcation). The simplest and optimal system to study with MD has an aspect ratio of 1 and involves 1500 particles only. Here, a stable single-roll pattern is observed. The hydrodynamic analysis, though, displays less good quantitative agreement with MD, which is due to the stronger influence of microscopic boundary effects.