We consider the instability of the steady, axisymmetric thermocapillary convection in cylindrical liquid bridges. Finite-difference method is applied to compute the steady axisymmetric basic solutions, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for liquid bridges of O(1) aspect ratio Γ (=length/radius) the first instability of the basic state is through either a regular bifurcation (stationary) or Hopf bifurcation (oscillatory), depending on the Prandtl number of the liquid. The bifurcation points and the corresponding eigenfunctions are predicted precisely by solving appropriate extended systems of equations. For very small Prandtl numbers, i.e. P r < 0.06, the instability is of hydrodynamical origin that breaks the azimuthal symmetry of the basic state. The critical Reynolds number, for unit aspect ratio and insulated free surface, tends to be constant, Re c → 1784, as P r → 0, the most dangerous mode being m = 2. While for P r ≥ 0.1, the instability takes the form of a pair of hydrothermal waves traveling azimuthally. The most dangerous mode is m = 3 for 0.1 ≤ P r ≤ 0.8 and m = 2 for P r ≥ 0.9. Dependence of the critical Reynolds number on other parameters is also presented. Our results confirm in large part the recent lineartheory results of Wanschura et al. [7] and provide a more complete stability diagram for the finite half-zone with a non-deformable free surface.

Period-doubling transitions to chaos, periodic windows, strange attractors, and intermittencies are observed in direct numerical simulations of convection in a closed cavity with differentially heated vertical walls. The cavity contains a Newtonian-Boussinesq fluid and is subject to horizontal oscillatory displacements with a frequency ⍀. The transitions occur through a sequence of bifurcations that exhibit the features of a Feigenbaum-type scenario. The first transition from a single-frequency response to a two-frequency response occurs through a parametric excitation of the subharmonic mode ⍀/2 by the driving frequency ⍀. Bifurcation diagrams also exhibit periodic windows and reveal the self-similar structure of the ''period-doubling tree.'' Intermittent flows show characteristics corresponding to a Pomeau-Manneville type-I intermittency.

No abstract

No abstract