Bifurcation analysis of the thermocapillary convection in cylindrical liquid bridges

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 Gamma (= 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. Pr < 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 Pr --> 0, the most dangerous mode being m = 2. While for Pr greater than or equal to 0.1, the instability takes the form of a pair of hydrothermal waves traveling azimuthally. The most dangerous mode is in = 3 for 0.1 less than or equal to Pr less than or equal to 0.8 and m = 2 for Pr greater than or equal to 0.9. Dependence of the critical Reynolds number on other parameters is also presented. Our results confirm in large part the recent linear-theory results of Wanschura et al. [7] and provide a more complete stability diagram for the finite half-zone with a non-deformable free surface.