Floquet analysis of spatially periodic thermocapillary convection in a low-Prandtl-number liquid bridge

Abstract

Secondary instability of thermocapillary convection was investigated for a half-zone liquid bridge with low Prandtl number fluids. The liquid bridge was suspended between two cylindrical flat disks, which were maintained at different temperatures. The thermocapillary-driven flow formed an axisymmetric steady toroidal vortex. If the temperature difference between the two disks exceeded a certain threshold, the axisymmetric flow transitioned to an azimuthal periodic steady flow. The objective in the analysis of secondary instability is to examine the stability limit of this steady flow. Employing Floquet analysis, we obtained the neutral stability curves as a function of the aspect ratio (radius/height) of the liquid bridge and identified the critical modes. The critical modes of the Floquet analysis are referred to as critical Floquet modes and are classified by Floquet parameter β, which characterizes a set of azimuthal wave numbers of the Floquet mode. The Newton–Krylov method and Arnoldi method are implemented to solve the large-scale nonlinear equations and generalized eigenvalue problems, respectively. Two critical Floquet modes were observed having different Floquet parameters, β = 0 and β = 1, which appear preferential in the liquid bridge for high and low aspect ratios, respectively. The mode with β = 1 was steady for low aspect ratios and was able to change to an oscillatory mode with a frequency of oscillation that rapidly increased with increasing aspect ratio. We visualized the temperature and azimuthal velocity distributions of the critical Floquet modes and its Fourier components. We concluded that the mode with β = 1 leads to a pulsating oscillation, denoted as “P-type,” whereas that with β = 0 leads to a twisting oscillation, denoted as “T-type.” Both types of oscillations were reported in previous studies. The present neutral stability curves are for the most part in good agreement with critical Reynolds numbers previously obtained in numerical simulations.