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.

The buoyancy effect on the stability of axisymmetric buoyant-thermocapillary flow is investigated in a laterally heated high-Prandtl-number liquid bridge using linear stability analysis. Target geometry is the so-called full-zone (FZ) model in which the liquid is sustained between the coaxial cylindrical disks of the same diameter. The disks are maintained at the same temperature, and the mid part of the liquid bridge is heated, resulting in a non-uniform temperature distribution over the free surface. In that model, axisymmetric basic flow exhibits reflection symmetry around the midplane, and two identical toroidal vortices are formed in the upper and lower halves in zero-gravity conditions. However, the buoyancy breaks this symmetry in gravity conditions. There are two different types of perturbation in the FZ model, the symmetric and antisymmetric modes around the mid plane of the liquid bridge. When increasing the Rayleigh number Ra, the buoyancy strongly stabilizes the basic flow for the antisymmetric oscillatory mode and has a weak destabilizing effect on the symmetric oscillatory mode. Therefore, when Ra exceeds a certain threshold value, the most dangerous mode switches from the antisymmetric oscillatory mode, the most dangerous mode under zero-gravity conditions, to the symmetric oscillatory mode. The neutral stability curve of the symmetric oscillatory mode folds with increasing Ra, wherein the critical Reynolds number suddenly drops. We reveal that such an abrupt change in the neutral curve is caused by the transition of the instability source from the vortex in the upper half of the liquid bridge to the one in the lower half by increasing the buoyancy effect. With a further increase in the Ra, the most dangerous mode switches from the symmetric oscillatory mode to the antisymmetric steady mode.

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We investigate the secondary instability of thermocapillary-driven convection in a high-Pr liquid bridge (Pr = 4) of half-zone geometry via direct numerical simulation. The convection is known to exhibit a three-dimensional time-dependent 'oscillatory' state with a distinct azimuthal modal structure, that is, spatio-temporally periodic state after the onset of the primary instability. We indicate that the convection exhibits another transition to spatiotemporally quasi-periodic states by increasing the intensity of the thermocapillary effect. The proper orthogonal decomposition (POD) is employed in order to extract the variation of the flow structures before/after the secondary instability. After the primary instability, one finds the oscillatory flow with a fundamental azimuthal modal number. It is indicated that the flow field consists of the primary component with the fundamental modal structure and the components with fundamental modal structures of higher harmonics of the primary components. Those components dominate the whole flow field. After the onset of secondary instability, additional components emerge in the flow; those components consist of the the fundamental azimuthal modal structures, which are different from higher harmonics of the primary flow field. We determine the onset condition or critical Reynolds number for the secondary instability Re c (2) by monitoring the development of the energy of the newley arisen components. It is found that the secondary instability evaluated through decomposed flow structures via nonlinear simulation corresponds to that predicted through Floquet modes via linear stability analysis.

The effect of ambient air flow on flow-transition points in thermocapillary convection was investigated using a floatingzone method (full-zone liquid bridge) with a high Prandtl number fluid (Pr = 28.1) under normal gravity conditions. In the liquid bridge, convection changes from two-dimensional steady flow to three-dimensional unsteady flow at a flowtransition point. A pair of partition plates was employed to suppress the ambient air flow. To understand the flow and thermal fields of the ambient air, flow was visualized using smoke and temperature was measured using a thermocouple. Thermocapillary convection was stabilized by suppressing ambient air flow. The primary stabilization factor is heat transfer from the ambient air to the liquid bridge through the free surface. These results suggest that flow-transition point was controllable by modifying ambient air temperature.