Linear stability of buoyant thermocapillary convection for a high-Prandtl number fluid in a laterally heated liquid bridge

Abstract

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.

Stability of thermocapillary flow in liquid bridges fully coupled to the gas phase

“…2015; Motegi, Fujimura & Ueno 2017 a ) investigations, only a few of which can be cited here. Investigations of the full-zone problem are sparse (see, however, Wanschura, Kuhlmann & Rath 1997 a ; Kasperski, Batoul & Labrosse 2000; Lappa 2003, 2004, 2005; Hu, Tang & Li 2008; Motegi, Kudo & Ueno 2017 b ).…”

“…As these flow instabilities are related to the striations found in crystals produced by the floating-zone technique, much effort has been devoted to flow instabilities in the half-zone model (Kuhlmann 1999), which led to numerous experimental (Preisser et al 1983;Velten, Schwabe & Scharmann 1991;Takagi et al 2001;Ueno, Tanaka & Kawamura 2003;Gaponenko, Mialdun & Shevtsova 2012;Yano et al 2017;Kang et al 2019) and numerical (Wanschura et al 1995;Leypoldt, Kuhlmann & Rath 2000;Levenstam, Amberg & Winkler 2001;Lappa, Savino & Monti 2001;Shevtsova, Gaponenko & Nepomnyashchy 2013;Li et al 2015;Motegi, Fujimura & Ueno 2017a) investigations, only a few of which can be cited here. Investigations of the full-zone problem are sparse (see, however, Wanschura, Kuhlmann & Rath 1997a;Kasperski, Batoul & Labrosse 2000;Lappa 2003Lappa , 2004Lappa , 2005Hu, Tang & Li 2008;Motegi, Kudo & Ueno 2017b).…”

“…The most popular approximation is to consider the interface indeformable (Shevtsova & Legros 1998; or even cylindrical (see e.g. Neitzel et al 1993;Wanschura et al 1995). Furthermore, the ambient atmosphere is often considered a passive gas that does not exert any viscous stresses on the interface and which may even be considered adiabatic.…”

Effect of the supporting disks shape on nonlinear flow dynamics in a liquid bridge

“…4,5 In a non-isothermal liquid bridge, the three-dimensional (3D) studies were mostly limited to the examination of a straight cylindrical column with flat solid supporting disks although with different aspect ratios. [6][7][8] From a theoretical point of view, such liquid bridges provide a unique system to test fundamental theories of nonlinear dynamics and chaos. [9][10][11] An oscillatory instability, which develops from the steady axisymmetric flow, may generate standing or traveling hydrothermal waves (HTW).…”

Hydrothermal waves in a liquid bridge subjected to a gas stream along the interface

“…In a one-phase model the interfacial heat transfer is neglected or, if considered, assumed in the form of Newton's law. The heat flux can be set directly (Motegi, Kudo & Ueno 2017) or, more often, through the Biot number and the temperature jump between the free surface and the reference temperature. The Biot number is defined as , where is the heat transfer coefficient, is the linear scale, is the thermal conductivity of the liquid.…”