On the onset of multi-wave patterns in laterally heated floating zones for slightly supercritical conditions

Abstract

This analysis follows and integrates the line of inquiry started in past author’s works [M. Lappa, “Three-dimensional numerical simulation of Marangoni flow instabilities in floating zones laterally heated by an equatorial ring,” Phys. Fluids 15(3), 776–789 (2003) and “Combined effect of volume and gravity on the three-dimensional flow instability in non-cylindrical floating zones heated by an equatorial ring,” ibid. 16(2), 331–343 (2004)] about the typical instabilities of the Marangoni flow and associated hierarchy of bifurcations in laterally heated floating zones with various shapes and aspect ratios. The main motivation for re-examining this kind of problems, which have attracted so much attention over the last twenty years, is the recent discovery [M. Kudo et al., “Transition of thermocapillary convection in a full-zone liquid bridge,” Trans. JSME (in Japanese) 80(812), TEP0095 (2014)] of a chaotic state in the region of the space of parameters where on the basis of existing theories and earlier results for the classical liquid-bridge problem with organic fluids, the flow should be relatively regular in time and with a simple structure in space. Axisymmetric computations are used to obtain the steady basic state, and then the Navier Stokes equations are solved in their complete, three-dimensional, time-dependent, and non-linear formulation to investigate the evolution of azimuthal disturbances. It is shown that the “apparent” doubling or quadrupling of the azimuthal wavenumber in the equatorial plane, previously reported for the case of floating zones of liquid metals, is replaced for high-Prandtl-number liquids by the complex interaction of disturbances with distinct spatial and temporal scales. These disturbances become critical at relatively comparable values of the Marangoni number. The unexpected multiplicity of waveforms and competition of spatial modes are explained according to the increased complexity of the considered system in terms of flow topology and structure with respect to the classical half-zone configuration.