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 (Phys. Fluids, 15(3): 776-789, 2003, and Phys. Fluids 16(2): 331-343, 2004) about the typical instabilities of 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 so much attention have attracted over the last twenty years, is the recent discovery (Kudo, Ueno and Kawamura, (2014), in Japanese, DOI: 10.1299/transjsme.2014tep0095) of a chaotic state in 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, threedimensional, 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 is 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.

Hydrothermal waves in two-dimensional liquid layers with sudden changes in the available cross-section

“…Along these lines, let us recall that in the presence of a free liquid/gas interface, the so-called Marangoni (thermocapillary) flow often plays a dominant role and, therefore, the aforementioned undesired effects must be ascribed primarily to the typical instability modes of this kind of convection. In many circumstances, these modes correspond to the so-called hydrothermal waves (HTWs) originally predicted by Smith and Davis (1983) for the idealized case of a layer of infinite extent and observed by many other authors in more complex geometrical configurations (Bucchignani, 2004; Shevtsova et al , 2001, 2003, 2009, 2011; Lappa et al , 2000, 2003; Ueno et al , 2003; Melnikov et al , 2004, 2005; Lappa, 2013, 2014, 2016; Schwabe et al , 1992, 2003; Li et al , 2003, 2006; Shi and Imaishi, 2006).…”

On the oscillatory hydrodynamic instability of gravitational thermal flows of liquid metals in variable cross-section containers

“…Some interesting analogies could be perhaps identified with a completely different phenomenon, typically occurring in high-Pr fluids subjected to thermally induced surfacetension gradients (namely, the so-called hydrothermal waves of the Marangoni flow, Smith and Davis, 50 Shevtsova et al, 51 and Lappa [52][53][54] ). Such waves, which also typically manifest as a chain of traveling cells, however, do not display the local roll coalescence phenomena seen here.…”

On the formation and propagation of hydrothermal waves in liquid layers with phase change

“…However, these authors limited to the case of semiconductor melts (Pr<<1) in floating zones. It is a well-known fact (see, e.g., Lappa [32]) that for these values of the Prandtl number, the typical instabilities of Marangoni flow display notable differences with respect to those expected for high-Pr materials such as oxide melts or succinonitrile.…”