Abstract
It is often assumed that twodimensional flow can be used to model with an acceptable degree of approximation the preferred mode of instability of thermogravitational flows and thermocapillary flows in laterally heated shallow cavities for a relatively wide range of substances and conditions (essentially pure or compound semiconductor materials in liquid state for the case of buoyancy convection and molten oxide materials or salts and a variety of organic liquids for the case of Marangoni convection). In line with the general spirit of this book, such assumption is challenged by comparing twodimensional and threedimensional results expressly produced for such a purpose. More precisely, we present a general mathematical and numerical framework specifically developed to 1) explore the sensitivity of such phenomena to geometrical “irregularities” affecting the liquid container and 2) take advantage of a reduced number of spatial degrees of freedom when this is possible. Sudden variations in the shape of the container are modelled as a single backwardfacing or forwardfacing step on the bottom wall or a combination of both features. The resulting framework is applied to a horizontally extended configuration with undeformable free top liquidgas surface (representative of the Bridgman crystal growth technique) and for two specific fluids pertaining to the abovementioned categories of materials, namely molten silicon (Pr<1) and silicone oil (Pr>1). The assumption of flat interface is justified on the basis of physical reasoning and a scaling analysis. The overall model proves successful in providing useful insights into the stability behaviour of these fluids and the departure from the approximation of twodimensional flow. It is shown that the presence of a topography in the bottom wall can lead to a variety of situations with significant changes in the emerging waveforms.
Original language  English 

Title of host publication  Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics 
Editors  Alexander Gelfgat 
Place of Publication  Cham. 
Publisher  Springer 
Chapter  12 
Pages  419458 
Number of pages  40 
Volume  50 
ISBN (Print)  9783319914930 
DOIs  
Publication status  Epub ahead of print  7 Jul 2018 
Publication series
Name  Springer Mathematical Series 

Keywords
 twodimensional flow
 thermogravitational flows
 oscillatory buoyancy
 shallow cavities
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Marcello Lappa
Person: Academic