A mathematical and numerical framework for the simulation of oscillatory buoyancy and Marangoni convection in rectangular cavities with variable cross section

It is often assumed that two-dimensional 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 two-dimensional and three-dimensional 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 backward-facing or forward-facing 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 liquid-gas surface (representative of the Bridgman crystal growth technique) and for two specific fluids pertaining to the above-mentioned 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 two-dimensional 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.