Oscillatory and turbulent flows of liquid metals in differentially heated systems with horizontal and non-horizontal walls

Non isothermal flows of liquid metals induced by buoyancy are central to many advanced technological applications in materials science, often at the cutting-edge of modern engineering. They have indeed a significant impact on the production of many materials obtained via the solidification of a melt. The quality and mechanical or electrical properties of the resulting solids and crystals are adversely affected by thermogravitational convection as it can induce defects in their atomic or molecular structure (this is the case, e.g., of typical crystal-growth techniques such as the horizontal Bridgman (HB), the Floating zone (FZ) or the Czochralski (CZ) methods). The present chapter aims to present a focused review of landmark (past) and very recent contributions on the nature, structure and hierarchy of instabilities of this type of convection. In particular, starting from simple situations corresponding to steady and laminar flows and moving towards fully developed turbulence, we present the typical hydrodynamic and hydrothermal disturbances emerging in differentially heated liquid metals and clarify the relationship among their properties and general influential factors such as: the degree of confinement (aspect ratio), morphology (wall orientation in space) and spatial degrees of freedom (number of active dimensions) of the domain hosting the melt. Manifestations of these modes of convection (including, but not limited to, transverse waves travelling in the downstream or in the upstream direction, standing waves, modulated pulso-traveling disturbances, longitudinal waves and multi-wave patterns) are discussed in detail. More complex situations are placed in the context of existing theories on turbulence in fluids and treated using concepts, methods and tools typical of the chaotic systems analysis.