Why do solid particles form quadric hypersurfaces in thermovibrational fluid flow?

Recent experiments conducted in space have confirmed the discovery of a new category of inertial particle attractors in fluid flow, originally predicted (a decade ago) on the basis of numerical simulations. Although the underlying physical mechanism have been clarified over the years through a modeling hierarchy based on a wide range of model types, spanning from basic to intricate, where different processes were activated or deactivated, and the outcomes carefully diagnosed, characterization of these phenomena under a precise “geometrical” perspective is still missing due to the inherent difficulties related to any attempt to unveil the underlying (hidden) “mathematical order”. This study represents the first attempt to fill this gap. In particular, in our endeavor to meet this objective, the discussion is articulated along two different threads, that is, 1) creating the right connections between the emerging particle formations and the geometry of quadratic surfaces, which are generally represented as level sets of second-degree polynomial expressions, and 2) defining the affine and/or projective transformations that can be used to turn one surface into another as the physical conditions that can produce them are varied accordingly. Thereby a general mathematical framework is defined that can encapsulate in a single treatment the outcomes of CFD (Computational Fluid Dynamics) simulations based on a hybrid Eulerian-Lagrangian approach and the idealized mathematical shapes mimicking the particle formations obtained accordingly. The resulting framework not only represents a more fundamental description of this subject than what preceded it, it also provides a rich context for a more rational and thorough characterization of these phenomena, thereby giving this discovery a firm mathematical grounding.