Towards a computer-assisted global linear stability analysis of fluid particles

For many technical processes, the dynamics of rising or falling liquid particles are of paramount importance. Their paths and forms show a complex spectrum of physical phenomena, involving instabilities and bifurcations. Despite numerous numerical and experimental investigations, many of these phenomena remain to be understood. While experimental access to local quantities is often not feasible without interfering with the flow, numerical studies allow their extraction without disturbance. The present work, therefore, is concerned with the development and implementation of an algorithm capable of global linear stability analysis of deformable fluid particles. In mathematical terms, stability is assessed by eigenvalues, which are calculated by a combination of a numerical flow solver and linear algebra. For this purpose, a stationary state of the system is superimposed with a small perturbation and numerically integrated to obtain the evolution of the perturbation. In the context of two-phase flows, the presence of dynamic interfaces requires a transformation onto a reference configuration, whose mathematical foundations are derived. The stability of two relevant prototypical configurations - spherical particles in zero gravity and droplets freely rising in an ambient fluid - is investigated, whereby the modes belonging to the unstable eigenvalues are described both quantitatively and qualitatively. For small perturbations, initialization of the numerical flow solver requires highly accurate volume fractions, whose computation in three spatial dimensions is mathematically demanding and resorts to a novel method developed in the context of this work. Exploiting the appropriate divergence theorems for hypersurfaces, in combination with an approximation motivated by differential geometry, allows transforming the integrals in three spatial dimensions to line integrals, implying a considerable reduction of complexity for their numerical quadrature. The consideration of topological configurations of the intersected computational cells allows for a robust implementation, that is numerically tested for spheres, ellipsoids, and disturbed spheres. The global errors of the volume approximation show up to fourth-order convergence with spatial resolution. Employing numerical methods for stability analysis of physical systems implies that the spectra found contain both numerical and physical elements. Surface tension is crucial for the flows considered here, the numerical treatment of which requires the calculation of mean curvatures from discrete volume fractions. A substantiated interpretation of the results of stability analysis, therefore, requires accurate knowledge of the properties of the implemented numerical schemes. A theoretical and numerical investigation of the height function method is conducted for several technically relevant hypersurfaces, with statistical processing flanking the interpretation. Provided sufficiently accurate volume fractions, the theoretical errors exhibit second-order convergence for smooth hypersurfaces. However, numerical experiments show that the quality of the volume fractions (in terms of global volume conservation) is not sufficient for the expected convergence. This is directly related to the unstable eigenvalues found