Fundamental Theories for Bulk-Surface Interactions: Evolving Interfaces, Phase-Field Adhesion and Mixed-Dimensional Flows - Boschman, Anne M. (2023)
In a wide range of engineering and biological applications, processes at a stationary or moving surface affect the dynamics in the bulk material. Examples of these processes include cellular migration caused by protein interactions at cell membranes, or the decrease in surface tension due to the presence of surface active compounds in cleaning products. Computational models are essential to gain insight into these bulk-surface interactions and to make accurate predictions about the behaviour of these complex systems. From a mathematical perspective, models proposed for the study of these phenomena are based on either a sharp or diffuse interpretation of the surface. Yet, the exact coupling of the bulk and surface dynamics is challenging due to its non-linear character, and many of the current bulk-surface models lack a comprehensive continuum theory as a foundation. This thesis aims to arrive at fundamental theories for bulk-surface interactions. To this end, it first outlines several frameworks used for the construction of models for moving surfaces. Although each framework offers distinct insights and motivation for the system’s dynamics, it is demonstrated that for particular modelling choices the same models can be established. Secondly, a framework is proposed to describe adhesive interactions between surfaces. This framework is based on a diffuse description of the moving surfaces. The form of the system’s underlying energy is key to establishing the adhesive interaction, more specifically, it includes terms that only play a role in the diffuse region representing the surface. In a novel way, equilibrium states of the adhesion model are characterized, which connect the proposed diffuse framework with a sharp theory for adhesion. Numerical experiments provide further geometrical insights into the proposed adhesion model. As a third contribution, a continuum theory based on a sharp description of the surface is presented for bulk-surface fluids. In particular, a bulk fluid coupled to a surface fluid is considered, which both deform in an incompressible way. This framework also accounts for additional mechanics, such as phase separation and species transport in the bulk-surface material. Here, the framework is founded on a so-called bulk-surface principle of virtual power, which reflects the energetic structure of the coupled system. Another energy-based argument provides the dynamic coupling conditions between the bulk and surface fluid, thereby complementing the framework in a consistent manner. Finally, the frameworks developed within this thesis can be tailored to diverse bulk-surface applications. In combination with efficient computational tools, these frameworks are capable of improving our understanding of the interplay between bulk and surface within complex materials.