The last decade has seen significant progress in the integration and application of fluid mechanical principles in many domains. Examples include geophysical, industrial, biomimetic, and soft robotic applications. These new interdisciplinary efforts have underscored the need to study fluid flow of, within, or over complex geometric configurations. In these systems, different geometric scales may be present, in the form of global curvature, local fine-scale asperities, and gradients of these. Examples include gravity currents over complex topography, cleaning of rough surfaces, or spin coating of patterned objects. The shape of a substrate or channel can vary dynamically in response to mechanical pressure, deposition of sediment or suspended particles, or simply through the relative motion of curved surfaces. Natural and engineered surfaces have microscopic surface textures, which influence flow: Small surface asperities lead to large-scale coating defects, and surface textures strongly affect friction and adhesion between lubricated (wet) materials. Interfacial geometry plays an important role in the rheological response of multiphase mixtures. Flows with free surfaces often exhibit instabilities and entail dynamically evolving geometric features such as pinch-off singularities, leading to the generation of droplets and bubbles.

The course will introduce the rich phenomenology and fundamental tools to characterize, study, and model the geometry of evolving fluid flows, with applications to interdisciplinary topics of current research interest. Participants will be exposed to a range of approaches to analyze and model flows with, through, and over complex geometries, including systems where the geometry evolves dynamically with the flow. While the course will focus on theoretical ideas, connections will be made to experimental findings and applications in engineering, physics, and other fields. By the end of the course, students will possess the necessary knowledge of the phenomena, an understanding of the state of the field including the identification of open research questions, and a new set of tools to synthesize and recombine in order to address these questions.

The course will deliver a broad discussion of flow and geometry, and their applications in a wide range of interdisciplinary topics. It is intended for graduate students and postdoctoral researchers in physics, engineering (mechanical/chemical), mechanics, and applied mathematics. The course will also be of interest to faculty seeking to expand into new interdisciplinary research areas at the intersection of mechanics, geometry, and the applied sciences.

The team of lecturers consists of researchers working on varied aspects of the topics mentioned above. Each will deliver six hours of lectures. While lecturers will focus on specific areas aligned with their expertise, these areas will be thematically united, so that participants can appreciate connections between various ideas.