Interfacial flow phenomena are observed in everyday life on nanoscopic to geophysically large scales (wetting, Leidenfrost phenomenon, Lotus effect, teapot effect, sheet formation and breakup, free-surface waves, etc.) and are vital for many biological and industrial processes. Despite the demand for their thorough understanding and control, their rigorous description still poses a major challenge, given the various disparate spatial and temporal scales involved. Deep insight into their beauty and formidable, rich complexity is gained by pushing the governing dimensionless key parameters to their limits and applying asymptotic techniques. These allow for the systematic approximative examination of self-similar phenomena and singularities and their regularisation. The appropriate scaling of the physical problem and identification of the relevant scales enables its formal reduction to a simpler canonical one that already captures the essential effects; higher-order corrections account rationally for less important ones. The required mastery of advanced analytical/numerical tools is rewarded by a conclusive comprehension of the multi-scale physics at play, far beyond that pure full-scale simulations provide. Asymptotic methods have proven their strengths in more traditional areas of fluid mechanics for long, but applying them to interfacial flows is a relatively new thrust of research, which has already led to exciting recent advances.

The course addresses doctoral students, post-docs and early-career researchers interested in theoretical fluid mechanics and its mathematical foundations. It focusses on the state of the art of perturbation and related mathematical methods and how to apply them to a representative variety of interfacial flows – and to almost any physical or engineering problem. The following building blocks of interfacial-flow phenomena are covered:

1. Drop dynamics.
• Sessile drops: large/small Bond numbers, hydrophobic limit.
• Quasi-spherical drops: on an inclined plane, contact-line hysteresis, sticking, sliding, rolling.
• Drop–vapour interaction (levitated drops, Leidenfrost phenomenon).

2. Free-surface instabilities and singularities.
• Rayleigh–Plateau (long-wave limit), Saffman–Taylor and other instabilities, geometrical influence of bounding solid phase.
• Instability series might trigger highly nonlinear singular phenomena, due to the vanishing of a length scale described by
• Local similarity solutions (exactly solvable problems upon rescaling)
• Examples: droplet pinching, in non-Newtonian flows, soft-/condensed-matter physics, interplay liquid sheets/elasticity of wetted substrates

3. Free-surface potential flows, gravity waves, sloshing problems. Playing a significant role in geophysical/maritime sciences and applications, their complex nonlinear dynamics is analysed through.
• Advanced conformal-mapping methods.
• Multi-modal analysis.
• Near-resonance asymptotics.

4. Slender viscous sheets. The emerging multiple length and time scales render them ideally suited for an asymptotic analysis:
• Lubrication approximation.
• Coating flows (Landau–Levich problem) .
• Droplet coalescence, defragmentation (Savart sheets), film-on-film spreading, soap films.
• Thin films on rotating cylinders, on textured/superhydrophobic surfaces, rivulet flows.