Interfacial Flows—The Power and Beauty of Asymptotic Methods

We exemplify the rigorous treatment of capillarity-driven free-surface flows by considering the “teapot effect” as an appealing daily-life phenomenon. To this end, we formulate the problem in full, elucidate its structure for small viscous influence in a first step and then refine the results by scrutinising the underlying potential flow in due detail. Finally, we address the subtleties arising when it comes to the rational inclusion of viscous effects so as to sort the real flow out of a one-parametric class of inviscid-flow solutions (so-called selection problem). This approach shall demonstrate the successful, systematic treatment of complex flow problems, involving a variety of disparate length scales. Amongst others, it is demonstrated how the correctly performed abstraction process can unveil unexpected mechanisms and deepen the understanding of known physical phenomena. Many intriguing questions associated with the effect in focus, but also neighbouring fields, are found as not settled conclusively yet. This calls for further analytical and numerical progress.

The lecture notes present a ‘guided tour’ over the so-called nonlinear multimodal method in the liquid sloshing dynamics. The method is, in fact, a version of the Reduced Order Modelling based on combining variational and asymptotic approaches to the original free-surface (sloshing) problem. I start with mathematical fundamentals of how to derive a discrete approximate model of sloshing from the original fluid dynamics formulation by using the Bateman-Luke variational principle. The derived Euler-Lagrange equations (here, the Miles-Lukovsky-type modal system) couple the generalised hydrodynamic coordinates and velocities associated with time-dependent coefficients in functional series representing the free surface and velocity potential, respectively. Linearising the modal system yields an infinite set of uncoupled linear oscillators which, naturally, have an analytical solution for prescribed tank motions. The linear modal theory becomes physically irrelevant for resonant excitations. Analytical approaches to nonlinear resonant sloshing can be based on asymptotic methods reducing the infinite-dimensional Miles-Lukovsky-type system to finite dimensions. The subsequent text centres around how to construct and analyse those finite-dimensional asymptotic modal equations for rectangular and square base tanks. A particular focus is on the so-called Moiseev’s (Duffing-like) third-order asymptotic ordering and resonant steady-state wave regimes which are associated with asymptotic periodic solutions of modal equations when the forcing frequency is close to the lowest natural sloshing frequency. These solutions are (semi-) analytically constructed, their stability is examined, the results are validated by experiments.

These notes summarize five 45 min lectures the author presented at CISM in June 2023. The topics are centered around the theme of “thin fluid films,” which constitutes an area of (mostly low-Reynolds-number) fluid dynamics with wide applicability. It is also a set of topics where nonlinearity is common, yet analytical results, either in the form of scaling laws and/or the reduction of partial differential equations to ordinary differential equations, are possible. The lectures seek to highlight this intersection of physical problems, scaling laws, analyses, including similarity solutions and detailed results, spanning traditional coating flows and surfactant-mediated dynamics, as well as thin-film descriptions common to dynamics of cellular membranes, which gives a link to biophysics. To start the article, we survey a few problems where surface tension is important and where dimensional analysis yields insights and quantitative results. Then, in turn, we analyze the differential equations and boundary conditions that describe physically common thin-film flows, with emphasis on analytical insights and the steps towards developing, where possible, similarity solutions. The motion of a particle in a viscous membrane constitutes the last lecture. In preparing the notes the author filled in various steps and other explanations that time did not allow for during the actual lectures.

I use interfacial flows as an introduction to self-similar phenomena and scaling, my main examples being optics (wave fronts), and thin film dynamics. I describe how similarity solutions can be used to describe singular behavior in higher dimensions, where in general different spatial directions are characterized by different scaling behavior. I then show how singular solutions develop complexity through a sequence of instabilities. Combining chaotic dynamics with a higher-dimensional description, one obtains a mechanism for spatial complexity, as it is characteristic for turbulent flows.

The classical theory of lubrication is dedicated to the flow of a viscous fluid in a narrow gap between two solid boundaries, with the specific goal of computing the resulting lubrication forces. These forces can be modified by elasticity: the solid boundary can deform elastically and/or the fluid can exhibit viscoelastic stress. Here we focus on the influence of elasticity inside the fluid, using the second-order fluid model, which for steady flows can be viewed as an expansion for small elasticity. We demonstrate how viscoelasticity changes the lubrication forces in various two-dimensional flow geometries, for which analytical solutions are presented. These examples illustrate the physical mechanisms of viscoelastic lubrication at small Deborah numbers. We also discuss the challenges of using the second-order fluid in unsteady flows and large Deborah numbers, and briefly comment on how the modelling framework can get extended to viscoelastic thin-film flows with free boundaries.

The aim of these lecture notes is to illustrate how the judicious application of asymptotic methods to relatively simple mathematical models can give considerable insight into three paradigm fluid mechanical problems, namely coating and rimming flow on a horizontal cylinder, rivulet flow, and the evaporation of a sessile droplet.