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Veranstaltungen
Workshop on liquid thin films

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

26.08.24 30.08.24

Workshop on liquid thin films

The theme are capillarity-driven interfaces and surfaces in fluids, with a specific focus on their singularities. Phenomena include contact-angle dynamics and the no-slip paradox, film rupture and thermal fluctuation. The goal is to trigger exchange between novel regularity theory, asymptotic modelling, and multi-scale physics.

Speakers

Helmut Abels

University Regensburg

Andrea Bertozzi

University of California, Los Angeles

Gabriele Brüll

Lund University

Peter Constantin

Princeton University
Participation cancelled!

Jens Eggers

University of Bristol, BRISTOL

Julian Fischer

Institute of Science and Technology Austria, Klosterneuburg

Francisco Gancedo

Universidad de Sevilla
Participation cancelled!

Benjamin Gess

MPI MIS Leipzig & Universität Bielefeld

Günther Grün

Friedrich-Alexander-Universität Erlangen-Nürnberg

Hangjie Ji

North Carolina State University, Raleigh

Inwon Kim

UCLA

Herbert Koch

Universität Bonn
Participation cancelled!

Christina Lienstromberg

University of Stuttgart

Juan Jose Lopez Velazquez

University of Bonn

Nader Masmoudi

New York University
Participation cancelled!

Katerina Nik

Universität Wien
Participation cancelled!

Laurel Ohm

University of Wisconsin - Madison, Madison, Wisconsin

Dirk Peschka

Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin

David Quéré

ESPCI Paris

Howard Stone

Princeton University

Roman Taranets

Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Cherkasy

Ian Tice

Carnegie Mellon University, Pittsburgh, PA

Barbara Wagner

WIAS Berlin
Participation cancelled!

Maria G. Westdickenberg

RWTH Aachen University

Anouk Wisse

TU Delft

Thomas Witelski

Duke University, Durham, North Carolina

Chengxi Zhao

University of Science and Technology of China, Hefei

Program

10:30 - 11:00	Welcome
11:00 - 11:45	David Quéré (ESPCI Paris) Tiny drops on tiny texture Texturing solids is a way to provide original properties towardsl liquids. Then, drops can be made less adhesive and/or less frictional, as observed on water repellent or liquid infused materials. Generally, the scale of the liquid is much larger than that of the texture, so that properties (wetting or friction properties for instance) are averaged over a large collection of « defects ». Cases where the drop scale is comparable with that of the texture were less studied, despite their practical interest. For instance, it was shown that adding macro-texture to micro-texture does shorten the rebound time of droplets. Also, a liquid repelled by a texture will be stuck to it if it is (!) built by condensation, that is, by increasing the drop size from a minute volume to a macroscopic one. We plan to discuss how the use of tiny texture allows us to modify this picture (and give rise to anti-fogging materials), and a few things we learn from the observation of such situations.
12:00 - 14:00	Lunch Break
14:00 - 14:45	Howard Stone (Princeton University) Two types of thin film flows: (1) Similarity solutions near corners and (2) Soap films with surfactant gradients I will discuss two types of problems involving thin films, one for a film on a planar impermeable substrate and the other for situations involving soap films. First, I will illustrate an experimentally motivated solution of the thin-film equations that involves three independent variables. In particular, we consider a thin film draining along an edge of a vertical plate and report the three-dimensional film shape. Then, we construct an analytical solution of the PDEs involve three independent variables and show it can be reduced to a nonlinear ordinary differential equation, which can be compared with the experimental measurements. Second, for situations such as soap films we consider the thin-film equations with Marangoni effects (surface tension varies along the surface) but allowing for asymmetry, e.g., top to bottom differences. We discuss some aspects of the dynamics and show the coupling to deformation of the centerline (the film bends).
15:00 - 15:30	Coffee Break
15:30 - 16:15	Juan Jose Lopez Velazquez (University of Bonn) Solutions of the thin film equation with shrinking support In this talk I will discuss several solutions of the thin film equation, with the usual mobility h^3, for which the support of the solutions shrinks in finite time. The meaning of the dissipation of energy formula will be discussed in detail. Some particular self-similar solutions describing mass aggregation will be described too. (Joint work with Hans Knuepfer).
16:30 - 17:15	Laurel Ohm (University of Wisconsin - Madison) Free boundary dynamics of an elastic filament in 3D Stokes flow Motivated by biophysical applications, we consider a free boundary problem for a thin elastic filament immersed in 3D Stokes flow. The 3D fluid is coupled to the quasi-1D filament dynamics via a novel type of angle-averaged Neumann-to-Dirichlet operator. Much of the difficulty in the analysis lies in understanding this operator. We show that the principal part of this NtD map is the corresponding operator about a straight, periodic filament, for which we derive an explicit symbol. It is then possible to establish local well-posedness for an immersed filament evolving via a simple elasticity law. This establishes a mathematical foundation for the myriad computational results based on slender body approximations for thin immersed elastic structures.

09:30 - 10:15	Jens Eggers (University of Bristol) A pinned contact line We consider the dynamics of a drop with a pinned contact line $h(0,t) = 0$, as described by the thin film equation $h_t + (h^n h_{xxx})_x = 0$. Starting from an arbitrary initial condition at $t=0$, the profile relaxes toward an equilibrium parabolic profile. Our focus is on the early-time dynamics, which for $n\ne 3$ is described by a local similarity solution. As a result, the contact angle changes like a power law as a function of the time $t$. In the case $n=3$, which corresponds to a no slip boundary condition, the problem becomes non-local, and the change in contact angle is of the form $e^{-A/t^\alpha}$ (with $A$ complex). We consider the implications of this result for Huh and Scriven's contact line paradox for the case $n=3$. (joint work with Marco Fontelos)
10:30 - 11:00	Coffee Break
11:00 - 11:45	Dirk Peschka (Weierstraß-Institut für Angewandte Analysis und Stochastik) Dissipative Processes in Thin Film Flows The study of thin film flows leads to the development of reduced hydrodynamic models derived from a Navier-Stokes free boundary problem. The key dissipation processes contributing to these models include viscous dissipation, Navier-slip, and contact line dissipation. In this talk, we first present a general approach to reformulating the metric of dissipative processes using saddle point structures. We then discuss the corresponding discretization of these free boundary problems that are based on a gradient flow structure. The impact of varying the scaling of the Navier-slip length on pattern formation in three-dimensional flows will be showcased. Finally, we explore the existence of a similar discussion and a hierarchy of models for different orders of contact line dissipation.
12:00 - 13:00	Lunch Break at MPI
13:00 - 14:00	Scientific discussion
14:00 - 14:45	Inwon Kim (UCLA) On the geometry of rate-independent drolpet motion We consider a toy model of rate-independent droplet motion on a surface with contact angle hysteresis, based on the one-phase Bernoulli free boundary problem. Taking advantage of two notions of weak solutions, energy- based and comparison-principle-based, we study the dynamic contact angle of moving contact lines and the geometry of de-pinning.
15:00 - 15:30	Coffee Break
15:30 - 16:15	Gabriele Brüll (Lund University) Thin fluid films heated from below Thin fluid films find various applications in industrial processes. These films, ranging from nanometers to micrometers in thickness, are often exposed to temperature gradients, which compete with stabilising surface tension effects. A mathematical model describing the evolution of thin fluid film's height resting on a horizontal, heated plate can be derived via lubrication approximation, constituting a quasilinear, degenerate, fourth-order partial differential equations. When a critical temperature disparity between the heated plate and the surrounding environment is reached, the flat steady state becomes spectrally unstable and periodic stationary solutions bifurcate. Using a Hamiltonian formulation of the stationary problem and analytic bifurcation theory, we prove the existence of a global bifurcation curve converging to a weak stationary periodic solution exhibiting film rupture. Based on a joint work with B. Hilder (Munich) and J. Jansen (Lund).
16:30 - 17:15	Christina Lienstromberg (University of Stuttgart) Dynamics of solutions to a rimming-flow equation Abstract
18:30 -	Dinner

09:30 - 10:15	Chengxi Zhao (University of Science and Technology of China) Fluctuating hydrodynamics of nanoscale liquid ligaments and thin-films The effects of thermal fluctuations on nanoscale flows are described using the framework of fluctuating hydrodynamics. We employ two types of stochastic lubrication equations (SLEs), derived from the fundamental equations of fluctuating hydrodynamics, to investigate fluctuation-driven dynamics in nanoscale liquid ligaments and films. Theoretical solutions of the linearized SLEs are developed to analyze the spectra of interface instabilities, which are found to be enhanced by thermal fluctuations. Additionally, the SLEs are solved numerically to capture the nonlinear behaviors. Both the theoretical and numerical results are validated against molecular dynamics simulations.
10:30 - 11:00	Coffee Break
11:00 - 11:45	Anouk Wisse (TU Delft) Classical solutions to the thin-film equation with general mobility in the perfect-wetting regime We prove well-posedness, partial regularity, and stability of the thin-film equation with general mobility m(h) = h^n and mobility exponent n in the union of (1,3/2) and (3/2,3) in the regime of perfect wetting (zero contact angle). After a suitable coordinate transformation to fix the free boundary (the contact line where liquid, air, and solid coalesce), the thin-film equation is rewritten as an abstract Cauchy problem and we obtain maximal L^p_t regularity for the linearized evolution. Partial regularity close to the free boundary is obtained by studying the elliptic regularity of the spatial part of the linearization. This yields solutions that are non-smooth in the distance to the free boundary, in line with previous findings for source-type self-similar solutions. In a scaling-wise quasi-minimal norm for the initial data, we obtain a well-posedness and asymptotic stability result for perturbations of traveling waves. The novelty of this work lies in the usage of L^p-estimates in time, where p is between one and infinity, while the existing literature mostly deals with p = 2 at least for nonlinear mobilities. This turns out to be essential to obtain for the first time a well-posedness result in the perfect-wetting regime for all physical nonlinear slip conditions except for a strongly degenerate case at n = 3/2 and the well-understood Greenspan-slip case n = 1.
12:00 - 14:00	Lunch Break
14:00 - 14:45	Günther Grün (Friedrich-Alexander-Universität Erlangen-Nürnberg) Finite Speed of Propagation for a Class of Stochastic Thin-Film Equations We present an energy method to prove a qualitative result on finite speed of propagation for energy-dissipating solutions to stochastic thin-film equations with conservative nonlinear multiplicative noise under periodic boundary conditions. Physically, the equations under consideration correspond to weak slippage, i.e. the case of mobility exponents $n\in (2,3)$. Analytically, our approach is based on two main ingredients. First, a new weighted decay estimate for appropriate powers of the solution which is set up in combination with Bernis inequalities and energy estimates. Secondly, tailored filtering techniques inspired by the methods which were presented for stochastic porous-medium equations with source-term noise in (J. Fischer and G. Grün, SIAM J. Math. Anal. 47:825-854, 2015). This is joint work with L. Klein (Erlangen).
15:00 - 15:30	Coffee Break
15:30 - 16:15	Julian Fischer (Institute of Science and Technology Austria) A rigorous approach to the Dean-Kawasaki equation of fluctuating hydrodynamics Fluctuating hydrodynamics provides a framework for approximating density fluctuations in interacting particle systems by suitable SPDEs. The Dean-Kawasaki equation - a strongly singular SPDE - is perhaps the most basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N diffusing weakly interacting particles in the regime of large particle numbers N. The strongly singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification: Besides being non-renormalizable by approaches like regularity structures, it has recently been shown to not even admit nontrivial martingale solutions. In this talk, we give an overview of recent quantitative results for the justification of fluctuating hydrodynamics models. In particular, we give an interpretation of the Dean-Kawasaki equation as a "recipe" for accurate and efficient numerical simulations of the density fluctuations for weakly interacting diffusing particles, allowing for an error that is of arbitarily high order in the inverse particle number. based on joint works with Federico Cornalba, Jonas Ingmanns, and Claudia Raithel
16:30 - 17:15	Benjamin Gess (MPI MIS Leipzig & Universität Bielefeld) Thermodynamically consistent discretization of the stochastic thin film equation We begin by informally demonstrating how the gradient flow structure of the deterministic thin film equation, which encodes the balance between driving capillary forces and limiting viscous forces, can be used as a foundation for the thermodynamically consistent introduction of fluctuations. This approach is then pursued at the level of a spatial discretization of the gradient flow structure, specifically, by discretizing the energy functional and the dissipative metric tensor. This leads us to derive a stochastic differential equation that can be interpreted as an approximation of the stochastic thin film equation. Subsequently, we show that a suitable choice of the discrete mobility function results in solutions that strictly preserve positivity. The talk closes by discussing the probability of solutions to the stochastic thin film equation to approach zero.

09:30 - 10:15	Maria G. Westdickenberg (RWTH Aachen University) Quantifying the shape of energy landscapes and timescale separation in a few model problems We will explore the connection between the shape of an energy landscape and the dynamics of the corresponding gradient flow for three classical model problems, the Allen-Cahn equation (in d=1), the Cahn-Hilliard equation (in d=1), and the Mullins-Sekerka evolution (in d=3). In all cases we are particularly interested in initial perturbations that are not small. Along the way we will comment on a few questions (solved and open) about stochastically perturbed versions of these models. An underlying issue of interest in the Allen-Cahn and Cahn-Hilliard equations is the phenomenon of metastability. The variational techniques developed for and explained within the context of the model problems are more generally applicable.
10:30 - 11:00	Coffee Break
11:00 - 11:45	Roman Taranets (Institute of Applied Mathematics and Mechanics of the NAS of Ukraine) Existence and asymptotic behaviour of solutions for a non-local thin-film equation We prove existence of solutions to a family of fractional thin-film equations in a bounded domain. A non-local operator is given by the spectral fractional Laplacian. In the case of a "strong slippage" regime with "complete wetting" interfacial conditions, we prove local entropy estimates which entails finite speed of propagation of the support and a lower bound for the waiting time phenomenon.
12:00 - 13:00	Lunch Break at MPI
13:00 - 14:00	Scientific discussion
14:00 - 14:45	Thomas Witelski (Duke University) Coarsening dynamics of condensing fluid thin films Thin films of viscous fluids on hydrophobic surfaces are unstable and rapidly break up then slowly evolve as interacting localized droplets. For mass-conserving dewetting dynamics, many aspects follow from results on phase separation in Cahn-Hilliard models from materials science. Considered over long times, the volatility of the fluid will ultimately influence the state of the system. In a narrow range of conditions, the disjoining pressure can balance with thermal effects and care is needed to determine if evaporation or condensation will dominate. A lubrication PDE model was used to examine the long-time dynamics in a regime with weak condensation. A reduced-order dynamical system was derived for the interactions between quasi-steady droplets. Dynamics for arrays of identical drops and pairwise droplet interactions were investigated to provide insights into the coarsening dynamics of macroscopic systems. Weak condensation was shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics compared to non-volatile mass-preserving fluids. This was joint work with Hangjie Ji (NCSU).
15:00 - 15:30	Coffee Break
15:30 - 16:15	Hangjie Ji (North Carolina State University) Mean field control of thin film droplet dynamics Interfacial instabilities in volatile liquid thin films on a hydrophobic substrate can lead to complex droplet dynamics such as droplet merging, splitting, and transport. We study a class of mean field control formulations for these droplet dynamics. An optimal control problem is designed by formulating the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. As an example, we consider a thin volatile liquid film laden with an active suspension, with control achieved through its activity field. Numerical examples, including droplet transport, bead-up/spreading, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism. This is joint work with Guosheng Fu (Notre Dame), Will Pazner (PSU), and Wuchen Li (USC).
16:30 - 17:15	Andrea Bertozzi (University of California, Los Angeles) Particle-Laden Thin Film Flows This talk will showcase the challenge in the modeling and mathematics for particle-laden thin film flows. Such flows arise in many applications including the food industry, mining, and environmental cleanup. These flows are notoriously difficult to model because the dominant physics, especially for viscous flows, is due to many body interactions of the particles. There are no "first principles" continuum models for the physics and instead modelers rely on semi-empirical rules for particle settling and migration. In the lubrication limit, the mathematical equations are a system of conservation laws with fluxes that need to be estimated numerically. The speaker will review our current understanding about such systems and will discuss some open problems. Our group at UCLA is just beginning a five-year NSF-sponsored project in this research area and would welcome visitors and collaborators interested in working with us on these problems.

09:30 - 10:15	Ian Tice (Carnegie Mellon University) The traveling wave problem for the shallow water equations with drag The shallow water equations describe the effective dynamics of an incompressible fluid in the regime in which the horizontal scale is much larger than the vertical scale. These equations may be obtained from the free-boundary incompressible Navier-Stokes system via a formal asymptotic expansion in a parameter that measures this vertical-to-horizontal scale, and as such can model a number of interesting physical phenomena present in the NS system. In this talk we focus on the SW system obtained from NS with Navier slip conditions imposed at the fluid bottom, which results in a useful and interesting drag term in the SW system. We will discuss the construction of traveling wave solutions for the SW system as well as the limits obtained by sending the viscosity and surface tension parameters to zero. This is joint work with Noah Stevenson.
10:30 - 11:00	Coffee Break
11:00 - 11:45	Helmut Abels (University Regensburg) Sharp Interface Limit of a Navier-Stokes/Allen-Cahn System with Vanishing Mobility We consider the sharp interface limit of a Navier-Stokes/Allen-Cahn system, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. In dependence on the mobility coefficient in the Allen-Cahn equation in dependence on $\varepsilon>0$ different limit systems or non-convergence can occur. In the case that the mobility vanishes as $\varepsilon$ tends to zero slower than quadratic we prove convergence of solutions to a smooth solution of a classical sharp interface model for well-prepared and sufficiently smooth initial data. The proof is based on a relative entropy method and the construction of sufficiently smooth solutions of a suitable perturbed sharp interface limit system. This is a joint work with Julian Fischer and Maximilian Moser (ISTA Klosterneuburg, Austria)

Participants

Helmut Abels

University Regensburg

Andrea Bertozzi

University of California, Los Angeles

Gabriele Brüll

Lund University

Mario Bukal

University of Zagreb

Sebastian Bürger

Leipzig University

Kateryna Buryachenko

Humboldt University of Berlin

Zhaodong Ding

University of Bristol

Elias Döhrer

TU Chemnitz

Riccardo Durastanti

University of Naples Federico II

Jens Eggers

University of Bristol

Julian Fischer

Institute of Science and Technology Austria

Benjamin Gess

MPI MIS Leipzig & Universität Bielefeld

Lorenzo Giacomelli

Sapienza Università di Roma

Günther Grün

Friedrich-Alexander-Universität Erlangen-Nürnberg

Rishabh Gvalani

ETH Zurich

Hangjie Ji

North Carolina State University

Juri Joussen

Universität Stuttgart

Konstantin Kalinin

Max Planck Institute for Mathematics in the Sciences

Malte Kampschulte

Universität Leipzig

Inwon Kim

UCLA

Hans Knüpfer

Ruprecht-Karls-Universität Heidelberg

Erika Kokavcová

Institute of Science and Technology Austria

King Ming Lam

TU Delft

Christina Lienstromberg

University of Stuttgart

Juan Jose Lopez Velazquez

University of Bonn

Laurel Ohm

University of Wisconsin - Madison

Felix Otto

Max Planck Institute for Mathematics in the Sciences

Dirk Peschka

Weierstraß-Institut für Angewandte Analysis und Stochastik

David Quéré

ESPCI Paris

Philipp Reiter

Chemnitz University of Technology

Elena Salguero

Max Planck Institute for Mathematics in the Sciences

Jonas Sauer

Friedrich-Schiller-Universität Jena

Max Sauerbrey

TU Delft

Christian Seis

Universität Münster

Akram Sharif

TU Dresden

Sandeep Kumar Soni

University of Zagreb

Howard Stone

Princeton University

Roman Taranets

Institute of Applied Mathematics and Mechanics of the NAS of Ukraine

Markus Tempelmayr

Universität Münster

Ian Tice

Carnegie Mellon University

Joshua Utley

FAU Erlangen

Axel Voigt

TU Dresden

Maria G. Westdickenberg

RWTH Aachen University

Mathias Wilke

Martin-Luther-Universität Halle-Wittenberg

Anouk Wisse

TU Delft

Thomas Witelski

Duke University

Espen Xylander

Universität Stuttgart

Chengxi Zhao

University of Science and Technology of China

Scientific Organizers

Lorenzo Giacomelli

Sapienza Università di Roma

Hans Knüpfer

Ruprecht-Karls-Universität Heidelberg

Felix Otto

Max Planck Institute for Mathematics in the Sciences

Christian Seis

Universität Münster

Administrative Contact

Katja Heid

Max Planck Institute for Mathematics in the Sciences Contact via Mail