Thin films with moving contact lines frequently form a capillary rim that are susceptible to an instability that corrugates the rim and the contact line in the spanwise direction and grows into finger-like structures. This instability is observed in gravity- or marangoni-driven liquids as well as in films that dewet from an hydrophobic surface. The investigation of this instability via thin film models involves a variety of interesting mathematical structures and techniques some of which will be presented and discussed in this talk, such as undercompressive shocks, the derivation and analysis of sharp interface models for the capillary rims.

Many processes in nature can be considered as rate-independent processes. Typical examples are the evolution of damage and fracture in brittle materials or elastoplasticity.

In the corresponding models it is assumed that the actual state of the material can be completely characterized by the displacement field and a collection of internal variables that characterize for instance the damage state or the plastic deformation. The model describing the evolution of these quantities with respect to external loadings consist of a (quasistatic) balance of forces resulting in a system of linear elliptic equations that is coupled with an evolutionary variational inequality describing the changes of the internal variables. This coupled system can be derived from an energy functional and a dissipation potential, which, due to rate-independence, is positively homogeneous of degree one.

Since in many applications the energy functional is not simultaneously convex with respect to the displacements and the internal variables, solutions of the evolution system might be discontinuous in time. Hence, weak formulations and suitable jump criteria are needed that lead to physically reasonable solutions.

In this lecture we discuss different possibilities to define weak solutions and illustrate the predictions with a model describing the propagation of a single crack in a brittle elastic material.

The mathematical modelling of collective behaviour of organisms is receiving increasing attention in the recent years. Examples are the formation of consenus in human societies, herding of sheep and of market traders, insect swarming, bird flocking, or collective cell migration and chemotaxis. Although those processes go on on different scales and produce a variety of complex patterns, they share a surprisingly similar structure.

This talk will provide an overview of the modelling from a mathematician's viewpoint and demonstrate that (besides pre-existing patterns in the environment) the main source of pattern formation is an interaction of two effects, namely long-range attraction and short-range repulsion. In macroscopic models based on partial differential equations, this typically means a combination of a (nonlinear) diffusion term with an integral operator.

In order to understand this interplay minimal mathematical models will be derived and analyzed, which contribute strongly to the understanding of basic mechanisms of pattern formation and phase transitions with respect to system parameters as found in real data and detailed microscopic simulations.

The ensemble average process originally proposed by Drew and Passman is used to derive a multiphase model for non-Brownian suspensions. The model allows for the simulation of highly dilute up to near solid suspensions. In the limiting regimes of pure fluid and pure solid, the model becomes the Navier-Stokes equations with Newtonian stress tensor and Darcy's law, respectively. We will reduce the model for the two-dimensional shear and tube flow in order to get analytic results. Further, numeric solutions of the full 2D and 3D problem are discussed.

In this talk we consider linear stability of growing rims that appear in thin solid and liquid films as they retract from a solid substrate. The mathematical models in both cases lead to mass conserving free boundary problems for degenerate and non-degenerate thin film equations whose common feature is the presence of non-constant base states in the stability analysis. Hence the base state cannot be expressed by a simple traveling wave or self-similar form which impedes the straightfoward use of normal modes. However, large rims evolve on a slower time scale than the typical perturbations and we systematically exploit this time scale separation to derive asymptotic approximations for the base states and for solutions of the linear stability problem. Our results enable us to track the evolution of the dominant wavelength for long times and to assess the validity of the asymptotic approximations.

In this talk we consider the dynamics of a binary mixture that is confined between two walls of infinite horizontal extent. The evolution of the mixture is described by a purely diffusive phase-field model and we account for the energetic interactions between the walls and the constituent materials. That is, we assume both walls have a slight preference for one, but not the same, material. We show that by slowly cooling the system the onset of spinodal decomposition can be suppressed and the bias of the walls will induce the formation of a bilayer, where each layer is of nearly constant composition. The two layers are separated by a thin interfacial region where the composition changes rapidly. The long length of the bilayer "interface" suggests that this is not the lowest-energy state, and we show that locally perturbing the system can initiate a cascade of rupture events resulting in a topological transition from the bilayer state to a columnar state. The dynamics of the transition are investigated by reducing the phase-field model to a sharp-interface model which can be further simplified to a thin-film equation. This is joint work with A. Muench, B. Wagner, V. Burlakov, and A. Goriely.

In nature we often encounter situations where two immiscible fluids move on each other, for instance the different layers in human tear films or simply droplets of fat in a chicken soup. Also for technical applications liquid bilayers play a role. Here, for example, the dewetting of a nanoscopic polymer film from a liquid substrate is an interesting topic. In this talk we consider liquid bilayers with certain geometric properties. In particular, we assume that the height of these bilayers is small compared to the characteristic length scale of evolving patterns which, for example, is true for nanoscopic polymer films. We derive systems of thin film equations for the evolution of such bilayers and discuss the dependence of stability of these systems subject to interfacial slip and intermolecular forces. Furthermore, we study stationary solutions for these thin film models and compare our results with experimental data.

Reduced ODE models describing coarsening dynamics of droplets in nanometric polymer film interacting on solid substrate in the presence of large slippage at the liquid/solid interface are derived from one-dimensional lubrication equations. In the limiting case of the infinite slip length corresponding to the free suspended films a collision/absorption model then arises and is solved explicitly. The exact collision law is derived. Existence of a threshold at which the collision rates switch from algebraic to exponential ones is shown.

Around forty years ago, mixed finite element methods for the incompressible Navier-Stokes equations were introduced and are nowadays regarded as the standard discretization approach for incompressible flows in numerical mathematics. The success of mixed methods relies mainly on the relaxation of the divergence constraint. But only recently a systematic research was started, in order to understand what kind of disadvantages in practice and theory can arise by this relaxation. As a result it has turned out that mixed methods suffer from a lack of robustness, whenever large irrotational forces in the momentum equations of the Navier-Stokes equations have to balanced. In such cases, large spurious oscillations in the discrete velocities can spoil the discrete solution completely. The reason for this unpleasant behavior can be understood by investigating the Helmholtz decomposition of the discrete momentum equations. Irrotational and divergence-free forces should each build a separate force balance, but in classical mixed methods they interact in a non-physical way. Besides analysing the stability properties of classical mixed methods, a remedy is proposed which is based on the new theoretical insight, allowing the construction of new classes of cheap, divergence-free flow solvers in 2D and 3D.