Swalbe.jl: A lattice Boltzmann solver for thin film hydrodynamics

Small amounts of liquid deposited on a substrate are an everyday phenomenon. From atheoretical point of view this represents a modelling challenge, due to the multiple scalesinvolved: from the molecular interactions among the three phases (solid substrate, liquidfilm and surrounding vapor) to the hydrodynamic flows. An efficient way to deal with thismultiscale problem is the thin-film equation:ℎ = ∇ ⋅ ( (ℎ)∇), (1)where ℎ is the film thickness, (ℎ) is a thickness dependent mobility and is the pressure atthe liquid-vapor interface. Solving the thin film equation directly is a difficult task, because it isa fourth order degenerate PDE (Becker et al., 2003). Swalbe.jl approaches this problem froma different angle. Instead of directly solving the thin film equation we use a novel method basedon a class lattice Boltzmann models (Krüger et al., 2016), originally developed to simulateshallow water flows (Salmon, 1999). This approach serves two benefits, on the one hand theease of implementation where the lattice Boltzmann method essentially comprises of two steps:collision and streaming. On the other hand due to the simple algorithm a straightforwardapproach to parallelize the code and run it on accelerator devices. Choosing appropriate forcesit is possible to simulate complex problems. Among them is the dewetting of a patternedsubstrates as shown in Figure 1. Beyond films, low contact angle droplets can be studied andcompared to relaxation experiments, e.g. the Cox-Voinov or Tanner’s law (Bonn et al., 2009).Due to a disjoining pressure model for the three phase contact line droplets can not only relaxtowards their equilibrium they can slide as well (Zitz et al., 2019). All of this can be coupledwith thermal fluctuations to study the stochastic thin film equation (Shah et al., 2019; Zitz etal., 2021)

Dyson–Schwinger equations and

$$\mathcal{N}=4$$ Super Yang–Mills theory is a highly constrained theory, and therefore a valuable tool to test the understanding of less constrained Yang–Mills theories. Our aim is to use it to test our understanding of both the Landau gauge beyond perturbation theory and the truncations of Dyson–Schwinger equations in ordinary Yang–Mills theories. We derive the corresponding equations within the usual one-loop truncation for the propagators after imposing the Landau gauge. We find a conformal solution in this approximation, which surprisingly resembles many aspects of ordinary Yang–Mills theories. We furthermore discuss which role the Gribov–Singer ambiguity in this context could play, should it exist in this theory