Mathematical Aspects of Hydrodynamics

The workshop dealt with the partial differential equations that describe fluid motion, namely the Euler equations and the Navier-Stokes equations. This included topics in both inviscid and viscous fluids in two and three dimensions. A number of the talks were connected with issues of turbulence. Some talks addressed aspects of fluid dynamics such as magnetohydrodynamics, quantum and high energy physics, liquid crystals and the particle limit governed by the Boltzmann equations

Four out-of-equilibrium lectures

A collection of published papers on the subject of classical nonequilibrium statistical mechanics. Mainly stochastic systems are considered, with special regard to applications in soft matter physic

Temperature dependent extensions of the Cahn-Hilliard equation

The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In this paper we focus on the dynamics of these binary media, when the underlying temperature is not constant. The aim of this paper is twofold. We first derive two distinct models that extend the classical Cahn-Hilliard equation with an evolutionary equation for the absolute temperature. Secondly, we analyse the local well-posedness of classical solutions for one of these systems. Our modelling introduces the systems of PDEs by means of a general and unified formalism. This formalism couples standard principles of mechanics together with the main laws of thermodynamics. Our work highlights how certain assumptions on the transport of the temperature effect the overall physics of the systems. The variety of these thermodynamically consistent models opens the question of which one should be more appropriate. Our analysis shows that one of the derived models might be more desirable to the well-posedness theory of classical solutions, a property that might be natural as a selection criteria

Multipseudopotential interaction : a lattice Boltzmann scheme for simulation of multiphase flows

In this study, a new pseudopotential Lattice Boltzmann (LB) scheme, multipseudopotential interaction (MPI), including boundary conditions is proposed for simulation of two-phase flows. It solves several drawbacks of available schemes such as being thermodynamically inconsistent with practical equations of state (EOSs), being limited to low-density ratios, not having an independently tunable interface width. The lattice interaction potential is described by a series of consistent sub-potentials. In theory, in addition to being intrinsically consistent with thermodynamics, the MPI-LB scheme is stable for a large range of density ratios (up to 106), and tunable for interface width. In engineering applications, the scheme is superior over the previous schemes of reproducing practical EOSs by removing the deficiency of creating unphysical potentials. The scheme is unlimited to implement the practical EOSs which can be expressed in a polynomial format. The scheme is studied and verified regarding liquid vapour circular and flat interfaces, Laplace law, Galilean invariance, and change of viscosity. Furthermore, the scheme is investigated for the effects of different wall boundary conditions on the hydrodynamics of non-ideal single-phase fluids. The suitable boundary condition is chosen based on density variation across the channel, and errors because of domain resolution, relaxation time, and compressibility

On the Properties of Self-Thermophoretic Janus Particles: From Hot Brownian Motion to Motility Landscapes

This thesis investigates several phenomena that are associated with (self-)thermophoretic Janus particles with hemispheres made from different materials serving as a paradigm for active propul- sion on the microscale. (i) The dynamics of a single Janus sphere in the external temperature field created by an immobilized heat source is studied. I show that the particle’s angular velocity is solely determined by the temperature profile on the equator between the Janus particle’s hemispheres and their phoretic mobility contrast. (ii) The distinct polarization-density patterns observed for active-particle suspensions in activity landscapes are addressed. The results of my approximate theoretical model agree well with exact numerical and measurement data for a thermophoretic microswimmer, and can serve as a template for more complex applications. The essential physics behind the formal results is robustly captured and elucidated by a schematic two-species “run- and-tumble” model. (iii) I investigate coarse-grained models of suspended self-thermo- phoretic microswimmers. Starting from atomistic molecular dynamics simulations, the coarse-grained de- scription of the fluid in terms of a local molecular temperature field is verified, and effective nonequilibrium temperatures characterizing the particle’s so called hot Brownian motion are mea- sured from simulations. They are theoretically shown to remain relevant for any further spatial coarse-graining towards a hydrodynamic description of the entire suspension as a homogeneous complex fluid.In dieser Arbeit untersuche ich mehrere Phänomene, die im Zusammenhang mit (selbst-)thermo- phoretischen Janusteilchen auftreten. Diese Teilchen bestehen aus zwei Halbkugeln mit unter- schiedlichen Materialeigenschaften und dienen in dieser Arbeit als Musterbeispiel für aktive Fort- bewegung auf der Mikroskala. (i) Die Dynamik eines einzelnen Janusteilchens im externen Temper- aturfeld einer ortsfesten Heizquelle wird untersucht. Es wird gezeigt, dass die Winkelgeschwindigkeit des Teilchens ausschließlich durch das Temperaturprofil am Äquator zwischen den Hemisphären des Janusteilchens und dem Unterschied ihrer phoretischen Mobilitäten bestimmt wird. (ii) Ich befasse mich mit den charakteristischen Polarisations- und Dichteprofilen, die für aktive Teilchen in Aktivitätslandschaften beobachtet werden. Die Ergebnisse meines approximativen theoretis- chen Modells stimmen gut mit exakten numerischen Lösungen und Messdaten für einen ther- mophoretischen Mikroschwimmer überein und können als Vorlage für komplexere Anwendungen dienen. Die wesentliche Physik hinter den formalen Ergebnissen wird durch ein schematisches Zwei-Spezies-“Run-and-Tumble”-Modell erfasst und erklärt. (iii) Ich untersuche Coarse-Graining- Modelle von suspendierten selbst-thermophoretischen Mikroschwimmern. Ausgehend von atom- istischen molekulardynamischen Simulationen wird die grobkörnige (coarse-grained) Beschreibung des Fluids in Form eines lokalen molekularen Temperaturfeldes verifiziert. Anschließend berechne ich effektive Nichtgleichgewichtstemperaturen, die die sogenannte heiße Brownsche Bewegung der Teilchen charakterisieren, und vergleiche diese mit Simulationsdaten. Es wird gezeigt, dass diese effektiven Temperaturen für jede weitere räumliche Vergröberung hin zu einer hydrodynamischen Beschreibung der gesamten Suspension als homogenes komplexes Fluid relevant bleiben

Selected chapters (theoretical physics)

The book considers spontaneous and induced emission of particles and waves. The formation of coherent pulses near a detected new threshold of induced radiation is discussed. It is shown how modulation instabilities generate self-similar structures and anomalous waves. A comparison is made of the dynamics of instability of Langmuir oscillations in plasma and heating of ions in Silin and Zakharov models. Turbulent-wave instability is discussed and a new approach to the description of the Mossbauer effect is presented. The similarity of processes of superradiation and dissipative instability is noted. Structural transitions in the convective layer and the appearance of large-scale vortices during modulation instability of developed convection and other relevant problems are discussed. It is of interest to specialists, graduate students and students of physics departments