Temperature effects in thin droplets

Thin droplets spreading on a solid substrate are investigated, with a special focus on temperature effects. The aim is to manipulate the fingering instability which may occur in the spreading in a spin coating process. The analysis bases on lubrication approximation, valid for flat thin droplets, which usually is the case. The dynamic of the wetting is implemented by using a generalized law of Tanner, coupling the contact angle (CA) of the droplet at the (apparent) contact line (CL) with its speed. A one-way coupling is used to investigate, whether viscous heating has to be taken into account. It can be derived that its role is negligible in the spreading process of a thin droplet, even for a relatively large viscous influence (large capillary number). Analyzing the results of a linear stability analysis of the fingering instability and taking Marangoni-stresses (MS) into account reveals, that the instability may be suppressed by cooling the ambient gas or heating the substrate during the spreading. Unfortunately an comparison with experiments for spreading droplets in a heated gas shows deviations for larger spreading radii. The influence of temperature on density is investigated and on the way a criteria, from which it may be obtained whether a simple Boussinesq-approximation (BA) is appropriate or not. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A strict Positivstellensatz for enveloping algebras

Let G be a connected and simply connected real Lie group with Lie algebra . Semialgebraic subsets of the unitary dual of G are defined and a strict Positivstellensatz for positive elements of the universal enveloping algebra of is proved. An erratum to this article is available at .     

Radial spreading and stability of a thin rotating liquid droplet

Thin-film flows are involved in many coating processes, where it is desirable to achieve thin and homogeneous fluid layers. In the present investigations, we treat droplets, spreading on rotating solid substrates. Micro-scale effects appear, firstly, at the wetting front, where the film height tends to zero. Secondly, micro-scale effects may appear at other locations, where the free liquid/gas-interface approaches the solid substrate, as e.g. at film rupture. For such situations, molecular effects need to be considered, e.g. in form of the disjoining pressure (DJP), to get physically-correct solutions. Otherwise, the spreading can be modeled within the frame of continuum mechanics, augmented by the (empirical) law of Tanner to capture the contact-line dynamics. We present, on the one hand, an overview of several interesting issues, as (i) spreading with and without considering the DJP, (ii) spreading after central rupture, including hysteresis effects, and (iii) non-isothermal spreading, including temperature-dependent surface tension (Marangoni effect) and temperature-dependent density (Rayleigh-Bénard effect). On the other hand, we present results for the instability of the contact line, for which the contact line gets corrugated (under isothermal conditions). This instability goes along with a transition from (rotationally-symmetric) two-dimensional to three-dimensional behavior. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unit Distances and Diameters in Euclidean Spaces

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large depending on d. As a corollary, we determine the exact maximum number of unit distances for all even d≥6 and the exact maximum number of diameters for all d≥4 and all n sufficiently large depending on d. This material is based upon work supported by the South African National Research Foundation.     

Inquiry-based learning for students, teachers, researchers, and representatives of educational administration and policy: reflections on a nation-wide initiative fostering educational innovations

International comparative studies such as TIMSS and PISA have had a considerable influence on the national educational policy in many countries. In Austria, as a reaction to the disappointing TIMSS 1995 results at the upper secondary level, a national initiative with the aim to foster mathematics and science education was launched in 1998: the IMST project. Due to specific challenges of the Austrian educational system, it has undergone several adaptations, but is still running. One of the project’s basic interventions is to promote teachers’ investigation into their own work. It is assumed that this supports the teachers’ critical stance towards innovation and inquiry, which in turn is an important basis for disseminating inquiry-based learning. A general look at the whole project and a specific look into one research project of IMST are used as opportunities to reflect on the complex interconnection and natural tension between the goal of promoting students’ IBL and its sustainable dissemination on a large scale. The paper introduces the project’s theoretical framework and genesis, provides exemplary research results, and reflects on its impact. The paper finishes with five “lessons learnt” from IMST.     

Degrees of logics with Henkin quantifiers in poor vocabularies

We investigate some logics with Henkin quantifiers. For a given logic L, we consider questions of the form: what is the degree of the set of L–tautologies in a poor vocabulary (monadic or empty)? We prove that the set of tautologies of the logic with all Henkin quantifiers in empty vocabulary L* is of degree 0. We show that the same holds also for some weaker logics like L(H) and L(E). We show that each logic of the form L^(k)(Q), with the number of variables restricted to k, is decidable. Nevertheless – following the argument of M. Mostowski from [Mos89] – for each reasonable set theory no concrete algorithm can provably decide L^(k)(Q), for some (Q). We improve also some results related to undecidability and expressibility for logics L(H₄) and L(F₂) of Krynicki and M. Mostowski from [KM92].Mathematics Subject Classification (2000): 03C80, 03D35, 03B25Revised version: 28 August 2003     

Unraveling the electronic structure of transition metal dimers using resonant four‐wave mixing

The potential of two‐color resonant and degenerate four‐wave mixing spectroscopy for investigations of the complex spectra of transition metal dimers is explored. Two‐color resonant and degenerate four‐wave mixing spectroscopy scans of the well‐known A‐X and B‐X transitions in Cu₂ are reported and compared with previous experimental data obtained from standard single‐resonance techniques. The selectivity of the method is shown to enable the measurement of isotopologue pure spectra without the need for isotopically enriched metal targets. Specific subsets of the rovibronic structure are separated in a congested spectral region of overlapping transitions. The sensitivity of the method compares satisfactorily with linear spectroscopic methods such as laser‐induced fluorescence and cavity ring‐down. A new laser vaporization source for the production of transition metal dimers and clusters has been constructed. The new design aims for a high number density and maximum possible shot‐to‐shot stability. The possibilities of further applications of non‐linear four‐wave mixing spectroscopy to Cu₂ and other transition metal dimers are discussed. Copyright © 2015 John Wiley & Sons, Ltd.