Mathematische Analyse von Modellen zur Bildung fluider Strukturen an Grenzflächen

该项目的目标是分析、评估和改进均匀表面上薄液膜的脱湿和非均匀衬底上流体结构的形成的数学模型。在施沃朋克方案的第二阶段，将包括对蒸发和冷凝过程的调查。该项目包括三个主要部分，可概括如下：建模：基于润滑近似，应推导出非均匀表面上冷凝流体高度的演化方程。分析：应使用变分和偏微分方程组理论中的方法来获得关于油膜高度h和压力p的相应演化方程的解的存在性和定性行为的结果。除了它们对于更好地理解解的渐近行为的明显重要性之外，这些理论结果也是形成快速而可靠的数值模拟算法的关键因素。数值模拟：在过去的两年中，G.Grün和M.Rumpf成功地开发和分析了一种有限元/有限体积格式，该格式极大地减少了模拟扩散现象的计算时间。基于这一方案，应设计一个通用的有限元求解器，以实现对上述所有现象的数值模拟。评估将通过与实验数据的比较来进行；这在很大程度上依赖于与施沃朋克计划内的实验者的密切合作。

The goal of the project is to analyse, to evaluate, and to improve mathematical models for the dewetting of thin liquid films on homogeneous surfaces and the formation of fluid structures on inhomogeneous substrates. During the second period of the Schwerpunktprogramm, investigations on evaporation and condensation processes will be included. The proposed project consists of three main parts which can be summarized as follows:Modelling: Based on lubrication approximation, evolution equations for the height of condensing fluids on inhomogeneous surfaces shall be derived.Analysis: Methods from the calculus of variations and the theory of partial differential equations shall be used to obtain results on existence and qualitative behaviour of solutions to the corresponding evolution equations for film height h and pressure p. Besides their obvious importance for a better understanding of the asymptotic behaviour of solutions, these theoretical results are also the key ingredient to formulate fast and reliable algorithms for numerical sumulations.Numerical simulations: During the last two years, G. Grün and M. Rumpf succeded in developing and analysing a finite-element/finite-volume scheme which drastically reduces the computation time for the simulation of spreading phenomena. Based on this scheme, a general finite-element solver shall be designed to enable numerical simulations of all the phenomena mentioned above. The evaluation will be performed by comparison with experimental data; it strongly depends on an intense cooperation with experimentalists inside the Schwerpunktprogramm.