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液体薄膜的脱湿和非均匀基底上流体结构形成的数学模型。在schwerpunktprogram的第二阶段，将包括对蒸发和冷凝过程的研究。本项目主要包括以下三个部分：建模：基于润滑近似，推导非均匀表面上冷凝流体高度的演化方程。分析：利用变分法和偏微分方程理论应使用的方法，得到相应的膜高h和压力p演化方程解的存在性和定性行为的结果。这些理论结果除了对更好地理解解的渐近行为具有明显的重要性外，也是制定快速可靠的数值计算算法的关键因素。数值模拟：在过去两年中，G. gr<e:1> n和M. Rumpf成功地开发和分析了一种有限元/有限体积方案，该方案大大减少了模拟传播现象的计算时间。在此方案的基础上，设计一个通用的有限元求解器，对上述所有现象进行数值模拟。通过与实验数据的对比进行评价；它在很大程度上取决于与schwerpunktprogram内部的实验人员的密切合作。

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.