Sep 23, 2014
Charles University in Prague, Faculty of Mathematics and Physics, Prague, Czech Republic The float glass process (Pilkington process) is the standard industrial scale process for manufacturing flat glass. The first phase of the process is the flow of the glass melt down an inclined plane (spout), and its impact on the tin bath, which makes the process a practical example of a multicomponent physical system. Our objective is to develop a mathematical model for the process, and implement a numerical scheme that would allow us to perform computer simulations of the process. For the numerical simulations we use a Cahn-Hilliard-Navier-Stokes type model which conceptually belongs to the class of so-called diffuse interface models. These models treat the interface between the components as a thin layer across which the components can mix, and that, among others, automatically take into account the surface tension effects. This allows one to avoid highly specialized and difficult to implement interface tracking methods. The cost to pay is the need to use a very fine spatial resolution in particular at the interface between the components. We will discuss the numerical challenges that must be addressed in order to make the numerical simulations based on the Cahn-Hilliad-Navier-Stokes type model applicable in the modelling of Pilkington process.
MOdelling REvisited + MOdel REduction Modeling, analysis and computing in nonlinear PDEs. September 21-26, 2014, Chateau Liblice, Czech Republic. Recently developed implicit constitutive theory allows one to describe nonlinear response of complex materials in complicated processes and to model phenomena in both fluid and solid mechanics that have hitherto remained unexplained. The theory also provides a thermodynamically consistent framework for technologically important but so far only ad hoc engineering models without sound footing. The overall goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large-scale simulations for the models arising from the new theoretical framework. A natural part of the goal is rigorous mathematical analysis of the models. The nonstandard structure of constitutive relations arising from the new framework requires the reconsideration of many existing approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, basic notions such as the concept of the solution and its well-posedness need to be reconsidered. Further, the complexity of the constitutive relations calls for rigorous investigation on model reduction - the identification of simplified models that capture the chosen (practically relevant) information about the behaviour of the system and disregard irrelevant information. Reliable numerical simulations require the derivation of sharp a posteriori error estimates to control all possible sources of errors, including rarely studied but important algebraic errors. We believe that in solving difficult problems in mathematical modelling the individual aspects discussed above - physics, mathematical analysis and numerical analysis - are so closely interrelated that no breakthrough can be achieved without emphasising the holistic approach as the main principle. Our vision is to follow this principle: the entire process of modelling of complex materials will be revisited in an innovative manner.
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