Sep 23, 2014
Vít Pruša, Charles University in Prague, Faculty of Mathematics and Physics, Prague, Czech Republic Joint work with Tereza Perlácová The standard assumption in the theory of constitutive relations for non-Newtonian fluids is that the Cauchy stress tensor is a function of the symmetric part of the velocity gradient. By discussing experimental data available in the literature we show that the classical framework is overly restrictive. A simple framework that goes beyond the standard approach is the novel concept of implicit constitutive relations. Here, the basic assumption is that the relation between the stress and the symmetric part of the velocity gradient is given by an implicit tensorial equation. We demonstrate that the implicit type constitutive relations are adequate for fitting the one dimensional (shear stress versus shear rate) experimental data, and we speculate about the possible form of the corresponding three dimensional (Cauchy stress tensor versus symmetric part of the velocity gradient) implicit constitutive relations. Using the representation theorem for isotropic tensorial functions we conjecture that the implicit constitutive relations could lead to novel models capable to describe nonzero normal stress differences. Finally, we provide an example of a nontrivial thermodynamically and dynamically admissible implicit type tensorial constitutive relation. The simple model does predict nonzero normal stress difference, and shows that the conjecture is correct.
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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