Numerical simulation of some viscoelastic fluids

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Numerical simulation of the Oldroyd-B type viscoelastic fluids is a very challenging problem. \r\nThe well-known High Weissenberg Number Problem\" has haunted the mathematicians, computer scientists, and \r\nengineers for more than 40 years. \r\nWhen the Weissenberg number, which represents the ratio of elasticity to viscosity, \r\nexceeds some limits, simulations done by standard methods break down exponentially fast in time. \r\nHowever, some approaches, such as the logarithm transformation technique can significantly improve \r\nthe limits of the Weissenberg number until which the simulations stay stable. \r\n\r\nWe should point out that the global existence of weak solutions for the Oldroyd-B model is still open. \r\nLet us note that in the evolution equation of the elastic stress tensor the terms describing diffusive \r\neffects are typically neglected in the modelling due to their smallness. However, when keeping \r\nthese diffusive terms in the constitutive law the global existence of weak solutions in two-space dimension \r\ncan been shown. \r\n\r\nThis main part of the thesis is devoted to the stability study of the Oldroyd-B viscoelastic model. \r\nFirstly, we show that the free energy of the diffusive Oldroyd-B model as well as its \r\nlogarithm transformation are dissipative in time. \r\nFurther, we have developed free energy dissipative schemes based on the characteristic finite element and finite difference framework. \r\nIn addition, the global linear stability analysis of the diffusive Oldroyd-B model has also be discussed. \r\nThe next part of the thesis deals with the error estimates of the combined finite element \r\nand finite volume discretization of a special Oldroyd-B model which covers the limiting \r\ncase of Weissenberg number going to infinity. Theoretical results are confirmed by a series of numerical \r\nexperiments, which are presented in the thesis, too.

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