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Mathematical Analysis of Selected Problems for Complex Fluids
Los, Tomáš ; Málek, Josef (advisor) ; Kreml, Ondřej (referee) ; Süli, Endré (referee)
We study long-time and large-data existence theory of selected recently developed fluid mechanics models suitable for describing the mechanical behavior of materials with complex microstructure. In the first part of this work we focus on the Bingham type mod- els for granular materials with the activation parameter (critical value for the magnitude of the stress) dependent on the internal pore pressure. Our motivation comes from re- cent research concerning the implicitly constituted materials and also from an interesting paper by Chupin and Mathé [Chupin, Mathé, 2016], where the existence of weak solu- tions to the given problem was proved only in two spatial dimensions. Here we consider slightly different model (than in [Chupin, Mathé]) that we are able to derive from the basic governing equations of the theory of mixtures and we extend the existence result to three spatial dimensions. In the second part of this work we are concerned with fast developing field of viscoelastic materials. We study long-time and large-data existence of viscoelastic rate-type fluid models of higher order as they represent the simplest models suitable for describing the mechanical behavior of viscoelastic materials with complex microstructure. We are not aware of any long-time and large-data existence results for such models....
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Bingham-Korteweg fluids - modeling, analysis and computer simulations
Los, Tomáš ; Málek, Josef (advisor) ; Bulíček, Miroslav (referee)
Flow of granular materials is usually initiated when the shear stress is large enough and exceeds certain critical value. This can result in the presence of the dead-zones in which the flow itself does not take place. Motions of such materials are frequently described by Bingham model. Flows of granular fluids are frequently connected with the presence of free surface. In the thesis Bingham model is incorporated into a more general framework of Bingham-Korteweg fluids, which is a suitable way how to transfer free- boundary problems into the problems on fixed domains. A part of the thesis concerns mathematical analysis of interesting relevant problems for incompressible fluids. 1
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Bisectors
Los, Tomáš ; Johanis, Michal (advisor) ; Kalenda, Ondřej (referee)
This work deals with the study of bisectors (i.e. sets of points of equal distance from two given points) and the impact of their shape on the shape of the unit ball. It is known that if each bisector of two antipodal points on the sphere of a normed linear space lies in a hyperplane, then the norm is an inner product norm (for a special case of norm in R2 it is proved in Theorem 18). Here we generalise this statement in R2 for the case of (a priori) non-symmetric unit ball. In particular, we show that if the set of points x in the unit sphere, such that the bisector of x and −x is a line, has non-empty interior with respect to the sphere and the sphere is smooth, then the unit sphere is an ellipse centred at the origin. The work is based on the preprint [1]. 1
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