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Mathematical analysis of equations describing the flow of compressible heat conducting fluids
Axmann, Šimon ; Pokorný, Milan (advisor) ; Feireisl, Eduard (referee) ; Novotný, Antonín (referee)
Title: Mathematical analysis of equations describing the flow of compressible heat conducting fluids Author: Šimon Axmann Department: Mathematical Institute of Charles University Supervisor: doc. Mgr. Milan Pokorný, Ph.D., Mathematical Institute of Charles University Abstract: The present thesis is devoted to the mathematical analysis of equa- tions describing the flow of viscous compressible newtonian fluid in various time regimes. In particular, we present existence results for three problems arising as special cases of a general model derived in the introductory part. The first chap- ter deals with time-periodic solutions to the full Navier-Stokes-Fourier system for heat-conducting fluid. The second chapter contains the proof of existence of steady solutions to a system arising from phase field model for two-phase com- pressible fluid. Finally, in the last section we study steady strong solutions to the Navier-Stokes equations under the additional assumption that the fluid is suffi- ciently dense. For each problem a different concept of the solution is considered, on the other hand in all cases an essential role is played by the crucial quantity effective viscous flux. Keywords: compressible Navier-Stokes system; weak solution; entropy variational solution; large data
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Analysis of Point Clouds Representing Surfaces of Engineering Practice
Surynková, Petra ; Voráčová, Šárka (advisor) ; Pokorný, Milan (referee) ; Lávička, Miroslav (referee)
Title: Analysis of Point Clouds Representing Surfaces of Engineering Practice Author: Petra Surynková Department: Department of Mathematics Education Supervisor: Mgr. Šárka Voráčová, Ph.D., Faculty of Transportation Sciences, Czech Technical University in Prague Abstract: The doctoral dissertation Analysis of Point Clouds Representing Surfaces of Engineering Practice addresses the development and application of methods of digital reconstruction of surfaces of engineering and construction practice from point clouds. The main outcome of the dissertation is a presentation of new procedures and methods that contribute to each of the stages of the reconstruction process from the input point clouds. The work is mainly focused on the analysis of input clouds that describe special types of surfaces. Several completely new algorithms and improvements of existing algorithms that contribute to individual steps of surface reconstruction are presented. New procedures are based on geometrical characteristics of the reconstructed object. An important result of the dissertation is an analysis of not only synthetically generated point clouds but above all an analysis of real point clouds that have been obtained from measurements of real objects. The significant contribution of the dissertation is also an...
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Weak formulation of equations describing fluid flows
Dostalík, Mark ; Pokorný, Milan (advisor) ; Kaplický, Petr (referee)
The standard way of deriving the weak formulation of balance equations of continuum mechanics is derived from their localized form, and thus requires differentiability of functions involved in the corresponding balance law. However, the existence of classical solutions of these equations is often not known. It would be suitable to find a transition to the weak formulation of balance laws without the need of their differential form. The aim of this work is to show that the initial integral form of balance equations of continuum mechanics, provided relatively weak assumptions, directly implies their weak formulation, and thus that the weak solution is for these equations a more natural notion than the classical solution is.
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Matematická analýza regularizovaného modelu viskoelastické nenewtonovské tekutiny
Šalom, Pavel ; Pokorný, Milan (advisor) ; Bulíček, Miroslav (referee)
In this thesis we provide an existence result for a regularized model of viscoelastic non- newtonian fluid. We consider incompressible fluid with shear rate dependent viscosity and with Cauchy stress tensor capable to describe stress relaxation. An elastic part of the Cauchy stress tensor is governed by Oldroyd-type differential equation. In particular, we are interested in fluids with strong shear thinning effect. We prove that if the viscosity function µ (D) is such that tensor µ (D) D is p-coercive, monotone and has (p − 1)-growth for p > 6 5 and some other additional assumptions are satisfied, then there exists a solution to the system of PDEs describing the flow in a bounded domain. The proof is not simple because the convective term is not integrable with a high power. The problem is solved using Lipschitz truncation method for evolution PDEs. 1
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Regularity criteria for instationary incompressible Navier-Stokes equations
Axmann, Šimon ; Pokorný, Milan (advisor) ; Neustupa, Jiří (referee)
Title: Regularity criteria for instationary incompressible Navier-Stokes equations Author: Šimon Axmann Institute: Mathematical Institute of Charles University Supervisor: doc. Mgr. Milan Pokorný, Ph.D., Mathematical Institute of Charles University Abstract: In the present thesis we study the global conditional regularity of weak solutions to the Cauchy problem for instationary incompressible Navier-Stokes equations in three space dimensions. In the first section, we present an overview of known conditions implying the full regularity of the equations under conside- ration. For the sake of clarity, we expose only the regularity criteria on the scale of Lebesgue spaces, especially in terms of the velocity and its components, the gradient of the velocity and its components, the pressure and the vorticity. In the subsequent sections, we generalize four regularity criteria using two different techniques. We are able to replace one velocity component or its gradient, consi- dered in the known results, by a projection of the velocity into a general vector field. For the purpose of the second method, we also generalize the multiplicative Gagliardo-Nirenberg inequality.
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Compressible fluid motion in time dependent domains
Sýkora, Petr ; Feireisl, Eduard (advisor) ; Pokorný, Milan (referee)
In this work we study the existence of weak solutions for compressible Navier-Stokes equations in unbounded time dependent domains. Using the methods introduced in Feireisl E. Dynamics of Viscous Compressible Fluids we extend the results of article Feireisl E. Neustupa J. Stebel J., Convergence of a Brinkman-type penalization for compressible fluid flows, which studies the flow with a "no-slip" boundary condition on bounded domains. Next, we extend results of article Feireisl E. Kreml O. Nečasová Š. Neustupa J. Stebel J., Weak solutions to the barotropic Navier- Stokes system with slip boundary conditions in time dependent domains, which studies flow with compete Navier boundary condition. Finally, we discuss solutions for rotating fluid system. In this case, there are new members in momentum equation, representing the Coriolis and centrifugal force, which cause problems.
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Mathematical Analysis of Models for Viscoelastic Fluids
Kreml, Ondřej ; Pokorný, Milan (advisor) ; Skalák, Zdeněk (referee) ; Neustupa, Jiří (referee)
1 Title: Mathematical analysis of models for viscoelastic fluids Author: Ondřej Kreml Department: Mathematical Institute of Charles University Supervisor: Doc. Mgr. Milan Pokorný, Ph.D. Abstract: We consider several problems in the thesis. First we summarize key ideas of fluid mechanics theory and introduce several models describ- ing nonnewtonian behaviour of fluids. In the second chapter we prove local existence of solutions to the Oldroyd-type system achieved as a limit case with infinite relaxation and retardation times. We work with three types of boundary conditions, namely homogenous Dirichlet and periodic conditions and whole space, in 2D and 3D. We study also related system of PDE's which is equivalent to the Oldroyd-type system in 2D. In the third chapter we prove local existence of solutions to the system of PDE's describing the flow of a polymeric liquid. The polymer molecules are modeled as elastic dumbbells with spring force having the so-called FENE potential. Arising system con- sists of Navier-Stokes equations coupled with Fokker-Planck equation. In the fourth chapter we study asymptotic behaviour of solutions to equations desribing steady flow of a second grade fluid past an obstacle in three dimen- sions with prescribed nonzero velocity at infinity. Key point in the proof is using results of...
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Mathematical Analysis of Fluids in Large Domains
Poul, Lukáš ; Feireisl, Eduard (advisor) ; Pokorný, Milan (referee) ; Vodák, Rostislav (referee)
This thesis contains a set of articles concerned with flow of a viscous, compressible and heat conducting fluid in large domains. In the first part of the thesis, the existence of the weak solutions in unbounded domains is studied. The results follow each other in the way they were obtained through the time, and range from a simple extension to bounded domains with Lipschitz boundary up to the most general existence theorem for fluid flow in general open sets. The existence results are supplemented with the study of existence of weak solutions in the unbounded domain case with prescribed nonvanishing boundary conditions for density and temperature at infinity. The last contribution then concerns with the low Mach number limit in the compressible fluid flow.
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Generalized Stokes systems - theoretical analysis approach
Holeček, Martin ; Pokorný, Milan (referee) ; Málek, Josef (advisor)
We consider steady flows of homogenous incompressible fluid described by generalized Stokes system. We study two models, first with shear-rate dependent viskosity and second with pressure and shear-rate dependent viskosity. We investigate internal flows in bounded domains subject to Navier's boundary condition. First, to show the difference, we present proofs of existence and uniqueness of solutions for both systems. Then we investigate, what are the assumptions allowing to take the fluid mechanics limit, as Navier's boundary conditions approximate first no-slip and then (perfect) slip boundary conditions. Finally, we consider for simplicity specially periodic problem and show regularity result (integrability of the second derivatives of the velocity and the first derivatives of the pressure).
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