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Matematické modelování proudění krve
Michalová, Marie ; Hron, Jaroslav (advisor) ; Průša, Vít (referee)
At first we point out some important properties and behaviour of blood and blood particles. In the next part we approximate steady two-dimensional Pois- seiulle flow between two parallel plates. We show analytical solutions for New- tonian and power-law models. According to observations of Fahraeus-Lindquist effect we show two-layer models combining previously mentioned models. In the last section we model pulsatile Poisseuille flow and we show analytical and nu- merical results for some pressure and viscosity values. In the end we show our best model, where we shift pressure gradient and get closer to the real blood flow characteristics. 1
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Simple oscillatory flows of fluids with general boundary conditions
Košárková, Lenka ; Hron, Jaroslav (advisor) ; Průša, Vít (referee)
In mathematical models of a blood flow it is often a key role to choose appropriate boundary conditions imposed on the system of partial differential equations. Our aim is to investigate the oscillatory Hagen-Poiseuille flow in infinite pipe with Navier's slip condition on the boundary. Obtained velocity profiles can be then used as an inflow condition for numerical solutions of oscillatory flow in a finite pipe, where slip at the boundary is allowed. At first we recall the analytical solution for steady Hagen-Poiseuille flow in an infinite pipe with Navier's slip boundary condition. By replacing the constant pressure gradient with time periodic pressure gradient we get oscillatory Hagen-Poiseuille flow, where we derive in detail known analytical solution with respect to the no-slip boundary condition. Our contribution is in deriving analytical solution when considering the Navier's slip boundary condition, for which we show the velocity profiles and phase lag between the pressure gradient and the fluid's response. 1
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Metoda vrstevnic a modelování proudění nemísitelných nestlačitelných tekutin
Čapek, Marek ; Hron, Jaroslav (advisor) ; Maršík, František (referee)
The focus of this thesis are three-dimensional simulations of flow of two immiscible incompressible viscous fluids and distribution of concentration transported by the flow where the free interface between the fluids is realized by level-set method. It is applied to blood flow and coagulation. The blood flow is described using Navier-Stokes equations for incompressible fluids, numerical method is done by finite element method with Q2 - Q1 elements and the problem is solved as coupled. The distribution of concentration is described using transport equation, discretization is done using Q1 elements.
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Modelování proudění krve v geometrii aneuryzma
Zábojníková, Tereza ; Hron, Jaroslav (advisor) ; Feistauer, Miloslav (referee)
The aim of this work is to find a stable scheme which would solve the Stokes problem of the fluid flow, in which an elastic structure is immersed. Unlike most of the schemes solving fluid-structure interaction problems, in our scheme meshes of fluid and structure do not have to coincide. We have restricted ourselves to two-dimensional domain occupied by fluid with one-dimensional im- mersed structure. To describe a fluid-structure interaction, we have used an Immersed boundary method. At first we consider the strucure to be massless. We have modified an existing scheme and made it unconditionally stable, which was mathematically proven and numerically tested. Then we have proposed a modification where the structure is not massless and also proved the uncondi- tional stability in this case. The proposed schemes were implemented using the Freefem++ software and tested on aneurysm-like geometry. We have tested the behavior of our scheme in case when the qrowing aneurysm touches an obstacle, for example a bone (with no-slip condition on the bone boundary). Powered by TCPDF (www.tcpdf.org)
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Modelling of viscoelastic materials with temperature dependence
Miloš, Vojtěch ; Hron, Jaroslav (advisor) ; Tůma, Karel (referee)
Materials such as asphalt, polymers or the Earth's crust tend to behave in a way that can be described neither with a model of viscous fluid, nor a model from solid mechanics. There are indeed models capable of capturing these so called viscoelastic phenomena far better, but they are based on the presumption of constant temperature. In many cases, e.g. in the glass industry or in geophysics, the properties of a viscoelastic material strongly depend on temperature. That is why it is precisely these changes that need to be described. There are viscoelastic models used in practice that take into account the material parameters' dependence on temperature, however, they do not consider the viscoelastic nature of the material when describing the temperature evolution. The objective of this thesis is to derive thermodynamically consistent viscoelastic models with temperature dependent parameters and the appropriate evolution equation for temperature, implementation of the models and computing simple test simulations. Powered by TCPDF (www.tcpdf.org)
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Efficient methods for visualization of volumetric data
Holeček, Martin ; Hron, Jaroslav (advisor) ; Tůma, Miroslav (referee)
Title: Efficient methods for visualization of volumetric data Author: Martin Holeček Department: Mathematical Institute of Charles University Supervisor: RNDr. Ing. Jaroslav Hron, Ph.D., Mathematical Institute Abstract: The aim is to make an overview of and present implementation usefull for rendering of simulated datasets and CT and MR datasets. We will examine the methods of direct volume rendering of structured and unstructured grids and head to meaningful simultaneous realtime rendering of both types. In the first part, we briefly present the development, problems and targets of the field. Next, based on knowledge about existing algorithms, we choose one solution and present our own implementation and modification of algorithms. In detail, the object of our study will be numerical solutions of volume rendering integral by preintegration and paralelization of the process of projecting tetrahedra and perspective correction. For practical reasons we focus on the efficiency, that means computation time, used memory and usefullnes for medical presentation. The results will be compared with some other existing implementations. Keywords: volume rendering, preintegration, unstructured grid
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Computation of viscous flows due to an oscillating cylinder of rectangular cross section.
Outrata, Ondřej ; Hron, Jaroslav (advisor) ; Tůma, Karel (referee)
Incompressible flows due to an oscillating cylinder of rectangular cross section in viscous fluid are governed by Navier-Stokes equations. In this thesis, these equations will be reformulated in a weak sense and their solution approximated by Finite Element Method. Fictitious Boundary Method is used as a tool to handle time dependent boundary. Behavior of a fluid was computed using these methods and is illustrated for various parameters, especially a behavior of the vortices originated in liquid He II is compared to an experiment.
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Matematické modelování viskoplastických materiálů
Touška, Kryštof ; Hron, Jaroslav (advisor) ; Stebel, Jan (referee)
In the first chapter of the thesis we present an introduction to the visco- plasticity and overview of the presented problems. The constitutive relation for Bingham fluid is derived and in the second chapter. Further there is demon- strated a procedure of proving existence and uniqueness with classical varia- tional method. This method is compared with the same process using modern implicit theory. The last chapter starts with summary of used problem formu- lations and used software. It is then followed by the main part with results of numerical simulations, both for the purpose of used formulations comparison and then verification of the preferred one in more complicated simulations. We expect a possible application of tested approaches on different materials. 1
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Immersed boundary methods in air-flow models
Benešová, Stanislava ; Brechler, Josef (advisor) ; Hron, Jaroslav (referee)
This work focuses on the immersed boundary methods and their aplications to modeling flow in urban areas. In the first part Navier-Stokes equations are derived. In the second part we describe the main priciples of the immersed boundary methods. Two main groups of methods are introduced accompanied by examples. Based on the properties of these methods one method is selected which is suitable to describe the geometry in the modeling of airflow in urban areas. The method MIB from the article Mark et al. (2008) is chosen. This method is compared with the method introduced in the article Kim et al. (2001). Both methods are described in detail and compared according to several criteria in the third chapter. Such a criterion is the accuracy of the continuity equation. Simulations of a several test problems were carried out for the purpose of comparison of the chosen methods. Flow around a circular cylinder and a square cylinder for Reynolds numbers of 10 and 30 is solved. The divergence value of~the~velocity field in the modeled area was observed during the calculation of both methods. The results yields that MIB method solves the continuity equation more accurate.
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