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Analysis of spontaneous collapse in elastic tubes
Netušil, Marek ; Maršík, František (advisor) ; Horný, Lukáš (referee)
Interaction of fluid with elastic tube is complicated issue studied by many scientific departments around the world. Object of this thesis is to analyze simplified one-dimensional model. At the beginning, used balance equations and basics of hyper-elasticity are presented. Then we review three most common materials used for the description of blood vessels and other soft tissues. For these materials we introduce a method which we use to derive a relation between tube deformation and transmural pressure (i.e. difference between inner and outer pressure). In mathematical section we give brief review of theory of nonlinear hyperbolic equations and some relatively new results in the field of existence and uniqueness of a solution of one-dimensional hyperbolic system. The "building stone" of these results is a solution of the so-called Riemann problem. We use a method for finding exact solutions to the Riemann problem to analyze studied model of fluid-tube interaction and study dependence of the qualitative behavior of the solution on the material properties of the tube wall.
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Exact and approximate Riemann solvers for the Euler equations
Živčáková, Andrea ; Kučera, Václav (advisor) ; Felcman, Jiří (referee)
In this work we deal with the solution and implementation of the problem of solving a partial differential equation with a piecewise constant initial condition, the so-called Riemann's problem. Specifically, we study the equations of conservation laws describing inviscid adiabatic flow of an ideal gas - the Euler equations. After some investigation, we show that these equations can be transformed to a quasilinear hyperbolic partial differential equation of first order. We are especially interested in the one-dimensional Euler equations for which we want to get an analytically exact Riemann's solver. The solution is found by investigation of properties of waves, namely rarefaction waves, shock waves and contact discontinuities were treated. The output of this work is a program in C for finding the exact Riemann's solver for one-dimensional Euler equations. The program is based on a theoretical analysis summarized in the first two chapters, and is tested on standard test data. The theory is based on the books [1] and [2].
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Exact and approximate Riemann solvers for the Euler equations
Živčáková, Andrea ; Kučera, Václav (advisor) ; Felcman, Jiří (referee)
In this work we deal with the solution and implementation of the problem of solving a partial differential equation with a piecewise constant initial condition, the so-called Riemann's problem. Specifically, we study the equations of conservation laws describing inviscid adiabatic flow of an ideal gas - the Euler equations. After some investigation, we show that these equations can be transformed to a quasilinear hyperbolic partial differential equation of first order. We are especially interested in the one-dimensional Euler equations for which we want to get an analytically exact Riemann's solver. The solution is found by investigation of properties of waves, namely rarefaction waves, shock waves and contact discontinuities were treated. The output of this work is a program in C for finding the exact Riemann's solver for one-dimensional Euler equations. The program is based on a theoretical analysis summarized in the first two chapters, and is tested on standard test data. The theory is based on the books [1] and [2].
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Analysis of spontaneous collapse in elastic tubes
Netušil, Marek ; Maršík, František (advisor) ; Horný, Lukáš (referee)
Interaction of fluid with elastic tube is complicated issue studied by many scientific departments around the world. Object of this thesis is to analyze simplified one-dimensional model. At the beginning, used balance equations and basics of hyper-elasticity are presented. Then we review three most common materials used for the description of blood vessels and other soft tissues. For these materials we introduce a method which we use to derive a relation between tube deformation and transmural pressure (i.e. difference between inner and outer pressure). In mathematical section we give brief review of theory of nonlinear hyperbolic equations and some relatively new results in the field of existence and uniqueness of a solution of one-dimensional hyperbolic system. The "building stone" of these results is a solution of the so-called Riemann problem. We use a method for finding exact solutions to the Riemann problem to analyze studied model of fluid-tube interaction and study dependence of the qualitative behavior of the solution on the material properties of the tube wall.
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