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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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Vliv materiálových parametrů na stabilitu termální konvekce
Dostalík, Mark ; Matyska, Ctirad (advisor) ; Klika, Václav (referee)
The thesis is focused on the investigation of Rayleigh-Bénard problem in an extended setting approximating the conditions in the Earth's mantle. The aim is to evaluate the influence of depth- and temperature- dependent material parameters, dissipation, adiabatic heating/cooling and heat sources on the qualitative characteristics of thermal convection. We identify the critical values of dimensionless parameters that determine the onset of convection and characterize the dominating convection patterns in marginally supercritical states. These issues are addressed by the application of linear stability analysis and weakly non-linear analysis. It has been found that the character of convection differ substantially from the standard case of Rayleigh-Bénard convection. Powered by TCPDF (www.tcpdf.org)
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Nonlinear stability of steady states in thermomechanics of viscoelastic fluids
Dostalík, Mark ; Průša, Vít (advisor) ; Chupin, Laurent (referee) ; Neustupa, Jiří (referee)
We study nonlinear stability of steady state solutions of partial differential equations governing the thermomechanical evolution of viscoelastic fluids; materials that exhibit both viscous as well as elastic response when undergoing deformation. It is well-known that thermodynamical concepts can be gainfully exploited in the construction of Lya- punov functionals for nonlinear stability analysis of spatially homogeneous equilibrium rest states in thermodynamically closed systems. We show that the thermodynamically oriented approach can be utilized in the nonlinear stability analysis of spatially inhomo- geneous non-equilibrium steady states in thermodynamically open systems as well. The thesis consists of two parts. In the first part, we revisit the classical construction of Lyapunov functionals in thermodynamically closed systems and we apply the nonlinear stability theory to compressible heat-conducting viscoelastic fluids modeled by a multi- scale, as well as a purely macroscopic approach. In the second part, we focus on two special instances of thermodynamically open systems. First, we show that the spatially inhomogeneous non-equilibrium steady state of an incompressible heat-conducting vis- coelastic fluid, which occupies a mechanically isolated vessel with walls kept at spatially non-uniform...
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Vliv materiálových parametrů na stabilitu termální konvekce
Dostalík, Mark ; Matyska, Ctirad (advisor) ; Klika, Václav (referee)
The thesis is focused on the investigation of Rayleigh-Bénard problem in an extended setting approximating the conditions in the Earth's mantle. The aim is to evaluate the influence of depth- and temperature- dependent material parameters, dissipation, adiabatic heating/cooling and heat sources on the qualitative characteristics of thermal convection. We identify the critical values of dimensionless parameters that determine the onset of convection and characterize the dominating convection patterns in marginally supercritical states. These issues are addressed by the application of linear stability analysis and weakly non-linear analysis. It has been found that the character of convection differ substantially from the standard case of Rayleigh-Bénard convection. Powered by TCPDF (www.tcpdf.org)
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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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