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SPn approximation to the radiative heat transport equation
Krupa, Tobiáš ; Souček, Ondřej (advisor) ; Málek, Josef (referee)
This thesis is dedicated to a problem of heat transport where radiation is taken into account. Models for such setting, although complicated, are very important for industrial purposes. We provide a derivation and explanation of the fundamental physical model for ra- diative heat transport. The resulting system of radiative transfer equations (RTE) is then approximated with so-called SPn equations. Here, we focus on asymptotically de- riving a simple set of SP1 equations. Special attention is given to Marsak-type boundary conditions which we formulate in a more precise form than other sources. Inspired by the float-glass forming process, we look into problems with multiple do- mains with different refractive indices. For such an arrangement, there is a need for transition conditions describing the behaviour of the solutions on the interface between two domains of interest. By following an analogous procedure as for the boundary con- ditions, we have managed to identify a natural set of transition conditions that allow for discontinuity in the intensity variable. To our best knowledge, these conditions have not been yet presented in the literature. Finally, we present several numerical experiments of solving these equations in Wol- fram Mathematica software and compare them with benchmark results. 1
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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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Unidirectional flows for binary fluid mixtures
Púček, Jiří ; Málek, Josef (advisor) ; Tůma, Karel (referee)
In a recent work Málek a Souček (2021), the authors propose several classes of simple models suitable to describe flows of binary mixtures, such as bubbly liquids, emulsions, etc. The objective of the bachaler thesis is to investigate the potential of these model by their investigation in simple geometrical settings linked with the assumption that the admissible flows are isothermal and unidirectional. In addition we will compare our results with other models. 1
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Numerical Modeling of Ice Sheet Dynamics
Souček, Ondřej ; Martinec, Zdeněk (advisor) ; Málek, Josef (referee) ; Ivins, Erik R. (referee)
Title: Numerical modeling of ice sheet dynamics Author: Ondřej Souček Department: Department of Geophysics, Faculty of Mathematics and Physics, Charles University in Prague Supervisor: Prof. RNDr. Zdeněk Martinec, DrSc. Supervisor's e-mail address: zdenek@cp.dias.ie Abstract: The main topic of the presented thesis is the numerical modeling of large-scale ice-sheet evo- lution over several glacial cycles. For this purpose a "Shallow Ice Approximation" (SIA) is traditionally adopted, a scaling approximation utilizing the fact that the vertical-to-horizontal aspect ratio in ice sheets is typically a small parameter allowing for a perturbation expansion of the governing equations with respect to this parameter. We present a formulation of the boundary value problem for a polythermal ice sheet based on the rational thermodynamics of mixtures and derive its "shallow-ice" form. We then provide a novel SIA-I algorithm for iterative improvement of the SIA and thoroughly test its performance in a series of benchmarks including the ISMIP-HOM benchmark and a realistic simulation for the Dronning Maud Land region, Antarctica. The SIA-I algorithm is implemented to an evolutionary thermo-mechanical numerical ice-sheet model and this model is tested in the SIA regime in two EISMINT benchmarks, EISMINT - Effect of...
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Nonnewtonian fluid flow simulation using lattice Boltzmann method
Kuriščák, Pavel ; Hron, Jaroslav (advisor) ; Málek, Josef (referee)
Title: Non-newtonian fluid flow simulation using lattice Boltzmann method Author: Bc. Pavel Kuriščák Department: Mathematical Institute, Charles University Supervisor: RNDr. Ing. Jaroslav Hron Ph.D. Supervisor's e-mail address: Jaroslav.Hron@mff.cuni.cz Abstract: The aim of this thesis is to find and estabilish a modification to the Lattice Boltzmann Method, allowing it to simulate non-newtonian behaviour of fluids. In the theoretical part of thesis, there is introduced a derivation, based on the work of [22], that is capable of arriving to macroscopical Navier-Stokes equa- tions completely a priori from the Boltzmann equation, utilizing the Hermite basis expansion. This derivation is afterwards applied to the method suggested by [11], that uses the changed equilibrium distribution to fine-tune the local fluid viscosity according to the non-newtonian model. In the last part of thesis, this method is implemented in the form of lattice kinetic scheme and tested on three sample problems. Keywords: Lattice Boltzmann Method, non-newtonian fluids, Hermite expansion, lattice kinetic scheme
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Biochemical and mechanical processes in synovial fluid - modeling, analysis and computational simulations
Pustějovská, Petra ; Málek, Josef (advisor) ; Süli, Endré (referee) ; Jäger, Willi (referee) ; Maršík, František (referee)
vi Title: Biochemical and mechanical processes in synovial fluid - modeling, mathematical analysis and computational simulations Author: Petra Pustějovská (petra.pustejovska@karlin.mff.cuni.cz) Department: Matematický ústav UK, Univerzita Karlova v Praze Institut für Angewandte Mathematik, Universität Heidelberg Supervisors: prof. RNDr. Josef Málek CSc., DSc. (malek@karlin.mff.cuni.cz) Matematický ústav UK, Univerzita Karlova v Praze, Prof. Dr. Dr. h.c. mult. Willi Jäger (jaeger@iwr.uni-heidelberg.de) Institut für Angewandte Mathematik, Universität Heidelberg Abstract: Synovial fluid is a polymeric liquid which generally behaves as a viscoelastic fluid due to the presence of polysaccharide molecules called hyaluronan. In this thesis, we study the biological and biochemical properties of synovial fluid, its complex rheology and interaction with synovial membrane during filtration process. From the mathematical point of view, we model the synovial fluid as a viscous incompressible fluid for which we develop a novel generalized power-law fluid model wherein the power-law exponent depends on the concentration of the hyaluronan. Such a model is adequate to describe the flows of synovial fluid as long as it is not subjected to instantaneous stimuli. Moreover, we try to find a suitable linear viscoelastic model...
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Asymptotic Behavior of Solutions in Problems of the Mathematical Theory of Fluids
Kukučka, Peter ; Feireisl, Eduard (advisor) ; Málek, Josef (referee) ; Novotný, Antonín (referee)
This thesis contains a set of articles concerned with flow of a viscous, compressible and heat conducting fluids in several kinds of domains. The first part is devoted to the existence of weak solutions in domains that may contain cusps. Next chapter is focused on the asymptotic limit of the equations of magnetohydrodynamics consisting of Navier-Stokes-Fourier system describing the evolution of fluid coupled with Maxwell equations governing the behavior of magnetic field with the low Mach and Alfv'en number. At the end of the thesis, we study the asymptotic limit passage of the Navier-Stokes-Fourier system under the strong stratification defined in unbounded domain. Special attention is paid to the acoustic waves which analysis is based on local energy decay.
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Korteweg fluids - modeling, analysis and computer simulations
Blaškovičová, Monika ; Málek, Josef (advisor) ; Řehoř, Martin (referee)
We present two possible thermodynamical approaches towards a derivation of a model, proposed by Korteweg at the beginning of the 20th century, that is suitable to describe phase transitions liquid-vapor with non-sharp interfaces. The first approach (Dunn, Serrin (1985)) is based on classical rational continuum thermodynamics. The second approach (Heida, Málek (2010)) stems from the principles of classical nonequilibrium continuum thermodynamics. We compare both approaches in favor of the second one. The considered constitutive equation for the Cauchy stress is nonlinear. Nonlinearity and higher order derivatives of the density makes the analysis of relevant problems for the Navier-Stokes- Korteweg (NSK) fluid more difficult in comparison to problems concerning Navier-Stokes equations. Special attention is devoted to the appropriate choice of the boundary conditions. We also investigate the influence of compressibility on the stability of bubbles by comparing numerical simulations for compressible NSK fluid and its incompressible variant. Instabilities observed for a compressible NSK fluid are due to the pressure that has a different meaning for incompressible fluid. Powered by TCPDF (www.tcpdf.org)
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