|
|
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....
|
|
|
Motion of a body in a fluid with pressure dependent viscosity
Sláčík, Stanislav ; Průša, Vít (advisor) ; Süli, Endré (referee)
A lot of technologically relevant incompressible fluids exhibit a substantial variaton of viscosity with the pressure;a falling cylinder viscometer is frequently used for the measurements, determining the viscosity indirectly from the time it takes the sinker to fall a given distance. The relation between the sinker fall velocity and the fluid viscosity is, however, derived under the constant viscosity assumption. The objective of the present thesis is to perform a numerical simulation of the viscometric experiment, assuming an explicit form of the pressure-viscosity dependence and realistic parameter values and to quantitatively assess the difference in body motion to the Navier- Stokes model. The computational method proposed, handling both the nonlinear constitutive relation and the body motion, was tested on simple problems with analytical solutions. The semianalytical relation for the Navier-Stokes model, also re-derived here, is compared to the computational results.The validity of the assumptions used in the theoretical derivation, based on the results of the numerical simulation, is discussed regarding the geometry of the viscometer. Powered by TCPDF (www.tcpdf.org)
|
|
|
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...
|
|
|
Towards efficient numerical computation of flows of non-Newtonian fluids
Blechta, Jan ; Málek, Josef (advisor) ; Herzog, Roland (referee) ; Süli, Endré (referee)
In the first part of this thesis we are concerned with the constitutive the- ory for incompressible fluids characterized by a continuous monotone rela- tion between the velocity gradient and the Cauchy stress. We, in particular, investigate a class of activated fluids that behave as the Euler fluid prior activation, and as the Navier-Stokes or power-law fluid once the activation takes place. We develop a large-data existence analysis for both steady and unsteady three-dimensional flows of such fluids subject either to the no-slip boundary condition or to a range of slip-type boundary conditions, including free-slip, Navier's slip, and stick-slip. In the second part we show that the W−1,q norm is localizable provided that the functional in question vanishes on locally supported functions which constitute a partition of unity. This represents a key tool for establishing local a posteriori efficiency for partial differential equations in divergence form with residuals in W−1,q . In the third part we provide a novel analysis for the pressure convection- diffusion (PCD) preconditioner. We first develop a theory for the precon- ditioner considered as an operator in infinite-dimensional spaces. We then provide a methodology for constructing discrete PCD operators for a broad class of pressure discretizations. The...
|
|
|
Analysis of unsteady flows of incompressible heat-conducting rate-type viscoelastic fluids with stress-diffusion
Bathory, Michal ; Bulíček, Miroslav (advisor) ; Feireisl, Eduard (referee) ; Süli, Endré (referee)
We prove a global-in-time and large-data existence of a suitable weak solution to a system of partial differential equations describing an unsteady flow of homogeneous incom- pressible viscoelastic rate-type fluid. The material parameters are continuous functions of temperature and, in particular, the dependence of the shear modulus is assumed to be linear. It is shown that studied models obey the fundamental laws of thermodynamics. The key step towards the existence proof is derivation of the balance of entropy. This in- equality is paramount in the analysis and as its consequence, we obtain sufficient a priori estimates, positivity of temperature and also regularity of the elastic deformation. The second part of the thesis deals with the existence analysis for the isothermal case, however using a completely different method, which is of independent interest. 1
|
|
|
Analysis of unsteady flows of incompressible heat-conducting rate-type viscoelastic fluids with stress-diffusion
Bathory, Michal ; Bulíček, Miroslav (advisor) ; Feireisl, Eduard (referee) ; Süli, Endré (referee)
We prove a global-in-time and large-data existence of a suitable weak solution to a system of partial differential equations describing an unsteady flow of homogeneous incom- pressible viscoelastic rate-type fluid. The material parameters are continuous functions of temperature and, in particular, the dependence of the shear modulus is assumed to be linear. It is shown that studied models obey the fundamental laws of thermodynamics. The key step towards the existence proof is derivation of the balance of entropy. This in- equality is paramount in the analysis and as its consequence, we obtain sufficient a priori estimates, positivity of temperature and also regularity of the elastic deformation. The second part of the thesis deals with the existence analysis for the isothermal case, however using a completely different method, which is of independent interest. 1
|
|
|
Mathematical analysis of models arising in continuum mechanics with implicitly given rheology and boundary conditions
Maringová, Erika ; Bulíček, Miroslav (advisor) ; Gwiazda, Piotr (referee) ; Süli, Endré (referee)
In the thesis, we study the Navier-Stokes-like and the Navier-Stokes-Fourier- like problems for the flows of homogeneous incompressible fluids. In the first part of the thesis, we introduce a new type of boundary condition for the shear stress tensor, which includes the time derivative of the velocity. Therefore, we are able to capture the dynamic response of the fluid on the boundary. As the second part of the thesis, we include the published journal article co-authored by J. Žabenský on the Navier-Stokes-Fourier-like problem formulated in the complete thermodynamic setting. In both parts, the constitutive relations are formulated implicitly with the use of maximal monotone graphs. The main result of the thesis is the existence analysis for the above mentioned problems.
|
|
|
Towards efficient numerical computation of flows of non-Newtonian fluids
Blechta, Jan ; Málek, Josef (advisor) ; Herzog, Roland (referee) ; Süli, Endré (referee)
In the first part of this thesis we are concerned with the constitutive the- ory for incompressible fluids characterized by a continuous monotone rela- tion between the velocity gradient and the Cauchy stress. We, in particular, investigate a class of activated fluids that behave as the Euler fluid prior activation, and as the Navier-Stokes or power-law fluid once the activation takes place. We develop a large-data existence analysis for both steady and unsteady three-dimensional flows of such fluids subject either to the no-slip boundary condition or to a range of slip-type boundary conditions, including free-slip, Navier's slip, and stick-slip. In the second part we show that the W−1,q norm is localizable provided that the functional in question vanishes on locally supported functions which constitute a partition of unity. This represents a key tool for establishing local a posteriori efficiency for partial differential equations in divergence form with residuals in W−1,q . In the third part we provide a novel analysis for the pressure convection- diffusion (PCD) preconditioner. We first develop a theory for the precon- ditioner considered as an operator in infinite-dimensional spaces. We then provide a methodology for constructing discrete PCD operators for a broad class of pressure discretizations. The...
|
|
|
Motion of a body in a fluid with pressure dependent viscosity
Sláčík, Stanislav ; Průša, Vít (advisor) ; Süli, Endré (referee)
A lot of technologically relevant incompressible fluids exhibit a substantial variaton of viscosity with the pressure;a falling cylinder viscometer is frequently used for the measurements, determining the viscosity indirectly from the time it takes the sinker to fall a given distance. The relation between the sinker fall velocity and the fluid viscosity is, however, derived under the constant viscosity assumption. The objective of the present thesis is to perform a numerical simulation of the viscometric experiment, assuming an explicit form of the pressure-viscosity dependence and realistic parameter values and to quantitatively assess the difference in body motion to the Navier- Stokes model. The computational method proposed, handling both the nonlinear constitutive relation and the body motion, was tested on simple problems with analytical solutions. The semianalytical relation for the Navier-Stokes model, also re-derived here, is compared to the computational results.The validity of the assumptions used in the theoretical derivation, based on the results of the numerical simulation, is discussed regarding the geometry of the viscometer. Powered by TCPDF (www.tcpdf.org)
|
|
|
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...
|
guest ::
