Page 542 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 542
PARTIAL DIFFERENTIAL EQUATIONS DESCRIBING FAR-FROM-EQUILIBRIUM
OPEN SYSTEMS (MS-51)
On planar flows of viscoelastic fluids of the Burgers type
Tomas Los, losjine@seznam.cz
Charles University, Czech Republic
Coauthors: Miroslav Bulícˇek, Yong Lu, Josef Málek
Viscoelastic rate-type fluid models involving the stress and its observer-invariant time deriva-
tives of higher order are used to describe a large class of viscoelastic mixtures - geomaterials
like asphalt, biomaterials such as vitreous in the eye, synthetic rubbers such as SBR. A standard
model that belongs to the category of viscoelastic rate-type fluid models of the second order is
the model due to Burgers, which can be viewed as a mixture of two Oldroyd-B models of the
first order. This viewpoint allows one to develop the whole hierarchy of generalized models of
a Burgers type. We study one such generalization. Carrying on the study by Masmoudi (2011),
who briefly proved the weak sequential stability of weak solutions to the Giesekus model, we
prove long time and large data existence of weak solutions to a mixture of two Giesekus models
in two spatial dimensions.
A simple thermodynamic framework for heat-conducting flows of
mixtures of two mechanically interacting fluids
Josef Málek, malek@karlin.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics, Czech Republic
Within the theory of interacting continua, we develop a model for a heat conducting mixture of
two mechanically interacting fluids described in the terms of the densities and the velocities for
each fluid and the temperature field for the whole mixture. We use a general thermodynamic
framework that determines the response of the material from the knowledge of two pieces of
information, namely how the material stores the energy and how the energy of material is dis-
sipated. This information is expressed in the form of the constitutive equations for two scalars:
the Helmholtz free energy and the entropy production. Additionally, we follow the goal to de-
termine the response of a mixture from a small (minimal) set of material parameters (bulk and
shear viscosity, heat conductivity, the drag coefficient) that can be associated with the mixture
as the whole. The same thermodynamic approach is used to obtain the model when the whole
mixture responses as an incompressible material. For both the compressible and incompress-
ible variants, we investigate two variants stemming from different definitions of the velocity
associated with the mixture as a whole.
Global existence analysis of fractional cross-diffusion systems
Erika Maringová, erika.maringova@ist.ac.at
IST Austria, Austria
TBA
540
OPEN SYSTEMS (MS-51)
On planar flows of viscoelastic fluids of the Burgers type
Tomas Los, losjine@seznam.cz
Charles University, Czech Republic
Coauthors: Miroslav Bulícˇek, Yong Lu, Josef Málek
Viscoelastic rate-type fluid models involving the stress and its observer-invariant time deriva-
tives of higher order are used to describe a large class of viscoelastic mixtures - geomaterials
like asphalt, biomaterials such as vitreous in the eye, synthetic rubbers such as SBR. A standard
model that belongs to the category of viscoelastic rate-type fluid models of the second order is
the model due to Burgers, which can be viewed as a mixture of two Oldroyd-B models of the
first order. This viewpoint allows one to develop the whole hierarchy of generalized models of
a Burgers type. We study one such generalization. Carrying on the study by Masmoudi (2011),
who briefly proved the weak sequential stability of weak solutions to the Giesekus model, we
prove long time and large data existence of weak solutions to a mixture of two Giesekus models
in two spatial dimensions.
A simple thermodynamic framework for heat-conducting flows of
mixtures of two mechanically interacting fluids
Josef Málek, malek@karlin.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics, Czech Republic
Within the theory of interacting continua, we develop a model for a heat conducting mixture of
two mechanically interacting fluids described in the terms of the densities and the velocities for
each fluid and the temperature field for the whole mixture. We use a general thermodynamic
framework that determines the response of the material from the knowledge of two pieces of
information, namely how the material stores the energy and how the energy of material is dis-
sipated. This information is expressed in the form of the constitutive equations for two scalars:
the Helmholtz free energy and the entropy production. Additionally, we follow the goal to de-
termine the response of a mixture from a small (minimal) set of material parameters (bulk and
shear viscosity, heat conductivity, the drag coefficient) that can be associated with the mixture
as the whole. The same thermodynamic approach is used to obtain the model when the whole
mixture responses as an incompressible material. For both the compressible and incompress-
ible variants, we investigate two variants stemming from different definitions of the velocity
associated with the mixture as a whole.
Global existence analysis of fractional cross-diffusion systems
Erika Maringová, erika.maringova@ist.ac.at
IST Austria, Austria
TBA
540
