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Grants





Theory and numerical analysis of coupled problems in fluid dynamics (GA1701747S)
from 01/01/2017
to 31/12/2019 investigator

Objectives:
The project is focused on several important fields of today's rapidly developing mathematical fluid mechanics. The aim is to derive a series of results, from new regularity criteria, stability and robustness analysis of solutions, up to the low Mach and high Reylolds limits in a compressible fluid interacting with a solid structure. Beside the qualitative analysis of flow problems, a part of the project is the development and analysis of new, accurate and robust numerical methods for the solution of important and topical models of fluid dynamics. The attention will be paid to the development and analysis of high order methods for the solution of nonstationary nonlinear partial differential equations and compressible flow, based on the discontinuous Galerkin method. Particularly we have hpversions in mind. These methods will be applied to the numerical solution of fluidstructure interaction and multiphase flow. Another subject is the study of flow model with slip boundary conditions.



Thermodynamically consistent models for fluid flows: mathematical theory and numerical solution (GA1603230S)
from 01/01/2016
to 31/12/2018 investigator

Objectives:
Mathematical and numerical analysis and numerical solution of fluid flows belong to the most often studied problems of the theory of partial differential equations and their numerical solution. During the last decades, a big progress has been achieved in these fields which enables us to study models of complex fluids including the possibility to consider their dependence on temperature. This project is focused on the study of such models of fluid thermodynamics and mechanics with the aim to extend the knowledge in the field of the theoretical analysis of the corresponding systems of partial differential equations and numerical analysis of the methods for their solution. Computational simulations using specific numerical methods will be performed to support the analytical results concerning the wellposedness of the model problems and qualitative properties of their solutions. The proposed projects assumes a tight collaboration of specialists in these fields which is an important prerequisite for further development of mathematical and computational fluid thermodynamics.



Flow of fluids in domains with variable geometry (P201/11/1304)
from 01/01/2011
to 31/12/2013 investigator

Objectives:
The goal of the project is to get new relevant results concerning flow in domains with varying geometry. From the viewpoint of theoretical analysis, we will deal with flow of fluids (incompressible and compressible) around a rotating body (existence of weak or very weak solutions, asymptotic behaviour solutions, artificial boundary conditions) in case that the axis of rotation of the body and the velocity at infinity are parallel or not parallel. We will also investigate the related hydrodynamical potential theory. Moreover, we will investigate the case of motion of rigid bodies in viscous fluid (mostrly nonNewtonian incompressible and Newtonian compressible), in several cases we include the changes of temperature. Part of the problems mentioned above will be solved numerically. Finally, we perform the numerical simulation of flow of fluids in domains with complicated geometry corresponding to the flow of blood in healthy veins as well as in cases of cardiovascular diseases.



The motion of rigid bodies in liquid: mathematical analysis, numerical simulation and related problems (IAA100190804)
from 01/01/2008
to 31/12/2010 investigator

In the framework of the project we will study the steady flow around bodies. We will consider the case when the direction of the angular velocity and of the velocity at infinity are or are not parallel. We will extend the results from the previous project, where the angular and tranlation velocities were parallel. We will study the linear cases and NavierStokes equations. We will investigate the existence of solution, asymptotic behaviour, resolvent and spectrum problem. Further, we will study the motion of several bodies in the fluid. We will consider the influence of boundary conditions and possibility of collisions. In this part we will study the existence of weak solution for steady and nonsteady cases. We will investigate fluid flows described by NavierStokes equations as well as by nonNewtonian models. We will investigate the modeling of blood flow and related cardiovascular cases. Next to it the numerical simulation of severeal models will be performed.
