Title: Flows of incompressible fluids with pressure-dependent viscosity (and their application to modelling the flow in journal bearing) Author: Martin Lanzendörfer Department: Mathematical Institute of Charles University Supervisor: prof. RNDr. Josef Málek, DSc. Abstract: The viscosity of the fluids involved in hydrodynamic lubrication typically depends on pressure and shear rate. The thesis is concerned with steady isothermal flows of such fluds. Generalizing the recent results achieved in the case of homogeneous Dirichlet boundary conditions, the existence and uniqueness of weak solutions subject to the boundary conditions employed in practical applications will be established. The second part is concerned with numerical simulations of the lubrication flow. The experiments indicate that the presented finite element method is successful as long as certain restrictions on the constitutive model are met. Both the restrictions involved in the theoretical results and those indicated by the numerical experiments allow to accurately model real-world lubricants in certain ranges of pressures and shear rates. The last part quantifies those ranges for three representative lubricants. Keywords: existence and uniqueness of weak solutions, finite element method, pressure- thickening, shear-thinning, incompressible fluids,...

Qualitative properties of solutions to equations of fluid mechanics Mgr. Jakub Tichý Supervisor: doc. Mgr. Petr Kaplický, Ph.D. Department: Department of Mathematical Analysis Abstract This thesis is devoted to the boundary regularity of weak solutions to the system of nonlinear partial differential equations describing incompressible flows of a certain class of generalized Newtonian fluids in bounded domains. Equations of motion and continuity equation are complemented with perfect slip boundary conditions. For stationary generalized Stokes system in Rn with growth condi- tion described by N-function Φ the existence of the second derivatives of velocity and their regularity up to the boundary are shown. For the same system of equa- tions integrability of velocity gradients is proven. Lq estimates are obtained also for classical evolutionary Stokes system via interpolation-extrapolation scales. Hölder continuity of velocity gradients and pressure is shown for evolutionary generalized Navier-Stokes equations in R2 . Keywords Generalized Stokes and Navier - Stokes equations, incompressible fluids, perfect slip boundary conditions, regularity up to the boundary