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978-3-8439-0782-8, Reihe Mathematik

Dario Götz Three topics in fluid dynamics: Viscoelastic, generalized Newtonian, and compressible fluids

202 Seiten, Dissertation Technische Universität Darmstadt (2012), Softcover, B5

This thesis is concerned with the mathematical analysis of systems of partial differential equations describing the motion of fluids in various situations. The existence and uniqueness of strong solutions is investigated based upon the maximal regularity approach for parabolic equations and fixed point arguments.

Firstly, a general model for incompressible, viscoelastic flows in fixed domains is studied under both no-slip and pure slip boundary conditions. The class of admissible domains for the no-slip case is very general and includes the whole space, half-spaces, bounded domains, exterior domains, aperture domains, and more general unbounded domains. For the pure slip conditions, the half-space and bounded domains are considered.

The second part deals with a mathematical model describing the spin-coating process for incompressible, generalized Newtonian fluids which takes the form of a one-phase free boundary problem with surface tension under the influence of Coriolis forces and centrifugal forces. The considered fluid model allows for a shear-rate dependent viscosity, e.g. power-law fluids.

Finally, a linearization of the Navier-Stokes model for barotropic, compressible fluids in a free boundary situation without surface tension is investigated. In this situation, for the linear system of interest, inhomogeneous Neumann boundary conditions are prescribed. A maximal regularity result is shown for flows in the half-space.