Lp-theory for a class of viscoelastic fluids with and without a free surface.
Technische Universität, Darmstadt
[Ph.D. Thesis], (2012)
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|Item Type:||Ph.D. Thesis|
|Title:||Lp-theory for a class of viscoelastic fluids with and without a free surface|
This thesis deals with systems of nonlinear partial differential equations, which describe the motion of a certain class of non-Newtonian fluids. More precisely, the considered fluids are generalized Newtonian fluids as well as generalized viscoelastic fluids (a generalization of the Oldroyd-B fluid). By completing these models with appropriate initial and boundary conditions, we end up with a nonlinear system of partial differential equations. We investigate these on existence and uniqueness of strong Lp-solutions.
Firstly, the generalized viscoelastic fluid model is analyzed on a fixed domain with "no-slip" as well as "perfect-slip" boundary conditions. For bounded domains, local existence of a unique solution is shown. Under the additional assumption, that the viscosity is constant, this result is transferred to a large class of unbounded domains in the case of "no-slip" boundary conditions as well as on the half-space in case of "perfect-slip" boundary conditions.
Secondly, a two-phase problem with surface tension is investigated, where both phases consist of generalized Newtonian fluids. At the initial configuration, it is assumed that both fluids are separated by a hypersurface, which is given as a graph of a height function. We prove the existence and uniqueness of a strong solution on any finite time interval, provided the initial values are sufficiently small. It is shown, that the hypersurface, which separates the fluids, is given as the graph of a height function for all considered times.
Finally, we return to the generalized viscoelastic fluids. Neglecting the effect of surface tension, a corresponding one-phase problem in Lagrangian coordinates is analyzed. It is assumed that the boundary of the initial domain is compact. Local existence and uniqueness of strong solution of the problem in the Lagrangian formulation is proven.
|Place of Publication:||Darmstadt|
|Classification DDC:||500 Naturwissenschaften und Mathematik > 510 Mathematik|
|Divisions:||04 Department of Mathematics
04 Department of Mathematics > Analysis
|Date Deposited:||23 Aug 2012 07:15|
|Last Modified:||07 Dec 2012 12:05|
|Referees:||Geißert, PD Dr. Matthias and Saal, Prof. Dr. Jürgen and Shibata, Prof. Dr. Yoshihiro|
|Refereed:||16 April 2012|