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978-3-8439-1001-9, Reihe Mathematik

Maike Lorenz On a Viscoelastic Fibre Model - Asymptotics and Numerics

152 Seiten, Dissertation Technische Universität Kaiserslautern (2013), Softcover, A5

In this thesis we asymptotically derive a dynamic, viscoelastic, one-dimensional so called string model for a curved fibre. Fibres can be found in a variety of products, they are for example parts of diapers, filter or insulation material. Initially, we consider a three-dimensional flow problem with an upper convected Maxwell model for an isothermal incompressible fluid. The reduction to a one-dimensional model proceeds as a strict asymptotic expansion in the slenderness parameter given by the ratio of nozzle diameter and typical fibre length. The strict asymptotic derivation is performed without imposing prior assumptions on the velocity profile, the shape, symmetries or the stress tensor.

The three parts of the thesis address the derivation of the model, the analysis of steady states and the numerical simulation of the dynamic problem for inflow-outflow processes. In the first part we derive using a strict asymptotic expansion in the slenderness parameter the leading order systems from the three-dimensional flow problem. The system is close by cross-sectionally averaging the balance laws. The resulting model is mixed elliptic-hyperbolic, it includes inertia and outer forces but neglects temperature effects and surface tension. It is applicable for curved fibres and includes the special cases known from literature of a viscous or uniaxial fibre.

In the second part the steady states are studied. For both the uniaxial and the curved stationary case we determine the regime of existence of solutions. Besides a study on the effect of the die swell we also investigate the influence of the occurring dimensionless parameters on the fibre.

Finally, the third part deals with simulations of time-dependent inflow-outflow problems. The underlying equations are a set of mixed elliptic-hyperbolic equations coupled with and ordinary differential equation for the arc-length parametrisation of the fibre. For numerical solution of the simplified uniaxial system we use explicit schemes derived for Hamilton-Jacobi equations. For the curved fibre we employ an implicit scheme also used for Hamilton-Jacobi equations. Finally, the influence of the dimensionless parameters is also studied for the time-dependent case.