Kinetic and Related Models (KRM)

A smooth model for fiber lay-down processes and its diffusion approximations

Pages: 489 - 502, Volume 2, Issue 3, September 2009      doi:10.3934/krm.2009.2.489

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Michael Herty - Fachbereich Mathematik, RWTH Aachen University, Templergraben 55, D-52074 Aachen, Germany (email)
Axel Klar - Fachbereich Mathematik, Technische Universität Kaiserslautern, PO Box 3049, D-67653 Kaiserslautern, Germany (email)
Sébastien Motsch - Institut Mathématique de Toulouse IMT, Université Paul Sabatier deToulouse, 118, route de Narbonne. F-31062 TOULOUSE Cedex, France (email)
Ferdinand Olawsky - Fraunhofer ITWM, Fraunhofer-Platz 1, D-67663 Kaiserslautern, Germany (email)

Abstract: In this paper we improve and investigate a stochastic model and its associated Fokker-Planck equation for the lay-down of fibers on a conveyor belt in the production process of nonwoven materials which has been developed in [2]. The model is based on a stochastic differential equation taking into account the motion of the fiber under the influence of turbulence. In the present paper we remove an obvious drawback of the model, namely the non-differentiability of the paths of the process. We develop a model with smoother trajectories and investigate the relations between the different models looking at different scalings and diffusion approximations. Moreover, we compare the numerical results to simulations of the full physical process.

Keywords:  fiber dynamics, Fokker-Planck equations, diffusion approximation.
Mathematics Subject Classification:  Primary: 35F10, Secondary: 65Mxx.

Received: March 2009;      Revised: May 2009;      Available Online: July 2009.