A Stochastic Model and Associated Fokker-Planck Equation for the Fiber Lay-Down Process in Nonwoven Production Processes

In this paper we present 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. The model is based on a stochastic differential equation taking into account the motion of the fiber under the influence of turbulence. A reformulation as a stochastic Hamiltonian system and an application of the stochastic averaging theorem lead to further simplifications of the model. Finally, the model is used to compute the distribution of functionals of the process that are important for the quality assessment of industrial fabrics.

[1]  R. Khas'minskii,et al.  The behavior of a conservative system under the action of slight friction and slight random noise , 1964 .

[2]  J. Weidmann.  Linear Operators in Hilbert Spaces , 1980 .

[3]  Axel Klar,et al.  A Semi-Lagrangian Method for a Fokker-Planck Equation Describing Fiber Dynamics , 2009, J. Sci. Comput..

[4]  Lakshminarayanan Mahadevan,et al.  Coiling of flexible ropes , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[5]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[6]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[7]  B. N. Cox,et al.  Handbook of Analytical Methods for Textile Composites , 1997 .

[8]  Sergio Albeverio,et al.  Longtime Behaviour of Stochastic Hamiltonian Systems: The Multidimensional Case , 2000 .

[9]  Nicole Marheineke,et al.  Fiber Dynamics in Turbulent Flows: Specific Taylor Drag , 2007, SIAM J. Appl. Math..

[10]  Nicole Marheineke,et al.  Modeling and Numerical Simulation of Fiber Dynamics , 2005 .

[11]  Sergio Albeverio,et al.  Long time behavior of nonlinear stochastic oscillators: The one-dimensional Hamiltonian case , 1994 .

[12]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[13]  Y. Kutoyants.  Statistical Inference for Ergodic Diffusion Processes , 2004 .

[14]  Michael Scholz,et al.  Analysis and Simulation of Nonwoven Textures , 1999 .

[15]  C. Villani,et al.  On the trend to global equilibrium in spatially inhomogeneous entropy‐dissipating systems: The linear Fokker‐Planck equation , 2001 .

[16]  Nicole Marheineke,et al.  Fiber Dynamics in Turbulent Flows: General Modeling Framework , 2006, SIAM J. Appl. Math..

[17]  Iosif Ilitch Gikhman,et al.  Introduction to the theory of random processes , 1969 .

[18]  T. M. Damiani,et al.  Micromechanics of fabric reinforced composites with periodic microstructure , 2005 .

[19]  J. Koko.  Modeling of textile composites with warp/weft frictional contact , 2000 .

[20]  Axel Klar,et al.  Ergodicity and Rate of Convergence for a Nonsectorial Fiber Lay-Down Process , 2008, SIAM J. Math. Anal..