A smooth 3D model for fiber lay-down in nonwoven production processes

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March 2012, 5(1): 97-112. doi: 10.3934/krm.2012.5.97

A smooth 3D model for fiber lay-down in nonwoven production processes

Axel Klar ^1, , Johannes Maringer ^1, and Raimund Wegener ^2,

1.	Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany, Germany
2.	Fraunhofer ITWM, Kaiserslautern, Germany

Received March 2011 Revised August 2011 Published January 2012

Abstract
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In this paper we develop an improved three dimensional stochastic model for the lay-down of fibers on a moving conveyor belt in the production process of nonwoven materials. The model removes a drawback of a previous 3D model, that is the non-smoothness of the fiber paths. A similar result in the 2D case has been presented in [12]. The resulting equations are investigated for different limit situations and numerical simulations are presented.

Keywords: diffusion approximation., Fokker-Planck equations, stochastic processes, Fiber dynamics.

Mathematics Subject Classification: 37H10, 35Q84, 41A6.

Citation: Axel Klar, Johannes Maringer, Raimund Wegener. A smooth 3D model for fiber lay-down in nonwoven production processes. Kinetic and Related Models, 2012, 5 (1) : 97-112. doi: 10.3934/krm.2012.5.97

References:

[1]	W. Albrecht, H. Fuchs and W. Kittelmann, "Nonwoven Fabrics," Wiley, 2003.
[2]	L. Arnold, "Stochastic Differential Equations," Springer, 1978.
[3]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.
[4]	L. Bonilla and T. Götz, A. Klar, N. Marheineke and R. Wegener, Hydrodynamic limit of a Fokker-Planck equation describing fiber lay-down processes, SIAM J. Appl. Math., 68 (2007/08), 648-665. doi: 10.1137/070692728.
[5]	J.-A. Carrillo, T. Goudon and P. Lafitte, Simulation of fluid and particles flows: Asymptotic preserving schemes for bubbling and flowing regimes, J. Comp. Phys., 227 (2008), 7929-7951. doi: 10.1016/j.jcp.2008.05.002.
[6]	P. Degond and S. Motsch, Large-scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021. doi: 10.1007/s10955-008-9529-8.
[7]	J. Dolbeault, A. Klar, C. Mouhot and C. Schmeiser, Hypocoercivity and a Fokker-Planck equation for fiber lay-down, preprint.
[8]	J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Acad. Sci. Paris, 347 (2009), 511-516.
[9]	T. Götz, A. Klar, N. Marheineke and R. Wegener, A stochastic model and associated Fokker-Planck equation for the fiber lay-down process in nonwoven production processes, SIAM J. Appl. Math., 67 (2007), 1704-1717. doi: 10.1137/06067715X.
[10]	T. Goudon, Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci., 15 (2005), 737-752. doi: 10.1142/S021820250500056X.
[11]	J. W. Hearle, M. A. Sultan and S. Govender, The form taken by threads laid on a moving belt, Part I-III, Journal of the Textile Institute, 67 (1976), 373-386. doi: 10.1080/00405007608630170.
[12]	M. Herty, A. Klar, S. Motsch and F. Olawsky, A smooth model for fiber lay-down processes and its diffusion approximations, KRM, 2 (2009), 489-502. doi: 10.3934/krm.2009.2.489.
[13]	A. Klar, N. Marheineke and R. Wegener, Hierarchy of mathematical models for production processes of technical textiles, ZAMM Z. Angew. Math. Mech., 89 (2009), 941-961. doi: 10.1002/zamm.200900282.
[14]	A. Klar, J. Maringer and R. Wegener, A 3D model for fiber lay-down processes in non-woven production processes, to appear in MMMAS.
[15]	Y. Kutoyantz, "Statistical Inference for Ergodic Diffusion Processes," Springer, 2004.
[16]	L. Mahadevan and J. B. Keller, Coiling of flexible ropes, Proc. R. Soc. Lond. Ser. A, 452 (1996), 1679-1694. doi: 10.1098/rspa.1996.0089.
[17]	N. Marheineke and R. Wegener, Fiber dynamics in turbulent flows: General modeling framework, SIAM J. Appl. Math., 66 (2006), 1703-1726. doi: 10.1137/050637182.
[18]	N. Marheineke and R. Wegener, Modeling and application of a stochastic drag for fibers in turbulent flows, International Journal of Multiphase Flow, 37 (2011), 136-148. doi: 10.1016/j.ijmultiphaseflow.2010.10.001.
[19]	E. Nelson, "Dynamical Theories of Brownian Motion," Princeton University Press, Princeton, N.J., 1967.
[20]	M. R. D'Orsogna, V. Panferov and J. A. Carrillo, Double milling in self-propelled swarms from kinetic theory, Kinetic and Related Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363.

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References:

[1]	W. Albrecht, H. Fuchs and W. Kittelmann, "Nonwoven Fabrics," Wiley, 2003.
[2]	L. Arnold, "Stochastic Differential Equations," Springer, 1978.
[3]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.
[4]	L. Bonilla and T. Götz, A. Klar, N. Marheineke and R. Wegener, Hydrodynamic limit of a Fokker-Planck equation describing fiber lay-down processes, SIAM J. Appl. Math., 68 (2007/08), 648-665. doi: 10.1137/070692728.
[5]	J.-A. Carrillo, T. Goudon and P. Lafitte, Simulation of fluid and particles flows: Asymptotic preserving schemes for bubbling and flowing regimes, J. Comp. Phys., 227 (2008), 7929-7951. doi: 10.1016/j.jcp.2008.05.002.
[6]	P. Degond and S. Motsch, Large-scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021. doi: 10.1007/s10955-008-9529-8.
[7]	J. Dolbeault, A. Klar, C. Mouhot and C. Schmeiser, Hypocoercivity and a Fokker-Planck equation for fiber lay-down, preprint.
[8]	J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Acad. Sci. Paris, 347 (2009), 511-516.
[9]	T. Götz, A. Klar, N. Marheineke and R. Wegener, A stochastic model and associated Fokker-Planck equation for the fiber lay-down process in nonwoven production processes, SIAM J. Appl. Math., 67 (2007), 1704-1717. doi: 10.1137/06067715X.
[10]	T. Goudon, Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci., 15 (2005), 737-752. doi: 10.1142/S021820250500056X.
[11]	J. W. Hearle, M. A. Sultan and S. Govender, The form taken by threads laid on a moving belt, Part I-III, Journal of the Textile Institute, 67 (1976), 373-386. doi: 10.1080/00405007608630170.
[12]	M. Herty, A. Klar, S. Motsch and F. Olawsky, A smooth model for fiber lay-down processes and its diffusion approximations, KRM, 2 (2009), 489-502. doi: 10.3934/krm.2009.2.489.
[13]	A. Klar, N. Marheineke and R. Wegener, Hierarchy of mathematical models for production processes of technical textiles, ZAMM Z. Angew. Math. Mech., 89 (2009), 941-961. doi: 10.1002/zamm.200900282.
[14]	A. Klar, J. Maringer and R. Wegener, A 3D model for fiber lay-down processes in non-woven production processes, to appear in MMMAS.
[15]	Y. Kutoyantz, "Statistical Inference for Ergodic Diffusion Processes," Springer, 2004.
[16]	L. Mahadevan and J. B. Keller, Coiling of flexible ropes, Proc. R. Soc. Lond. Ser. A, 452 (1996), 1679-1694. doi: 10.1098/rspa.1996.0089.
[17]	N. Marheineke and R. Wegener, Fiber dynamics in turbulent flows: General modeling framework, SIAM J. Appl. Math., 66 (2006), 1703-1726. doi: 10.1137/050637182.
[18]	N. Marheineke and R. Wegener, Modeling and application of a stochastic drag for fibers in turbulent flows, International Journal of Multiphase Flow, 37 (2011), 136-148. doi: 10.1016/j.ijmultiphaseflow.2010.10.001.
[19]	E. Nelson, "Dynamical Theories of Brownian Motion," Princeton University Press, Princeton, N.J., 1967.
[20]	M. R. D'Orsogna, V. Panferov and J. A. Carrillo, Double milling in self-propelled swarms from kinetic theory, Kinetic and Related Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363.

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