# American Institute of Mathematical Sciences

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

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

Received  March 2011 Revised  August 2011 Published  January 2012

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.
Citation: Axel Klar, Johannes Maringer, Raimund Wegener. A smooth 3D model for fiber lay-down in nonwoven production processes. Kinetic & 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 (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 (): 648. 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. 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. 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. [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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.ijmultiphaseflow.2010.10.001. [19] E. Nelson, "Dynamical Theories of Brownian Motion,", Princeton University Press, (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. 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 (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 (): 648. 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. 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. 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. [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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.ijmultiphaseflow.2010.10.001. [19] E. Nelson, "Dynamical Theories of Brownian Motion,", Princeton University Press, (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. doi: 10.3934/krm.2009.2.363.
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