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.

show all references

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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