American Institute of Mathematical Sciences

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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

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 & 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).   Google Scholar

[2]

L. Arnold, "Stochastic Differential Equations,", Springer, (1978).   Google Scholar

[3]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,", Studies in Mathematics and its Applications, 5 (1978).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Dolbeault, A. Klar, C. Mouhot and C. Schmeiser, Hypocoercivity and a Fokker-Planck equation for fiber lay-down,, preprint., ().   Google Scholar

[8]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms,, C. R. Acad. Sci. Paris, 347 (2009), 511.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[15]

Y. Kutoyantz, "Statistical Inference for Ergodic Diffusion Processes,", Springer, (2004).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Nelson, "Dynamical Theories of Brownian Motion,", Princeton University Press, (1967).   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

W. Albrecht, H. Fuchs and W. Kittelmann, "Nonwoven Fabrics,", Wiley, (2003).   Google Scholar

[2]

L. Arnold, "Stochastic Differential Equations,", Springer, (1978).   Google Scholar

[3]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,", Studies in Mathematics and its Applications, 5 (1978).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Dolbeault, A. Klar, C. Mouhot and C. Schmeiser, Hypocoercivity and a Fokker-Planck equation for fiber lay-down,, preprint., ().   Google Scholar

[8]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms,, C. R. Acad. Sci. Paris, 347 (2009), 511.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[15]

Y. Kutoyantz, "Statistical Inference for Ergodic Diffusion Processes,", Springer, (2004).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Nelson, "Dynamical Theories of Brownian Motion,", Princeton University Press, (1967).   Google Scholar

[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.  Google Scholar

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