Abstract
A simplified Fokker-Planck model for the lay-down of fibers on a conveyor belt in the production process of nonwovens is investigated. It takes into account the motion of the fiber under the influence of turbulence. The emphasis in this paper is on the development of a numerical procedure to solve the model. We present a semi-Lagrangian scheme that accurately captures the fiber dynamics and conserves the mass. The scheme allows large time steps to be taken in numerical simulations and requires moderate computing times to obtain steady state solutions. Numerical results and examples are presented and compared for several selection of fiber parameters. The obtained results show that the semi-Lagrangian method is able to reproduce accurately the time development of functionals of the process that are important for the quality assessment of industrial fibers.
Similar content being viewed by others
References
Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Equ. 26(1–2), 43–100 (2001)
Bonilla, L., Götz, T., Klar, A., Marheineke, N., Wegener, R.: Hydrodynamic limit for the Fokker-Planck equation of fiber lay-down models. Preprint
Dawson, C.N.: Godunov-mixed methods for advective flow problems in one space dimension. SIAM J. Numer. Anal. 28, 1282–1309 (1991)
Desvilettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Commun. Pure Appl. Math. 54, 1–42 (2001)
Douglas, J., Russell, T.F.: Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite elements or finite differences. SIAM J. Numer. Anal. 19, 871–885 (1982)
Douglas, J., Huang, C., Pereira, F.: The modified method of characteristics with adjusted advection. Numer. Math. 83, 353–369 (1999)
Götz, T., Klar, A., Marheineke, N., Wegener, R.: A stochastic model and associated Fokker-Planck equation for the fiber lay-down process in nonwoven production processes. SIAM Appl. Math. 67(6), 1704–1717 (2007)
Grothaus, M., Klar, A.: Ergodicity and rate of convergence for a non-sectorial fiber lay-down process. SIAM Math. Anal. (2007, to appear)
Hearle, J.W., Sultan, M.A., Govender, S.: The form taken by threads laid on a moving belt, Part I–III. J. Text. Inst. 67, 373–386 (1976)
Marheineke, N., Wegener, R.: Fiber dynamics in turbulent flows: General modeling framework. SIAM J. Appl. Math. 66(5), 1703–1726 (2006)
Marheineke, N., Wegener, R.: Fiber dynamics in turbulent flows: Specific Taylor drag. SIAM J. Appl. Math. 68(1), 1–23 (2007)
Robert, A.: A stable numerical integration scheme for the primitive meteorological equations. Atmos. Ocean 19, 35–46 (1981)
Seaïd, M.: On the quasi-monotone modified method of characteristics for transport-diffusion problems with reactive sources. Comput. Methods Appl. Math. 2, 186–210 (2002)
Seaïd, M.: Semi-Lagrangian integration schemes for viscous incompressible flows. Comput. Methods Appl. Math. 4, 1–18 (2002)
Shu, C., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Klar, A., Reuterswärd, P. & Seaïd, M. A Semi-Lagrangian Method for a Fokker-Planck Equation Describing Fiber Dynamics. J Sci Comput 38, 349–367 (2009). https://doi.org/10.1007/s10915-008-9244-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-008-9244-2