Abstract
In non-woven manufacturing thousands of slender fibers are swirled by air flows before they lay down to form a web. The fiber-fluid interactions have a crucial influence on the quality of the final product. For the purpose of an efficient and fast computation of the multi-scale, two-way coupled interaction problem, we investigate classical homogenization strategies and a new continuum approach for very long fibers suspended in a fluid flow. We compare the results with Direct Numerical Simulation (DNS) and Immersed Boundary Methods for academic examples.
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References
Allaire, G.: Continuity of the Darcy’s law in the low-volume fraction limit. Ann. Scuola Norm. Sup. Pisa 18(4), 475–499 (1991)
Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. Part I + II. Arch. Rational Mech. Anal. 113, 209–298 (1991)
Andric, J.: Numerical modeling of air-fiber flows. Ph.D. thesis, Chalmers University of Technology, Göteborg (2014)
Arne, W., Marheineke, N., Meister, A., Wegener, R.: Numerical analysis of Cosserat rod and string models for viscous jets in rotational spinning processes. Math. Models Methods Appl. Sci. 20(10), 1941–1965 (2010)
Arne, W., Marheineke, N., Schnebele, J., Wegener, R.: Fluid-fiber-interactions in rotational spinning process of glass wool manufacturing. J. Math. Ind. 1(2), 1–26 (2011)
Brinkman, H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1947)
Cibis, T.: Homogenisierungsstrategien für Filament-Strömungs-Wechselwirkungen. Ph.D. thesis, FAU Erlangen-Nürnberg, Erlangen (2015)
Cibis, T., Marheineke, N., Wegener, R.: Asymptotic modeling framework for fiber-flow interactions in a two-way coupling. In: Fontes, M., et al. (eds.) Progress in Industrial Mathematics at ECMI 2012, pp. 109–117. Springer, Heidelberg (2014)
Cioranescu, D., Murat, F.: A strange term coming from nowhere. In: Cherkaev, A., Kohn, R. (eds.) Topics in the Mathematical Modelling of Composite Materials, pp. 45–93. Birkhäuser, Boston (1997)
Drummond, J., Tahir, M.: Laminar viscous flow through regular arrays of parallel solid cylinders. Int. J. Multiphase Flow 10, 515–540 (1983)
Durlofsky, L., Brady, J.: Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30(11), 3329–3341 (1987)
Hämäläinen, J., Lindström, S.B., Hämäläinen, T., Niskanen, H.: Papermaking fibre-suspension flow simulations at multiple scales. J. Eng. Math. 71(1), 55–79 (2011)
Hornung, U.: Homogenization and Porous Media. Springer, New York (1997)
Klar, A., Marheineke, N., Wegener, R.: Hierarchy of mathematical models for production processes of technical textiles. Z. Ang. Math. Mech. 89(12), 941–961 (2009)
Koch, D., Ladd, A.: Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J. Fluid Mech. 349, 31–66 (1997)
Marheineke, N., Wegener, R.: Modeling and application of a stochastic drag for fibers in turbulent flows. Int. J. Multiphase Flow 37, 136–148 (2011)
Marheineke, N., Liljo, J., Mohring, J., Schnebele, J., Wegener, R.: Multiphysics and multimethods problem of rotational glass fiber melt-spinning. Int. J. Num. Anal. Mod. B 3(3), 330–344 (2012)
Mark, A.: A novel immersed-boundary method for multiple moving and interacting bodies. Ph.D. thesis, Chalmers University of Technology, Göteborg (2007)
Nield, D.: The Beavers-Joseph boundary condition and related matters: a historical and critical note. Transp. Porous Media 78(3), 537–540 (2009)
Nield, D., Bejan, A.: Convection in Porous Media. Springer, New York (1992)
Peskin, C.: The immersed boundary method. Acta Numer. 11, 1–39 (2002)
Rubin, M.: Cosserat Theories: Shells, Rods and Points. Solid Mechanics and Its Applications. Springer, Dordrecht (2000)
Sangani, A., Acrivos, A.: Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiphase Flow 8, 193–206 (1982)
Svenning, E., Mark, A., Edelvik, F., Glatt, E., Rief, S., Wiegmann, A., Martinsson, L., Lai, R., Fredlund, M., Nyman, U.: Multiphase simulation of fiber suspension flows using immersed boundary methods. Nordic Pulp Paper Res. J. 27(2), 184–191 (2012)
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This work has been supported by German BMBF, 05M2010 and 05M2013.
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Cibis, T.M., Leithäuser, C., Marheineke, N., Wegener, R. (2016). Homogenization Strategies for Fiber Curtains and Bundles in Air Flows. In: Russo, G., Capasso, V., Nicosia, G., Romano, V. (eds) Progress in Industrial Mathematics at ECMI 2014. ECMI 2014. Mathematics in Industry(), vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-23413-7_136
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DOI: https://doi.org/10.1007/978-3-319-23413-7_136
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