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Mathematical model of falling of a viscous jet onto a moving surface

Published online by Cambridge University Press:  01 December 2007

A. HLOD ,
A. C. T. AARTS ,
A. A. F. van de VEN  and
M. A. PELETIER
A. HLOD
Affiliation:
Center for Analysis, Scientific computing and Applications, Eindhoven University of Technology, Eindhoven, The Netherlands
A. C. T. AARTS
Affiliation:
Center for Analysis, Scientific computing and Applications, Eindhoven University of Technology, Eindhoven, The Netherlands
A. A. F. van de VEN
Affiliation:
Center for Analysis, Scientific computing and Applications, Eindhoven University of Technology, Eindhoven, The Netherlands
M. A. PELETIER
Affiliation:
Center for Analysis, Scientific computing and Applications, Eindhoven University of Technology, Eindhoven, The Netherlands
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Abstract

The stationary flow of a jet of a Newtonian fluid that is drawn by gravity onto a moving surface is analyzed. It is assumed that the jet has a convex shape and hits the moving surface tangentially. The flow is modelled by a third-order ODE on a domain of unknown length and with an additional integral condition. By solving part of the equation explicitly, the problem is reformulated as a first-order ODE with an integral constraint. The corresponding existence region in the three-dimensional parameter space is characterized in terms of an easily calculable quantity. In a qualitative sense, the results from the model are found to correspond with experimental observations.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 6 , December 2007 , pp. 659 - 677
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Abramowitz, M. & Stegun, I. A. (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.Google Scholar
[2]Chiu-Webster, S. & Lister, J. R. (2006) The fall of a viscous thread onto a moving surface: a ‘fluid-mechanical sewing machine’. J. Fluid Mech., 569, 89–111.CrossRefGoogle Scholar
[3]Coddington, E. A. & Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York-Toronto-London.Google Scholar
[4]Cruickshank, J. O. (1980) Viscous Fluid Buckling: A Theoretical and Experimental Analysis with Extensions to General Fluid Stability. PhD thesis, Iowa State University, Ames IA, USA.Google Scholar
[5]Dyson, R. J.Howell, P. D.Breward, C. J. W.Herdman, P. & Brander, J. (2005) Mathematical modelling of curtain coating. In: Proceedings of the 6th European Coating Symposium ECS2005.Google Scholar
[6]Hlod, A., Aarts, A. C. T., van de Ven, A. A. F. & Peletier, M. A. Boundary conditions for the three flow regimes of a viscous jet falling onto a moving surface, in preparation.Google Scholar
[7]Polyanin, A. D., Zaitsev, V. F. & Moussiaux, A. (2002) Handbook of First Order Partial Differential Equations, volume 1 of Differential and Integral Equations and Their Applications. Taylor & Francis, London.Google Scholar
[8]Quarteroni, A., Sacco, R. & Saleri, F. (2000) Numerical Mathematics, volume 37 of Texts in Applied Mathematics. Springer-Verlag, New York.Google Scholar
[9]Ribe, N. M. (2003) Periodic folding of viscous sheets. Phys. Rev. E, 68, 036305.CrossRefGoogle ScholarPubMed
[10]Ribe, N. M. (2004) Coiling of viscous jets. Proc. R. Soc. London A, 460, 3223–3239.Google Scholar
[11]Ribe, N. M., Lister, J. R. & Chiu-Webster, S. (2006) Stability of a dragged viscous thread: Onset of stitching in a fluid-mechanical sewing machine. Phys. Fluids, 18, 124105 (2006) (8 pages).CrossRefGoogle Scholar
[12]Rudin, W. (1976) Principles of Mathematical Analysis. McGraw-Hill, New York, 3rd ed., International Series in Pure and Applied Mathematics.Google Scholar
[13]Skorobogatiy, M. & Mahadevan, L. (2000) Folding of viscous sheets and filaments. Europhys. Lett., 52 (5), 532–538.CrossRefGoogle Scholar
[14]Taylor, G. I. (1969) Instability of jets, threads, and sheets of viscous fluid. In Proceedings of the 12th International Congress of Applied mechnaics (Stanford, 1968), pp. 382–388. Springer-Verlag, New York.Google Scholar
[15]Tchavdarov, B. M., Yarin, A. L. & Radev, S. (1993) Buckling of thin liquid jets. J. Fluid Mech., 253, 593–615.Google Scholar
[16]Yarin, A. L. (1993) Free Liquid Jets and Films: Hydrodynamics and Rheology. Interaction of Mechanics and Mathematics Series. Longman Scientific & Technical, Harlow.Google Scholar
[17]Yarin, A. L. & Tchavdarovz, B. M. (1996) Onset of folding in plane liquid films. J. Fluid Mech., 307, 85–99.Google Scholar