Collapsed heteroclinic snaking near a heteroclinic chain in dragged meniscus problems

  • D. Tseluiko
  • M. GalvagnoEmail author
  • U. Thiele
Regular Article
Part of the following topical collections:
  1. Irreversible Dynamics: A topical issue dedicated to Paul Manneville


A liquid film is studied that is deposited onto a flat plate that is inclined at a constant angle to the horizontal and is extracted from a liquid bath at a constant speed. We analyse steady-state solutions of a long-wave evolution equation for the film thickness. Using centre manifold theory, we first obtain an asymptotic expansion of solutions in the bath region. The presence of an additional temperature gradient along the plate that induces a Marangoni shear stress significantly changes these expansions and leads to the presence of logarithmic terms that are absent otherwise. Next, we numerically obtain steady solutions and analyse their behaviour as the plate velocity is changed. We observe that the bifurcation curve exhibits collapsed (or exponential) heteroclinic snaking when the plate inclination angle is above a certain critical value. Otherwise, the bifurcation curve is monotonic. The steady profiles along these curves are characterised by a foot-like structure that is formed close to the meniscus and is preceded by a thin precursor film further up the plate. The length of the foot increases along the bifurcation curve. Finally, we prove with a Shilnikov-type method that the snaking behaviour of the bifurcation curves is caused by the existence of an infinite number of heteroclinic orbits close to a heteroclinic chain that connects in an appropriate three-dimensional phase space the fixed point corresponding to the precursor film with the fixed point corresponding to the foot and then with the fixed point corresponding to the bath.

Graphical abstract


Topical issue: Irreversible Dynamics: A topical issue dedicated to Paul Manneville 


  1. 1.
    S.J. Weinstein, K.J. Ruschak, Annu. Rev. Fluid Mech. 36, 29 (2004).ADSCrossRefGoogle Scholar
  2. 2.
    F.C. Morey, J. Res. Nat. Bur. Stand. 25, 385 (1940).CrossRefGoogle Scholar
  3. 3.
    J.J. Rossum, Appl. Sci. Res. Sect. A 7, 121 (1958).Google Scholar
  4. 4.
    R.P. Spiers, C.V. Subbaraman, W.L. Wilkinson, Chem. Eng. Sci. 29, 389 (1974).CrossRefGoogle Scholar
  5. 5.
    J.H. Snoeijer, B. Andreotti, G. Delon, M. Fermigier, J. Fluid Mech. 579, 63 (2007).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    G. Delon, M. Fermigier, J.H. Snoeijer, B. Andreotti, J. Fluid Mech. 604, 55 (2008).ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    M. Maleki, M. Reyssat, F. Restagno, D. Quéré, C. Clanet, J. Colloid Interface Sci. 354, 359 (2011).CrossRefGoogle Scholar
  8. 8.
    L. Landau, B. Levich, Acta Physicochim. U.R.S.S. 17, (1942) reprint in Pelc88.Google Scholar
  9. 9.
    P. Groenveld, Chem. Eng. Sci. 25, 1259 (1970).CrossRefGoogle Scholar
  10. 10.
    P. Groenveld, Chem. Eng. Sci. 25, 1579 (1970).CrossRefGoogle Scholar
  11. 11.
    S.D.R. Wilson, J. Eng. Math. 16, 209 (1981).CrossRefGoogle Scholar
  12. 12.
    J. Ziegler, J.H. Snoeijer, J. Eggers, Eur. Phys. J. ST 166, 177 (2009).CrossRefGoogle Scholar
  13. 13.
    E.S. Benilov, S.J. Chapman, J.B. McLeod, J.R. Ockendon, V.S. Zubkov, J. Fluid Mech. 663, 53 (2010).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    B. Jin, A. Acrivos, A. Münch, Phys. Fluids 17, 103603 (2005).ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    J.H. Snoeijer, J. Ziegler, B. Andreotti, M. Fermigier, J. Eggers, Phys. Rev. Lett. 100, 244502 (2008).ADSCrossRefGoogle Scholar
  16. 16.
    A. Münch, P.L. Evans, Physica D 209, 164 (2005).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    H. Riegler, K. Spratte, Thin Solid Films 210, 9 (1992).ADSCrossRefGoogle Scholar
  18. 18.
    M.H. Köpf, S.V. Gurevich, R. Friedrich, L.F. Chi, Langmuir 26, 10444 (2010).CrossRefGoogle Scholar
  19. 19.
    M.H. Köpf, S.V. Gurevich, R. Friedrich, U. Thiele, New J. Phys. 14, 023016 (2012).CrossRefGoogle Scholar
  20. 20.
    U. Thiele, Adv. Colloid Interface Sci. 206, 399 (2014).CrossRefGoogle Scholar
  21. 21.
    L.P. Shilnikov, Sov. Math. Dokl. 6, 163 (1965).Google Scholar
  22. 22.
    P. Glendinning, C. Sparrow, J. Stat. Phys. 35, 645 (1984).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    J. Knobloch, T. Wagenknecht, Physica D 206, 82 (2005).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Y.P. Ma, J. Burke, E. Knobloch, Physica D 239, 1867 (2010).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    L.P. Shilnikov, Sov. Math. Dokl. 8, 5458 (1967).Google Scholar
  26. 26.
    J. Knobloch, T. Wagenknecht, SIAM J. Appl. Dyn. Syst. 7, 1397 (2008).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    M. Chen, Appl. Anal. 75, 213 (2000).CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    G.W. Hunt, M.A. Peletier, A.R. Champneys, P.D. Woods, M. Ahmer Wadee, C.J. Budd, G.J. Lord, Nonlinear Dyn. 21, 3 (2000).CrossRefzbMATHGoogle Scholar
  29. 29.
    A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys. 69, 931 (1997).ADSCrossRefGoogle Scholar
  30. 30.
    U. Thiele, Structure formation in thin liquid films. In S. Kalliadasis, U. Thiele (Editors), Thin Films of Soft Matter, (Springer, Wien, 2007) pp. 25--93.Google Scholar
  31. 31.
    P.-G. de Gennes, Rev. Mod. Phys. 57, 827 (1985).ADSCrossRefGoogle Scholar
  32. 32.
    V.M. Starov, M.G. Velarde, J. Phys.: Condens. Matter 21, 464121 (2009).ADSGoogle Scholar
  33. 33.
    U. Thiele, J. Phys.: Condens. Matter 22, 084019 (2010).ADSGoogle Scholar
  34. 34.
    A.M. Cazabat, F. Heslot, S.M. Troian, P. Carles, Nature 346, 824 (1990).ADSCrossRefGoogle Scholar
  35. 35.
    B. Scheid, E.A. van Nierop, H.A. Stone, Appl. Phys. Lett. 97, 171906 (2010).ADSCrossRefGoogle Scholar
  36. 36.
    B. Scheid, E.A. van Nierop, H.A. Stone, Phys. Fluids 24, 032107 (2012).ADSCrossRefGoogle Scholar
  37. 37.
    J. Carr, Applications of Centre Manifold Theory, Vol. 35 in Applied Mathematical Sciences (Springer-Verlag, Berlin, 1981).Google Scholar
  38. 38.
    I.U.A. Kuznetsov, Elements of Applied Bifurcation Theory, Vol. 112 in Applied Mathematical Sciences (Springer, New York, 1998).Google Scholar
  39. 39.
    E. Doedel, H.B. Keller, J.P. Kernevez, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 493 (1991).CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    E. Doedel, H.B. Keller, J.P. Kernevez, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 745 (1991).CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    H.A. Dijkstra, F.W. Wubs, A.K. Cliffe, E. Doedel, I.F. Dragomirescu, B. Eckhart, A.Y. Gelfgat, A. Hazel, V. Lucarini, A.G. Salinger, E.T. Phipps, J. Sanchez-Umbria, H. Schuttelaars, L.S. Tuckerman, U. Thiele, Commun. Comput. Phys. 15, 1 (2014).MathSciNetGoogle Scholar
  42. 42.
    U. Thiele, L. Brusch, M. Bestehorn, M. Bär, Eur. Phys. J. E 11, 255 (2003).CrossRefGoogle Scholar
  43. 43.
    P. Beltrame, U. Thiele, SIAM J. Appl. Dyn. Syst. 9, 484 (2010).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    D. Tseluiko, J. Baxter, U. Thiele, IMA J. Appl. Math. 78, 762 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    A.L. Bertozzi, A. Münch, X. Fanton, A.M. Cazabat, Phys. Rev. Lett. 81, 5169 (1998).ADSCrossRefGoogle Scholar
  46. 46.
    A.O. Parry, C. Rascon, E.A.G. Jamie, D.G.A.L. Aarts, Phys. Rev. Lett. 108, 246101 (2012).ADSCrossRefGoogle Scholar
  47. 47.
    M. Galvagno, D. Tseluiko, H. Lopez, U. Thiele, Phys. Rev. Lett. 112, 137803 (2014).ADSCrossRefGoogle Scholar
  48. 48.
    P. Pelce (Editor), Dynamics of curved fronts, 1st edition (Academic Press, London, 1988).Google Scholar
  49. 49.
    R.C. Robinson, An introduction to dynamical systems: continuous and discrete (Pearson Prentice Hall, Upper Saddle River, NJ, 2004).Google Scholar

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© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLoughborough UniversityLeicestershireUK
  2. 2.Institut für Theoretische PhysikWestfälische Wilhelms-Universität MünsterMünsterGermany

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