Advertisement

Computational and Applied Mathematics

, Volume 36, Issue 2, pp 903–913 | Cite as

Well-posed initial conditions and numerical methods for one-dimensional models of liquid dynamics in a horizontal capillary

Article

Abstract

In this paper, we undertake a mathematical and numerical study of liquid dynamics models in a horizontal capillary. In particular, we prove that the classical model is ill-posed at initial time, and we recall two different approaches in order to define a well-posed problem. Moreover, for an academic test case, we compare the numerical approximations, obtained by an adaptive initial value problem solver based on an one-step one-method approach, with new asymptotic solutions. This is a possible way to validate the adaptive numerical approach for its application to real liquids.

Keywords

Well-posed initial conditions Horizontal capillary Asymptotic solutions Adaptive numerical method 

Mathematics Subject Classification

34A12 65L05 76D45 

Notes

Acknowledgments

The first author gratefully thanks Chris Budd for fruitful discussions concerning the asymptotic solutions while he was visiting the Department of Mathematics, University of Bath, United Kingdom. This work was supported by the University of Messina and by the GNCS (Computational Science National Group) of the INDAM (Italian Institute of High Mathematics).

References

  1. Bosanquet CH (1923) On the flow of liquids into capillary tubes. Philos Mag Ser 6 45:525–531CrossRefGoogle Scholar
  2. Budd C, Huang H (2008) Private communication, BathGoogle Scholar
  3. Butcher JC (2003) Numerical methods for ordinary differential equations. Whiley, ChichesterCrossRefMATHGoogle Scholar
  4. Caturano G, Cavaccini G, Ciliberto A, Pianese V, Fazio R (2009) Liquid penetrant testing: industrial process. Commun SIMAI Congr 3:319.1–319.12. doi: 10.1685/CSC09319 MATHGoogle Scholar
  5. Cavaccini G, Pianese V, Iacono S, Jannelli A, Fazio R (2009) One-dimensional mathematical and numerical modeling of liquid dynamics in a horizontal capillary. J Comput Method Sci Eng 9:3–16MathSciNetMATHGoogle Scholar
  6. Chibbaro S (2008) Capillary filling with pseudo-potential binary lattice-Boltzmann model. Eur Phys J E 27:99–106CrossRefGoogle Scholar
  7. Clanet C, Quéré D (2002) Onset of menisci. J Fluid Mech 460:131–149MathSciNetCrossRefMATHGoogle Scholar
  8. de Gennes PG (1985) Wetting: statics and dynamics. Rev Mod Phys 57:827–890CrossRefGoogle Scholar
  9. de Gennes PG, Brochard-Wyart F, Quéré D (2004) Capillarity and wetting phenomena. Springer, New YorkCrossRefMATHGoogle Scholar
  10. Diotallevi F, Biferale L, Chibbaro S, Puglisi A, Succi S (2008) Front pinning in capillary filling of chemically coated channels. Phys Rev E 78:036305CrossRefGoogle Scholar
  11. Dussan EB (1979) On the spreading of liquids on solid surfaces: static and dynamic contact angles. Ann Rev Fluid Mech 11:371–400CrossRefGoogle Scholar
  12. Fazio R, Iacono S (2014) An analytical and numerical study of liquid dynamics in a one-dimensional capillary under entrapped gas action. Math Method Appl Sci 37(18):2923–2933MathSciNetCrossRefMATHGoogle Scholar
  13. Fazio R, Jannelli A (2010) Ill and well-posed one-dimensional models of liquid dynamics in a horizontal capillary. In: De Bernardis E, Spligher R, Valenti V (eds) Applied and industrial mathematics in Italy III. World Scientific, Singapore, pp 353–364Google Scholar
  14. Fazio R, Iacono S, Jannelli A, Cavaccini G, Pianese V (2007) A two immiscible liquids penetration model for surface-driven capillary flows. Proc Appl Math Mech 7:2150003–2150004CrossRefGoogle Scholar
  15. Fazio R, Iacono S, Jannelli A, Cavaccini G, Pianese V (2012) Extended scaling invariance of one-dimensional models of liquid dynamics in a horizontal capillary. Math Method Appl Sci 35:935–942MathSciNetCrossRefMATHGoogle Scholar
  16. Fisher LR, Lark PD (1979) An experimental study of the Washburn equation for liquid flow in very fine capillaries. J Colloid Interf Sci 69:486–492CrossRefGoogle Scholar
  17. Jannelli A, Fazio R (2006) Adaptive stiff solvers at low accuracy and complexity. J Comput Appl Math 191:246–258MathSciNetCrossRefMATHGoogle Scholar
  18. Kornev KG, Neimark AV (2001) Spontaneous penetration of liquids into capillaries and porous membranes revisited. J Colloid Interf Sci 235:101–113CrossRefGoogle Scholar
  19. Leger L, Joanny JF (1992) Liquid spreading. Rep Prog Phys 55:431–486CrossRefGoogle Scholar
  20. Liu C, Zhu S (2015) Laguerre pseudospectral approximation to the Thomas–Fermi equation. J Comput Appl Math 282:161–251MathSciNetCrossRefMATHGoogle Scholar
  21. Mann JA Jr, Romero L, Rye RR, Yost FG (1995) Flow of simple liquids down narrow V grooves. Phys Rev E 52:3967–3972CrossRefGoogle Scholar
  22. Martic G, Gentner F, Seveno D, Coulon D, De Coninck J, Blake TD (2002) A molecular dynamics simulation of capillary imbibition. Langmuir 18:7971–7976CrossRefGoogle Scholar
  23. Martic G, Gentner F, Seveno D, De Coninck J, Blake TD (2004) The possibility of different time scales in the dynamics of pore imbibition. J Colloid Interf Sci 270:171–179CrossRefGoogle Scholar
  24. Martic G, Blake TD, De Coninck J (2005) Dynamics of imbibition into a pore with a heterogeneous surface. Langmuir 21:11201–11207CrossRefGoogle Scholar
  25. Romero LA, Yost FG (1996) Flow in an open channel capillary. J Fluid Mech 322:109–129CrossRefMATHGoogle Scholar
  26. Rye RR, Yost FG, Mann JA Jr (1996) Wetting kinetics in surface capillary grooves. Langmuir 12:4625–4627CrossRefGoogle Scholar
  27. Rye RR, Yost FG, O’Toole EJ (1998) Capillary flow in irregular surface grooves. Langmuir 14:3937–3943CrossRefGoogle Scholar
  28. Szekely J, Neumann AW, Chuang YK (1979) Rate of capillary penetration and applicability of Washburn equation. J Colloid Interf Sci 69:486–492CrossRefGoogle Scholar
  29. Washburn EW (1921) The dynamics of capillary flow. Phys Rev 17:273–283CrossRefGoogle Scholar
  30. Yost FG, Rye RR, Mann JA Jr (1997) Solder wetting kinetics in narrow V-grooves. Acta Mater 45:5337–5345CrossRefGoogle Scholar
  31. Zhmud BV, Tiberg F, Hallstensson K (2000) Dynamics of capillary rise. J Colloid Interf Sci 228:263–269CrossRefGoogle Scholar
  32. Zhu S, Zhu H, Wu Q, Khan Y (2012) An adaptive algorithm for the Thomas–Fermi equation. Numer Algorithms 59(3):359–372MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of MessinaMessinaItaly

Personalised recommendations