A mixed formulation of a sharp interface model of stokes flow with moving contact lines

Shawn W. Walker

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 4, page 969-1009
  • ISSN: 0764-583X

Abstract

top
Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis and pinning through a variational inequality. Moreover, the formulation can be extended to include non-linear contact line motion models. We prove the well-posedness of the time semi-discrete system and fully discrete method using appropriate choices of finite element spaces. A formal energy law is derived for the semi-discrete and fully discrete formulations and preliminary error estimates are also given. Simulation results are presented for a droplet in multiple configurations to illustrate the method.

How to cite

top

Walker, Shawn W.. "A mixed formulation of a sharp interface model of stokes flow with moving contact lines." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.4 (2014): 969-1009. <http://eudml.org/doc/273319>.

@article{Walker2014,
abstract = {Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis and pinning through a variational inequality. Moreover, the formulation can be extended to include non-linear contact line motion models. We prove the well-posedness of the time semi-discrete system and fully discrete method using appropriate choices of finite element spaces. A formal energy law is derived for the semi-discrete and fully discrete formulations and preliminary error estimates are also given. Simulation results are presented for a droplet in multiple configurations to illustrate the method.},
author = {Walker, Shawn W.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {mixed method; Stokes equations; surface tension; contact line motion; contact line pinning; variational inequality; well-posedness},
language = {eng},
number = {4},
pages = {969-1009},
publisher = {EDP-Sciences},
title = {A mixed formulation of a sharp interface model of stokes flow with moving contact lines},
url = {http://eudml.org/doc/273319},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Walker, Shawn W.
TI - A mixed formulation of a sharp interface model of stokes flow with moving contact lines
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 4
SP - 969
EP - 1009
AB - Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis and pinning through a variational inequality. Moreover, the formulation can be extended to include non-linear contact line motion models. We prove the well-posedness of the time semi-discrete system and fully discrete method using appropriate choices of finite element spaces. A formal energy law is derived for the semi-discrete and fully discrete formulations and preliminary error estimates are also given. Simulation results are presented for a droplet in multiple configurations to illustrate the method.
LA - eng
KW - mixed method; Stokes equations; surface tension; contact line motion; contact line pinning; variational inequality; well-posedness
UR - http://eudml.org/doc/273319
ER -

References

top
  1. [1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, vol. 140 of Pure Appl. Math. Series, 2nd edn. Elsevier (2003). Zbl1098.46001MR2424078
  2. [2] V.I. Arnold, Lectures on Partial Differential Equations. Springer (2006). Zbl1076.35001MR2031206
  3. [3] J.-P. Aubin, Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by gelerkin’s and finite difference methods. Ann. Scuola Norm. Sup. Pisa21 (1967) 599–637. Zbl0276.65052MR233068
  4. [4] T.A. Baer, R.A. Cairncross, P.R. Schunk, R.R. Rao and P.A. Sackinger, A finite element method for free surface flows of incompressible fluids in three dimensions. Part II. Dynamic wetting lines. Int. J. Numer. Methods Fluids 33 (2000) 405–427. Zbl0989.76044
  5. [5] E. Bänsch, Finite element discretization of the navier-stokes equations with a free capillary surface. Numer. Math.88 (2001) 203–235. Zbl0985.35060MR1826211
  6. [6] E. Bänsch and K. Deckelnick, Optimal error estimates for the stokes and navier-stokes equations with slip-boundary condition. ESAIM: M2AN 33 (1999) 923–938. Zbl0948.76035MR1726716
  7. [7] E. Bänsch and B. Höhn, Numerical treatment of the navier-stokes equations with slip boundary condition. SIAM J. Sci. Comput.21 (2000) 2144–2162. Zbl0970.76056MR1762035
  8. [8] F.B. Belgacem, The Mortar finite element method with Lagrange multipliers. Numer. Math.84 (1999) 173–197. Zbl0944.65114MR1730018
  9. [9] T.D. Blake, The physics of moving wetting lines. J. Colloid Interface Sci.299 (2006) 1–13. 
  10. [10] T.D. Blake and Y.D. Shikhmurzaev, Dynamic wetting by liquids of different viscosity. J. Colloid Interface Sci.253 (2002) 196–202. 
  11. [11] D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd edn. Cambridge University Press (2001). Zbl0976.65099MR1827293
  12. [12] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, New York (2002). Zbl1135.65042MR1894376
  13. [13] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). Zbl0788.73002MR1115205
  14. [14] F. Brezzi, W.W. Hager and P.A. Raviart, Error estimates for the finite element solution of variational inequalities: Part II. Mixed methods. Num. Math. 31 (1978) 1–16. Zbl0427.65077MR508584
  15. [15] C.E. Brown, T.D. Jones and E.L. Neustadter, Interfacial flow during immiscible displacement. J. Colloid Interface Sci.76 (1980) 582–586. 
  16. [16] R. Burridge and J.B. Keller, Peeling, slipping and cracking–some one-dimensional free-boundary problems in mechanics. SIAM Review20 (1978) 31–61. Zbl0394.73093MR464828
  17. [17] C.H.A. Cheng, D. Coutand and S. Shkoller, Navier-stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal.39 (2007) 742–800. Zbl1138.74022MR2349865
  18. [18] S.K. Cho, H. Moon and C.-J. Kim, Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits. J. Microelectromech. Systems12 (2003) 70–80. 
  19. [19] P. Ciarlet, On korns inequality. Chin. Ann. Math. Ser. B31 (2010) 607–618. Zbl1200.49039MR2726058
  20. [20] P. Clément, Approximation by finite element functions using local regularization. R.A.I.R.O. Analyse Numérique9 (1975) 77–84. Zbl0368.65008
  21. [21] P.-P. Cortet, M. Ciccotti and L. Vanel, Imaging the stickslip peeling of an adhesive tape under a constant load. J. Stat. Mech. 2007 (2007) P03005. 
  22. [22] J. Cui, X. Chen, F. Wang, X. Gong and Z. Yu, Study of liquid droplets impact on dry inclined surface. Asia-Pacific J. Chem. Eng.4 (2009) 643–648. 
  23. [23] M.C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization. Vol. 4 of Adv. Des. Control. SIAM (2001). Zbl1002.49029MR1855817
  24. [24] T. Deng, K. Varanasi, M. Hsu, N. Bhate, C. Keimel, J. Stein and M. Blohm, Non-wetting of impinging droplets on textured surface. Appl. Phys. Lett. 94 (2009) 133109. 
  25. [25] S. Dodds, M.S. Carvalho and S. Kumar, The dynamics of three-dimensional liquid bridges with pinned and moving contact lines. J. Fluid Mech.707 (2012) 521–540. Zbl1275.76085
  26. [26] G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer, New York (1976). Zbl0331.35002MR521262
  27. [27] C. Eck, M. Fontelos, G. Grün, F. Klingbeil and O. Vantzos, On a phase-field model for electrowetting. Interf. Free Bound.11 (2009) 259–290. Zbl1167.35553MR2511642
  28. [28] J. Eggers and R. Evans, Comment on dynamic wetting by liquids of different viscosity, by t.d. blake and y.d. shikhmurzaev. J. Colloid Interf. Sci. 280 (2004) 537–538. 
  29. [29] R. Eley and L. Schwartz, Interaction of rheology, geometry, and process in coating flow. J. Coat. Technol. 74 (2002) 43–53. DOI: 10.1007/BF02697974. MR1933205
  30. [30] M.S. Engelman, R.L. Sani and P.M. Gresho, The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow. Int. J. Numer. Methods Fluids2 (1982) 225–238. Zbl0501.76001MR667793
  31. [31] L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, Rhode Island (1998). Zbl1194.35001MR1625845
  32. [32] R.S. Falk and S.W. Walker, A mixed finite element method for ewod that directly computes the position of the moving interface. SIAM J. Numer. Anal.51 (2013) 1016–1040. Zbl1268.76063MR3035483
  33. [33] E. Fermi, Thermodynamics. Dover (1956). Zbl1085.01518
  34. [34] M. Fontelos, G. Grün and S. Jörres, On a phase-field model for electrowetting and other electrokinetic phenomena. SIAM J. Math. Anal.43 (2011) 527–563. Zbl1228.35082MR2783215
  35. [35] G.P. Galdi,An introduction to the mathematical theory of the Navier-Stokes equations. I. Linearized steady problems. Vol. 38 of Springer Tracts in Natural Philosophy. Springer-Verlag, New York (1994). Zbl0949.35004MR1284205
  36. [36] J.-F. Gerbeau and T. Lelièvre, Generalized navier boundary condition and geometric conservation law for surface tension. Comput. Methods Appl. Mech. Eng.198 (2009) 644–656. Zbl1229.76037MR2498521
  37. [37] C.M. Groh and M.A. Kelmanson, Multiple-timescale asymptotic analysis of transient coating flows. Phys. Fluids 21 (2009) 091702. Zbl1183.76229
  38. [38] B. Guo and C. Schwab, Analytic regularity of stokes flow on polygonal domains in countably weighted sobolev spaces. J. Comput. Appl. Math.190 (2006) 487–519. Zbl1121.35098MR2209521
  39. [39] K.K. Haller, Y. Ventikos, D. Poulikakos and P. Monkewitz, Computational study of high-speed liquid droplet impact. J. Appl. Phys.92 (2002) 2821–2828. 
  40. [40] J. Haslinger and R.A.E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation. Vol. 7 of Adv. Des. Control. SIAM (2003). Zbl1020.74001MR1969772
  41. [41] C. Huh and L.E. Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interf. Sci.35 (1971) 85–101. 
  42. [42] Y. Hyon, D.Y. Kwak and C. Liu, Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete Contin. Dyn. Syst. Ser. A26 (2010) 1291–1304. Zbl05685503MR2600746
  43. [43] M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal.23 (1986) 562–580. Zbl0605.65071MR842644
  44. [44] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems, Vol. 1. Springer (1972). 
  45. [45] F. Mugele and J.-C. Baret, Electrowetting: from basics to applications. J. Phys.: Condensed Matter 17 (2005) R705–R774. 
  46. [46] J. Nam and M.S. Carvalho, Mid-gap invasion in two-layer slot coating. J. Fluid Mech.631 (2009) 397–417. Zbl1181.76011
  47. [47] J. Nitsche, Ein kriterium für die quasi-optimalität des ritzschen verfahrens. Numer. Math.11 (1968) 346–348. Zbl0175.45801MR233502
  48. [48] R.H. Nochetto, A.J. Salgado and S.W. Walker, A diffuse interface model for electrowettng with moving contact lines. Submitted (2012). Zbl1280.35114
  49. [49] R.H. Nochetto and S.W. Walker, A hybrid variational front tracking-level set mesh generator for problems exhibiting large deformations and topological changes. J. Comput. Phys.229 (2010) 6243–6269. Zbl1197.65133MR2660304
  50. [50] L. Onsager, Reciprocal relations in irreversible processes. I. Phys. Rev.37 (1931) 405–426. Zbl0001.09501
  51. [51] L. Onsager, Reciprocal relations in irreversible processes. II. Phys. Rev.38 (1931) 2265–2279. Zbl0004.18303
  52. [52] M. Orlt and A.-M. Sändig, Boundary Value Problems And Integral Equations In Nonsmooth Domains, chapter Regularity Of Viscous Navier-Stokes Flows In Nonsmooth Domains. Marcel Dekker, New York (1995) 185–201. Zbl0826.35095MR1301349
  53. [53] R.F. Probstein, Physicochemical Hydrodynamics: An Introduction, 2nd edn. John Wiley and Sons, Inc., New York (1994). 
  54. [54] T. Qian, X.-P. Wang and P. Sheng, Generalized navier boundary condition for the moving contact line. Commun. Math. Sci.1 (2003) 333–341. Zbl1160.76340MR1980479
  55. [55] T. Qian, X.-P. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics. J. Fluid Mech.564 (2006) 333–360. Zbl1178.76296MR2261865
  56. [56] W. Ren and W.E., Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2007) 022101. Zbl1146.76513
  57. [57] W. Ren, D. Hu and W.E., Continuum models for the contact line problem. Phys. Fluids 22 (2010) 102103. Zbl1308.76082
  58. [58] R.V. Roy, A.J. Roberts and M.E. Simpson, A lubrication model of coating flows over a curved substrate in space. J. Fluid Mech.454 (2002) 235–261. Zbl1015.76018MR1889448
  59. [59] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput.54 (1990) 483–493. Zbl0696.65007MR1011446
  60. [60] Y.D. Shikhmurzaev, Capillary Flows with Forming Interfaces. Chapman & Hall/CRC, Boca Raton, FL, 1st edition (2007). Zbl1165.76001MR2455379
  61. [61] Y.D. Shikhmurzaev and T.D. Blake, Response to the comment on [J. Colloid Interface Sci. 253 (2002) 196] by j. eggers and r. evans. J. Colloid Interf. Sci. 280 (2004) 539–541. 
  62. [62] D.N. Sibley, N. Savva and S. Kalliadasis, Slip or not slip? A methodical examination of the interface formation model using two-dimensional droplet spreading on a horizontal planar substrate as a prototype system. Phys. Fluids 24 (2012). 
  63. [63] L. Slimane, A. Bendali and P. Laborde, Mixed formulations for a class of variational inequalities. ESAIM: M2AN 38 (2004) 177–201. Zbl1100.65059MR2073936
  64. [64] J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization. Springer Ser. Comput. Math. Springer-Verlag (1992). Zbl0761.73003MR1215733
  65. [65] E. Stein, R. de Borst and T.J. Hughes, Encyclopedia of Computational Mechanics. 1 - Fundamentals. Wiley, 1st edition (2004). Zbl1190.76001MR2288275
  66. [66] R. Temam, Navier-Stokes Equations. Theory and numerical analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI (2001). Zbl0981.35001MR769654
  67. [67] E. Vandre, M.S. Carvalho and S. Kumar, Delaying the onset of dynamic wetting failure through meniscus confinement. J. Fluid Mech.707 (2012) 496–520. Zbl1275.76086
  68. [68] W. Velte and P. Villaggio, On the detachment of an elastic body bonded to a rigid support. J. Elasticity 27 (1992) 133–142. DOI: 10.1007/BF00041646. Zbl0774.73063MR1151544
  69. [69] R. Verfürth, Finite element approximation of incompressible navier-stokes equations with slip boundary condition. Numer. Math.50 (1987) 697–721. Zbl0596.76031MR884296
  70. [70] S.W. Walker, A. Bonito and R.H. Nochetto, Mixed finite element method for electrowetting on dielectric with contact line pinning. Interf. Free Bound.12 (2010) 85–119. Zbl1189.78056MR2595379
  71. [71] S.W. Walker and B. Shapiro, Modeling the fluid dynamics of electrowetting on dielectric (ewod). J. Microelectromech. Systems15 (2006) 986–1000. 
  72. [72] S.W. Walker, B. Shapiro and R.H. Nochetto, Electrowetting with contact line pinning: Computational modeling and comparisons with experiments. Phys. Fluids 21 (2009) 102103. Zbl1183.76554
  73. [73] S.J. Weinstein and K.J. Ruschak, Coating flows. Ann. Rev. Fluid Mech.36 (2004) 29–53. Zbl1081.76009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.