Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T14:23:47.531Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  30 June 2014

Shawn W. Walker
Shawn W. Walker*
Affiliation:
Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA. walker@math.lsu.edu
Get access

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.

Keywords

Mixed methodStokes equationssurface tensioncontact line motioncontact line pinningvariational inequalitywell-posedness
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 969 - 1009
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams and J.J.F. Fournier, Sobolev Spaces, vol. 140 of Pure Appl. Math. Series, 2nd edn. Elsevier (2003).
V.I. Arnold, Lectures on Partial Differential Equations. Springer (2006).
Aubin, J.-P., 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. Pisa 21 (1967) 599–637. Google Scholar
Baer, T.A., Cairncross, R.A., Schunk, P.R., Rao, R.R. and Sackinger, P.A., 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. Google Scholar
Bänsch, E., Finite element discretization of the navier-stokes equations with a free capillary surface. Numer. Math. 88 (2001) 203–235. Google Scholar
Bänsch, E. and Deckelnick, K., Optimal error estimates for the stokes and navier-stokes equations with slip-boundary condition. ESAIM: M2AN 33 (1999) 923–938. Google Scholar
Bänsch, E. and Höhn, B., Numerical treatment of the navier-stokes equations with slip boundary condition. SIAM J. Sci. Comput. 21 (2000) 2144–2162. Google Scholar
Belgacem, F.B., The Mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173–197. Google Scholar
Blake, T.D., The physics of moving wetting lines. J. Colloid Interface Sci. 299 (2006) 1–13. Google ScholarPubMed
Blake, T.D. and Shikhmurzaev, Y.D., Dynamic wetting by liquids of different viscosity. J. Colloid Interface Sci. 253 (2002) 196–202. Google Scholar
D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd edn. Cambridge University Press (2001).
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, New York (2002).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
Brezzi, F., Hager, W.W. and Raviart, P.A., Error estimates for the finite element solution of variational inequalities: Part II. Mixed methods. Num. Math. 31 (1978) 1–16. Google Scholar
Brown, C.E., Jones, T.D. and Neustadter, E.L., Interfacial flow during immiscible displacement. J. Colloid Interface Sci. 76 (1980) 582–586. Google Scholar
Burridge, R. and Keller, J.B., Peeling, slipping and cracking–some one-dimensional free-boundary problems in mechanics. SIAM Review 20 (1978) 31–61. Google Scholar
Cheng, C.H.A., Coutand, D. and Shkoller, S., Navier-stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 39 (2007) 742–800. Google Scholar
Cho, S.K., Moon, H. and Kim, C.-J., Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits. J. Microelectromech. Systems 12 (2003) 70–80. Google Scholar
Ciarlet, P., On korns inequality. Chin. Ann. Math. Ser. B 31 (2010) 607–618. Google Scholar
Clément, P., Approximation by finite element functions using local regularization. R.A.I.R.O. Analyse Numérique 9 (1975) 77–84. Google Scholar
Cortet, P.-P., Ciccotti, M. and Vanel, L., Imaging the stickslip peeling of an adhesive tape under a constant load. J. Stat. Mech. 2007 (2007) P03005. Google Scholar
Cui, J., Chen, X., Wang, F., Gong, X. and Yu, Z., Study of liquid droplets impact on dry inclined surface. Asia-Pacific J. Chem. Eng. 4 (2009) 643–648. Google Scholar
M.C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization. Vol. 4 of Adv. Des. Control. SIAM (2001).
Deng, T., Varanasi, K., Hsu, M., Bhate, N., Keimel, C., Stein, J. and Blohm, M., Non-wetting of impinging droplets on textured surface. Appl. Phys. Lett. 94 (2009) 133109. Google Scholar
Dodds, S., Carvalho, M.S. and Kumar, S., The dynamics of three-dimensional liquid bridges with pinned and moving contact lines. J. Fluid Mech. 707 (2012) 521–540. Google Scholar
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer, New York (1976).
Eck, C., Fontelos, M., Grün, G., Klingbeil, F. and Vantzos, O., On a phase-field model for electrowetting. Interf. Free Bound. 11 (2009) 259–290. Google Scholar
Eggers, J. and Evans, R., 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. Google ScholarPubMed
Eley, R. and Schwartz, L., Interaction of rheology, geometry, and process in coating flow. J. Coat. Technol. 74 (2002) 43–53. DOI: 10.1007/BF02697974. Google Scholar
Engelman, M.S., Sani, R.L. and Gresho, P.M., The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow. Int. J. Numer. Methods Fluids 2 (1982) 225–238. Google Scholar
L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, Rhode Island (1998).
Falk, R.S. and Walker, S.W., A mixed finite element method for ewod that directly computes the position of the moving interface. SIAM J. Numer. Anal. 51 (2013) 1016–1040. Google Scholar
E. Fermi, Thermodynamics. Dover (1956).
Fontelos, M., Grün, G. and Jörres, S., On a phase-field model for electrowetting and other electrokinetic phenomena. SIAM J. Math. Anal. 43 (2011) 527–563. Google Scholar
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).
Gerbeau, J.-F. and Lelièvre, T., Generalized navier boundary condition and geometric conservation law for surface tension. Comput. Methods Appl. Mech. Eng. 198 (2009) 644–656. Google Scholar
Groh, C.M. and Kelmanson, M.A., Multiple-timescale asymptotic analysis of transient coating flows. Phys. Fluids 21 (2009) 091702. Google Scholar
Guo, B. and Schwab, C., Analytic regularity of stokes flow on polygonal domains in countably weighted sobolev spaces. J. Comput. Appl. Math. 190 (2006) 487–519. Google Scholar
Haller, K.K., Ventikos, Y., Poulikakos, D. and Monkewitz, P., Computational study of high-speed liquid droplet impact. J. Appl. Phys. 92 (2002) 2821–2828. Google Scholar
J. Haslinger and R.A.E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation. Vol. 7 of Adv. Des. Control. SIAM (2003).
Huh, C. and Scriven, L.E., Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interf. Sci. 35 (1971) 85–101. Google Scholar
Hyon, Y., Kwak, D.Y. and Liu, C., Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete Contin. Dyn. Syst. Ser. A 26 (2010) 1291–1304. Google Scholar
Lenoir, M., Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986) 562–580. Google Scholar
J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems, Vol. 1. Springer (1972).
Mugele, F. and Baret, J.-C., Electrowetting: from basics to applications. J. Phys.: Condensed Matter 17 (2005) R705–R774. Google Scholar
Nam, J. and Carvalho, M.S., Mid-gap invasion in two-layer slot coating. J. Fluid Mech. 631 (2009) 397–417. Google Scholar
Nitsche, J., Ein kriterium für die quasi-optimalität des ritzschen verfahrens. Numer. Math. 11 (1968) 346–348. Google Scholar
R.H. Nochetto, A.J. Salgado and S.W. Walker, A diffuse interface model for electrowettng with moving contact lines. Submitted (2012).
Nochetto, R.H. and Walker, S.W., A hybrid variational front tracking-level set mesh generator for problems exhibiting large deformations and topological changes. J. Comput. Phys. 229 (2010) 6243–6269. Google Scholar
Onsager, L., Reciprocal relations in irreversible processes. I. Phys. Rev. 37 (1931) 405–426. Google Scholar
Onsager, L., Reciprocal relations in irreversible processes. II. Phys. Rev. 38 (1931) 2265–2279. Google Scholar
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.
R.F. Probstein, Physicochemical Hydrodynamics: An Introduction, 2nd edn. John Wiley and Sons, Inc., New York (1994).
Qian, T., Wang, X.-P. and Sheng, P., Generalized navier boundary condition for the moving contact line. Commun. Math. Sci. 1 (2003) 333–341. CrossRefGoogle Scholar
Qian, T., Wang, X.-P. and Sheng, P., A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564 (2006) 333–360. Google Scholar
Ren, W. and W.E., Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2007) 022101. Google Scholar
Ren, W., D. Hu and W.E., Continuum models for the contact line problem. Phys. Fluids 22 (2010) 102103. Google Scholar
Roy, R.V., Roberts, A.J. and Simpson, M.E., A lubrication model of coating flows over a curved substrate in space. J. Fluid Mech. 454 (2002) 235–261. Google Scholar
Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. Google Scholar
Y.D. Shikhmurzaev, Capillary Flows with Forming Interfaces. Chapman & Hall/CRC, Boca Raton, FL, 1st edition (2007).
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.
Sibley, D.N., Savva, N. and Kalliadasis, S., 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). Google Scholar
Slimane, L., Bendali, A. and Laborde, P., Mixed formulations for a class of variational inequalities. ESAIM: M2AN 38 (2004) 177–201. Google Scholar
J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization. Springer Ser. Comput. Math. Springer-Verlag (1992).
E. Stein, R. de Borst and T.J. Hughes, Encyclopedia of Computational Mechanics. 1 - Fundamentals. Wiley, 1st edition (2004).
R. Temam, Navier-Stokes Equations. Theory and numerical analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI (2001).
Vandre, E., Carvalho, M.S. and Kumar, S., Delaying the onset of dynamic wetting failure through meniscus confinement. J. Fluid Mech. 707 (2012) 496–520. Google Scholar
Velte, W. and Villaggio, P., On the detachment of an elastic body bonded to a rigid support. J. Elasticity 27 (1992) 133–142. DOI: 10.1007/BF00041646. Google Scholar
Verfürth, R., Finite element approximation of incompressible navier-stokes equations with slip boundary condition. Numer. Math. 50 (1987) 697–721. Google Scholar
Walker, S.W., Bonito, A. and Nochetto, R.H., Mixed finite element method for electrowetting on dielectric with contact line pinning. Interf. Free Bound. 12 (2010) 85–119. Google Scholar
Walker, S.W. and Shapiro, B., Modeling the fluid dynamics of electrowetting on dielectric (ewod). J. Microelectromech. Systems 15 (2006) 986–1000. Google Scholar
Walker, S.W., Shapiro, B. and Nochetto, R.H., Electrowetting with contact line pinning: Computational modeling and comparisons with experiments. Phys. Fluids 21 (2009) 102103. Google Scholar
Weinstein, S.J. and Ruschak, K.J., Coating flows. Ann. Rev. Fluid Mech. 36 (2004) 29–53. Google Scholar