Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T00:19:10.668Z Has data issue: false hasContentIssue false
  • English
  • Français

Unifying binary fluid diffuse-interface models in the sharp-interface limit

Published online by Cambridge University Press:  01 November 2013

David N. Sibley ,
Andreas Nold  and
Serafim Kalliadasis
David N. Sibley
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Andreas Nold
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Serafim Kalliadasis*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: s.kalliadasis@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent results published by Gugenberger et al. on surface diffusion (Phys. Rev. E, vol. 78, 2008, 016703), show that the sharp-interface limit of the phase field models often adopted in the literature fails to produce the appropriate boundary conditions. With this knowledge, we consider the sharp-interface limit of phase field models for binary fluids, obtained carefully, where hydrodynamic equations are coupled to phase field evolution based on Cahn–Hilliard or Allen–Cahn theories, in a variety of guises, and unify and contrast their forms and behaviours in the sharp-interface limit. In particular, a tensorial mobility model is analysed, which allows the bulk fluids in the outer region to satisfy classical Navier–Stokes type equations to all orders in the Cahn number.

JFM classification

Mathematical Foundations: General fluid mechanics Interfacial Flows (free surface): Interfacial Flows (free surface) Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 736 , 10 December 2013 , pp. 5 - 43
Copyright
©2013 Cambridge University Press 

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

Allen, S. M. & Cahn, J. W. 1979 A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (6), 1085–1095.Google Scholar
Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165.Google Scholar
Anderson, D., McFadden, G. & Wheeler, A. 2001 A phase-field model with convection: sharp-interface asymptotics. Physica D 151 (2–4), 305–331.Google Scholar
Antanovskii, L. K. 1995 A phase field model of capillarity. Phys. Fluids 7 (4), 747–753.Google Scholar
Biben, T. & Joly, L. 2008 Wetting on nanorough surfaces. Phys. Rev. Lett. 100, 186103.Google Scholar
Biben, T., Misbah, C., Leyrat, A. & Verdier, C. 2003 An advected-field approach to the dynamics of fluid interfaces. Europhys. Lett. 63 (4), 623.Google Scholar
Caginalp, G. & Fife, P. C. 1988 Dynamics of layered interfaces arising from phase boundaries. SIAM J. Appl. Maths 48 (3), 506–518.Google Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a non-uniform system. Part 1. Interfacial free energy. J. Chem. Phys. 28 (2), 258–267.Google Scholar
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.Google Scholar
Cox, R. G. 1998 Inertial and viscous effects on dynamic contact angles. J. Fluid Mech. 357, 249–278.Google Scholar
Ding, H. & Spelt, P. D. M. 2007 Wetting condition in diffuse interface simulations of contact line motion. Phys. Rev. E 75, 046708.Google Scholar
Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226 (2), 2078–2095.Google Scholar
Dupuis, A. & Yeomans, J. M. 2005 Modeling droplets on superhydrophobic surfaces: equilibrium states and transitions. Langmuir 21 (6), 2624–2629.Google Scholar
Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865–930.Google Scholar
Folch, R., Casademunt, J., Hernández-Machado, A. & Ramírez-Piscina, L. 1999 Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. Part 1. Theoretical approach. Phys. Rev. E 60, 1724–1733.Google Scholar
Fredholm, I. 1903 Sur une classe d’équations fonctionnelles. Acta Mathematica 27, 365–390.Google Scholar
Gugenberger, C., Spatschek, R. & Kassner, K. 2008 Comparison of phase-field models for surface diffusion. Phys. Rev. E 78, 016703.Google Scholar
Hohenberg, P. C. & Halperin, B. I. 1977 Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479.Google Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1), 96–127.Google Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.Google Scholar
Jiang, W., Bao, W., Thompson, C. V. & Srolovitz, D. J. 2012 Phase field approach for simulating solid-state dewetting problems. Acta Materialia 60 (15), 5578–5592.Google Scholar
Khatavkar, V. V., Anderson, P. D. & Meijer, H. E. H. 2006 On scaling of diffuse-interface models. Chem. Engng Sci. 61 (8), 2364–2378.Google Scholar
Khatavkar, V. V., Anderson, P. D. & Meijer, H. E. H. 2007 Capillary spreading of a droplet in the partially wetting regime using a diffuse-interface model. J. Fluid Mech. 572, 367–387.Google Scholar
Kim, J. 2012 Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12 (3), 613–661.Google Scholar
Korteweg, D. 1901 Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité. Arch. Néerl. des Sci. Exactes Nat. Ser. II 6, 1–24.Google Scholar
Kusumaatmaja, H., Blow, M. L., Dupuis, A. & Yeomans, J. M. 2008 The collapse transition on superhydrophobic surfaces. Europhys. Lett. 81 (3), 36003.Google Scholar
Laplace, P.-S. 1805 Traité de Mécanique Céleste, vol. 4, Courcier.Google Scholar
Le Roy, S., Søndergård, E., Nerbø, I. S., Kildemo, M. & Plapp, M. 2010 Diffuse-interface model for nanopatterning induced by self-sustained ion-etch masking. Phys. Rev. B 81, 161401(R).Google Scholar
Liu, C. & Shen, J. 2003 A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179 (3–4), 211–228.Google Scholar
Lowengrub, J. & Truskinovsky, L. 1998 Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454 (1978), 2617–2654.Google Scholar
Magaletti, F., Picano, F., Chinappi, M., Marino, L. & Casciola, C. M. 2013 The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids. J. Fluid Mech. 714, 95–126.Google Scholar
Nicoli, M., Plapp, M. & Henry, H. 2011 Tensorial mobilities for accurate solution of transport problems in models with diffuse interfaces. Phys. Rev. E 84, 046707.Google Scholar
Pego, R. L. 1989 Front migration in the nonlinear Cahn–Hilliard equation. Proc. R. Soc. Lond. A 422 (1863), 261–278.Google Scholar
Pismen, L. M. & Pomeau, Y. 2000 Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics. Phys. Rev. E 62, 2480–2492.Google Scholar
Qian, T., Wang, X.-P. & Sheng, P. 2006 A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360.Google Scholar
Queralt-Martín, M., Pradas, M., Rodríguez-Trujillo, R., Arundell, M., Corvera Poiré, E. & Hernández-Machado, A. 2011 Pinning and avalanches in hydrophobic microchannels. Phys. Rev. Lett. 106, 194501.Google Scholar
Rowlinson, J. S. 1979 The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J. Stat. Phys. 20, 200–244 Translation of the work of J. D. van der Waals.Google Scholar
Savva, N. & Kalliadasis, S. 2013 Droplet motion on inclined heterogeneous substrates. J. Fluid Mech. 725, 462–491.Google Scholar
Savva, N., Kalliadasis, S. & Pavliotis, G. A. 2010 Two-dimensional droplet spreading over random topographical substrates. Phys. Rev. Lett. 104, 084501.Google Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31 (1), 567–603.CrossRefGoogle Scholar
Schmuck, M., Pradas, M., Pavliotis, G. A. & Kalliadasis, S. 2012 Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains. Proc. R. Soc. Lond. A 468 (2147), 3705–3724.Google Scholar
Seppecher, P. 1996 Moving contact lines in the Cahn–Hilliard theory. Intl J. Engng Sci. 34 (9), 977–992.Google Scholar
Shikhmurzaev, Y. D. 2008 Capillary Flows with Forming Interfaces. Chapman & Hall/CRC.Google Scholar
Sibley, D. N., Nold, A., Savva, N. & Kalliadasis, S. 2013 On the moving contact line singularity: Asymptotics of a diffuse-interface model. Eur. Phys. J. E 36, 26.Google Scholar
Sibley, D. N., Savva, N. & Kalliadasis, S. 2012 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 (8), 082105.Google Scholar
Starovoitov, V. 1994 Model of the motion of a two-component liquid with allowance of capillary forces. J. Appl. Mech. Tech. Phys. 35, 891–897.Google Scholar
Swift, M. R., Orlandini, E., Osborn, W. R. & Yeomans, J. M. 1996 Lattice Boltzmann simulations of liquid–gas and binary fluid systems. Phys. Rev. E 54, 5041–5052.Google Scholar
Vellingiri, R., Savva, N. & Kalliadasis, S. 2011 Droplet spreading on chemically heterogeneous substrates. Phys. Rev. E 84, 036305.Google Scholar
Villanueva, W., Boettinger, W. J., McFadden, G. B. & Warren, J. A. 2012 A diffuse-interface model of reactive wetting with intermetallic formation. Acta Mater. 60 (9), 3799–3814.Google Scholar
van der Waals, J. D. 1893 Thermodynamische theorie der capillariteit in de onderstelling van continue dichtheidsverandering. Verh. K. Akad. Wet. Amsterdam 1, 1–56.Google Scholar
Wylock, C., Pradas, M., Haut, B., Colinet, P. & Kalliadasis, S. 2012 Disorder-induced hysteresis and nonlocality of contact line motion in chemically heterogeneous microchannels. Phys. Fluids 24 (3), 032108.Google Scholar
Yang, X., Feng, J. J., Liu, C. & Shen, J. 2006 Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. J. Comput. Phys. 218 (1), 417–428.Google Scholar
Yeon, D.-H., Cha, P.-R. & Grant, M. 2006 Phase field model of stress-induced surface instabilities: surface diffusion. Acta Mater. 54 (6), 1623–1630.Google Scholar
Young, T. 1805 An essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. 95, 65–87.Google Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317.Google Scholar
Yue, P., Zhou, C. & Feng, J. J. 2010 Sharp-interface limit of the Cahn–Hilliard model for moving contact lines. J. Fluid Mech. 645, 279–294.Google Scholar
Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006 Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing. J. Comput. Phys. 219 (1), 47–67.Google Scholar