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Surface thermodynamics and wetting condition in the multiphase lattice Boltzmann model with self-tuning equation of state

Published online by Cambridge University Press:  19 April 2022

Rongzong Huang*
School of Energy Science and Engineering, Central South University, 410083 Changsha, PR China
Qing Li
School of Energy Science and Engineering, Central South University, 410083 Changsha, PR China
Nikolaus A. Adams
Institute of Aerodynamics and Fluid Mechanics, Technical University of Munich, 85748 Garching, Germany
Email address for correspondence:


The surface thermodynamics and wetting condition are investigated for the recent multiphase lattice Boltzmann model with a self-tuning equation of state (EOS), where the multiphase EOS is specified in advance and the reduced temperature is set to a relatively low value. The surface thermodynamics is first explored starting from the free-energy functional of a multiphase system and a theoretical expression for the contact angle is derived for the general multiphase EOS. The conventional free-energy density for the solid surface, which is in linear form, is analysed, and it is found that the fluid density on the solid surface significantly deviates from that in the bulk phase when the reduced temperature is relatively low. A new free-energy density for the solid surface, which is in hyperbolic tangent form, is then proposed. Two independent parameters are introduced, which can dramatically reduce the density deviation and effectively adjust the contact angle, respectively. Meanwhile, the contact angle, surface tension and interface thickness can be independently adjusted in the present theoretical framework. Based on the analysed surface thermodynamics, a thermodynamically consistent treatment for the wetting condition is proposed for both straight and curved walls. Numerical tests of droplets on straight and curved walls validate the theoretical analysis of the surface thermodynamics and the present wetting condition treatment. As further applications, a moving droplet on an inclined wall, which is vertically and sinusoidally oscillated, and the evaporation of a droplet on an adiabatic substrate are simulated, and satisfying results consistent with previous studies are obtained.

JFM Papers
© The Author(s), 2022. Published by Cambridge University Press

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Benzi, R., Biferale, L., Sbragaglia, M., Succi, S. & Toschi, F. 2006 Mesoscopic modeling of a two-phase flow in the presence of boundaries: the contact angle. Phys. Rev. E 74, 021509.CrossRefGoogle ScholarPubMed
Biferale, L., Perlekar, P., Sbragaglia, M. & Toschi, F. 2012 Convection in multiphase fluid flows using lattice Boltzmann methods. Phys. Rev. Lett. 108, 104502.CrossRefGoogle ScholarPubMed
Bradshaw, J.T. & Billingham, J. 2018 Thick drops climbing uphill on an oscillating substrate. J. Fluid Mech. 840, 131153.CrossRefGoogle Scholar
Briant, A.J., Papatzacos, P. & Yeomans, J.M. 2002 Lattice Boltzmann simulations of contact line motion in a liquid-gas system. Phil. Trans. R. Soc. Lond. A 360, 485495.CrossRefGoogle Scholar
Briant, A.J., Wagner, A.J. & Yeomans, J.M. 2004 Lattice Boltzmann simulations of contact line motion. I. Liquid-gas systems. Phys. Rev. E 69, 031602.CrossRefGoogle ScholarPubMed
Brunet, P., Eggers, J. & Deegan, R.D. 2007 Vibration-induced climbing of drops. Phys. Rev. Lett. 99, 144501.CrossRefGoogle ScholarPubMed
Cahn, J.W. 1977 Critical point wetting. J. Chem. Phys. 66, 36673672.CrossRefGoogle Scholar
Carnahan, N.F. & Starling, K.E. 1969 Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51, 635636.CrossRefGoogle Scholar
Colosqui, C.E., Falcucci, G., Ubertini, S. & Succi, S. 2012 Mesoscopic simulation of non-ideal fluids with self-tuning of the equation of state. Soft Matt. 8, 37983809.CrossRefGoogle Scholar
Colosqui, C.E., Kavousanakis, M.E., Papathanasiou, A.G. & Kevrekidis, I.G. 2013 Mesoscopic model for microscale hydrodynamics and interfacial phenomena: slip, films, and contact-angle hysteresis. Phys. Rev. E 87, 013302.CrossRefGoogle ScholarPubMed
Connington, K. & Lee, T. 2013 Lattice Boltzmann simulations of forced wetting transitions of drops on superhydrophobic surfaces. J. Comput. Phys. 250, 601615.CrossRefGoogle Scholar
Connington, K.W., Lee, T. & Morris, J.F. 2015 Interaction of fluid interfaces with immersed solid particles using the lattice Boltzmann method for liquid-gas-particle systems. J. Comput. Phys. 283, 453477.CrossRefGoogle Scholar
Cox, R.G. 1986 The dynamics of the spreading of liquids on a solid surface Part 1. Viscous flow. J. Fluid Mech. 168, 169194.CrossRefGoogle Scholar
Deegan, R.D., Bakajin, O., Dupont, T.F., Huber, G., Nagel, S.R. & Witten, T.A. 1997 Capillary flow as the cause of ring stains from dried liquid drops. Nature 389, 827829.CrossRefGoogle Scholar
Dellar, P.J. 2014 Lattice Boltzmann algorithms without cubic defects in Galilean invariance on standard lattices. J. Comput. Phys. 259, 270283.CrossRefGoogle Scholar
Fakhari, A. & Bolster, D. 2017 Diffuse interface modeling of three-phase contact line dynamics on curved boundaries: a lattice Boltzmann model for large density and viscosity ratios. J. Comput. Phys. 334, 620638.CrossRefGoogle Scholar
Falcucci, G., Bella, G., Chiatti, G., Chibbaro, S., Sbragaglia, M. & Succi, S. 2007 Lattice Boltzmann models with mid-range interactions. Commun. Comput. Phys. 2, 10711084.Google Scholar
Geier, M. & Pasquali, A. 2018 Fourth order Galilean invariance for the lattice Boltzmann method. Comput. Fluids 166, 139151.CrossRefGoogle Scholar
Geier, M., Schönherr, M., Pasquali, A. & Krafczyk, M. 2015 The cumulant lattice Boltzmann equation in three dimensions: theory and validation. Comput. Math. Appl. 70, 507547.CrossRefGoogle Scholar
de Gennes, P.G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827863.CrossRefGoogle Scholar
Gunstensen, A.K., Rothman, D.H., Zaleski, S. & Zanetti, G. 1991 Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 43, 43204327.CrossRefGoogle ScholarPubMed
He, X., Chen, S. & Zhang, R. 1999 A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability. J. Comput. Phys. 152, 642663.CrossRefGoogle Scholar
He, X. & Doolen, G.D. 2002 Thermodynamic foundations of kinetic theory and lattice Boltzmann models for multiphase flows. J. Stat. Phys. 107, 309328.CrossRefGoogle Scholar
He, Q., Li, Y., Huang, W., Hu, Y. & Wang, Y. 2020 Lattice Boltzmann model for ternary fluids with solid particles. Phys. Rev. E 101, 033307.CrossRefGoogle ScholarPubMed
Hocking, L.M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36, 5569.CrossRefGoogle Scholar
Hu, H. & Larson, R.G. 2002 Evaporation of a sessile droplet on a substrate. J. Phys. Chem. B 106, 13341344.CrossRefGoogle Scholar
Huang, R., Lan, L. & Li, Q. 2020 Lattice Boltzmann simulations of thermal flows beyond the Boussinesq and ideal-gas approximations. Phys. Rev. E 102, 043304.CrossRefGoogle ScholarPubMed
Huang, R. & Wu, H. 2016 Total enthalpy-based lattice Boltzmann method with adaptive mesh refinement for solid-liquid phase change. J. Comput. Phys. 315, 6583.CrossRefGoogle Scholar
Huang, R., Wu, H. & Adams, N.A. 2018 Eliminating cubic terms in the pseudopotential lattice Boltzmann model for multiphase flow. Phys. Rev. E 97, 053308.CrossRefGoogle ScholarPubMed
Huang, R., Wu, H. & Adams, N.A. 2019 a Density gradient calculation in a class of multiphase lattice Boltzmann models. Phys. Rev. E 100, 043306.CrossRefGoogle Scholar
Huang, R., Wu, H. & Adams, N.A. 2019 b Lattice Boltzmann model with adjustable equation of state for coupled thermo-hydrodynamic flows. J. Comput. Phys. 392, 227247.CrossRefGoogle Scholar
Huang, R., Wu, H. & Adams, N.A. 2019 c Lattice Boltzmann model with self-tuning equation of state for multiphase flows. Phys. Rev. E 99, 023303.CrossRefGoogle ScholarPubMed
Huang, R., Wu, H. & Adams, N.A. 2021 Mesoscopic lattice Boltzmann modeling of the liquid-vapor phase transition. Phys. Rev. Lett. 126, 244501.CrossRefGoogle ScholarPubMed
Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G. & Viggen, E.M. 2017 The Lattice Boltzmann Method: Principles and Practice. Springer International Publishing.CrossRefGoogle Scholar
Kupershtokh, A.L. 2004 New method of incorporating a body force term into the lattice Boltzmann equation. In Proceedings of the 5th International EHD Workshop, pp. 241–246. University of Poitiers.Google Scholar
Lallemand, P. & Luo, L.-S. 2000 Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 61, 65466562.CrossRefGoogle ScholarPubMed
Laurila, T., Carlson, A., Do-Quang, M., Ala-Nissila, T. & Amberg, G. 2012 Thermohydrodynamics of boiling in a van der Waals fluid. Phys. Rev. E 85, 026320.CrossRefGoogle Scholar
Li, Q., Kang, Q.J., Francois, M.M., He, Y.L. & Luo, K.H. 2015 Lattice Boltzmann modeling of boiling heat transfer: the boiling curve and the effects of wettability. Intl J. Heat Mass Transfer 85, 787796.CrossRefGoogle Scholar
Li, Q., Luo, K.H., Kang, Q.J. & Chen, Q. 2014 Contact angles in the pseudopotential lattice Boltzmann modeling of wetting. Phys. Rev. E 90, 053301.CrossRefGoogle ScholarPubMed
Li, Q., Luo, K.H. & Li, X.J. 2012 Forcing scheme in pseudopotential lattice Boltzmann model for multiphase flows. Phys. Rev. E 86, 016709.CrossRefGoogle ScholarPubMed
Luo, L.-S. 1998 Unified theory of lattice Boltzmann models for nonideal gases. Phys. Rev. Lett. 81, 16181621.CrossRefGoogle Scholar
Lycett-Brown, D. & Luo, K.H. 2015 Improved forcing scheme in pseudopotential lattice Boltzmann methods for multiphase flow at arbitrarily high density ratios. Phys. Rev. E 91, 023305.CrossRefGoogle ScholarPubMed
Martys, N.S. & Chen, H. 1996 Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. Phys. Rev. E 53, 743750.CrossRefGoogle ScholarPubMed
Noblin, X., Kofman, R. & Celestini, F. 2009 Ratchetlike motion of a shaken drop. Phys. Rev. Lett. 102, 194504.CrossRefGoogle ScholarPubMed
Qian, Y.H., d'Humières, D. & Lallemand, P. 1992 Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17, 479484.CrossRefGoogle Scholar
Sbragaglia, M., Benzi, R., Biferale, L., Succi, S., Sugiyama, K. & Toschi, F. 2007 Generalized lattice Boltzmann method with multirange pseudopotential. Phys. Rev. E 75, 026702.CrossRefGoogle ScholarPubMed
Sbragaglia, M., Sugiyama, K. & Biferale, L. 2008 Wetting failure and contact line dynamics in a Couette flow. J. Fluid Mech. 614, 471493.CrossRefGoogle Scholar
Semprebon, C., Krüger, T. & Kusumaatmaja, H. 2016 Ternary free-energy lattice Boltzmann model with tunable surface tensions and contact angles. Phys. Rev. E 93, 033305.CrossRefGoogle ScholarPubMed
Shan, X. & Chen, H. 1993 Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 47, 18151819.CrossRefGoogle ScholarPubMed
Succi, S. 2015 Lattice Boltzmann 2038. Europhys. Lett. 109, 50001.CrossRefGoogle Scholar
Swift, M.R., Osborn, W.R. & Yeomans, J.M. 1995 Lattice Boltzmann simulation of nonideal fluids. Phys. Rev. Lett. 75, 830833.CrossRefGoogle ScholarPubMed
Thompson, P.A. & Robbins, M.O. 1989 Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63, 766769.CrossRefGoogle ScholarPubMed
Voinov, O.V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714721.CrossRefGoogle Scholar
Wagner, A.J. 2006 Thermodynamic consistency of liquid-gas lattice Boltzmann simulations. Phys. Rev. E 74, 056703.CrossRefGoogle ScholarPubMed
Zhang, X., Liu, H. & Zhang, J. 2020 A new capillary force model implemented in lattice Boltzmann method for gas-liquid-solid three-phase flows. Phys. Fluids 32, 103301.CrossRefGoogle Scholar
Zhang, J. & Tian, F. 2008 A bottom-up approach to non-ideal fluids in the lattice Boltzmann method. Europhys. Lett. 81, 66005.CrossRefGoogle Scholar