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Sliding and merging of strongly sheared droplets

Published online by Cambridge University Press:  06 October 2023

C. Ruyer-Quil Open the ORCID record for C. Ruyer-Quil [Opens in a new window] ,
D. Bresch Open the ORCID record for D. Bresch [Opens in a new window] ,
M. Gisclon Open the ORCID record for M. Gisclon [Opens in a new window] ,
G.L. Richard Open the ORCID record for G.L. Richard [Opens in a new window] ,
M. Kessar  and
N. Cellier Open the ORCID record for N. Cellier [Opens in a new window]
C. Ruyer-Quil*
Affiliation:
Université Savoie Mont Blanc, CNRS UMR 5271, LOCIE, 73376 Le Bourget du Lac, France
D. Bresch
Affiliation:
Université Savoie Mont Blanc, CNRS UMR 5127, LAMA, 73376 Le Bourget du Lac, France
M. Gisclon
Affiliation:
Université Savoie Mont Blanc, CNRS UMR 5127, LAMA, 73376 Le Bourget du Lac, France
G.L. Richard
Affiliation:
Univ. Grenoble Alpes, INRAE, IGE, 38000 Grenoble, France
M. Kessar
Affiliation:
Université Savoie Mont Blanc, CNRS UMR 5127, LAMA, 73376 Le Bourget du Lac, France
N. Cellier
Affiliation:
IMT, Mines d'Albi, 81 000 Albi, France
*
†Email address for correspondence: Christian.Ruyer-Quil@univ-smb.fr
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Abstract

A mathematical and numerical framework is proposed to compute the displacement and merging dynamics of sliding droplets under the action of a constant shear exerted by a gas flow. An augmented formulation is implemented to model surface tension including the full curvature of the free surface. A set of shallow-water evolution equations is obtained for the film thickness, the averaged velocity, an additional quantity (with dimension of a velocity) taking into account the capillary effects and a tensor called enstrophy. The enstrophy accounts for the deviation of the velocity profile from a constant velocity distribution. The formulation is consistent with the long-wave expansion of the basic equations with a conservative part and source terms including the effect of viscosity, in the form of a viscous friction and the effect of the shear stress. The model is hyperbolic with generalised diffusion terms due to capillarity. Finally, our model is completed with a disjoining pressure formulation that is able to account for the hysteresis of the static contact angle. In this formulation, the advancing or receding nature of the contact line is assessed by the accumulation or reduction of mass of the droplet at the contact line. Simulations of sliding water droplets are performed with periodic boundary conditions in a domain of limited size. Hysteresis of the static contact angle causes a slowdown of the drops and a delay in the sequence of coalescence of the drops.

JFM classification

Drops and Bubbles: Drops Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 972 , 10 October 2023 , A40
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Ahmed, G., Sellier, M., Jeremy, M. & Taylor, M. 2014 Modeling the effects of contact angle hysteresis on the sliding of droplets down inclined surfaces. Eur. J. Mech. B/Fluids 48, 218–230.CrossRefGoogle Scholar
Bertozzi, A.L. & Pugh, M. 1994 The lubrication approximation for thin viscous films: the moving contact line with a ‘porous media’ cut-off of van der Waals interactions. Nonlinearity 7 (6), 1535–1564.CrossRefGoogle Scholar
Brackbill, J.U., Kothe, D.B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.CrossRefGoogle Scholar
Brandon, S., Wachs, A. & Mamur, A. 1997 Simulated contact angle hysteresis of a three-dimensional drop on a chemically heterogeneous surface: a numerical example. J. Colloid Interface Sci. 191, 110–116.CrossRefGoogle ScholarPubMed
Bresch, D., Cellier, N., Couderc, F., Gisclon, M., Noble, P., Richard, G.L., Ruyer-Quil, C. & Vila, J.-P. 2020 Augmented skew-symmetric system for shallow-water system with surface tension allowing large gradient of density. J. Comput. Phys. 419, 109670.CrossRefGoogle Scholar
Bresch, D., Colin, M., Msheik, K., Noble, P. & Song, X. 2019 BD entropy and Bernis–Friedman entropy. C. R. Math. 357 (1), 1–6.CrossRefGoogle Scholar
Bresch, D., Couderc, F., Noble, P. & Vila, J.-P. 2016 A generalization of the quantum Bohm identity: hyperbolic CFL condition for Euler–Korteweg equations. C. R. Math. 354 (1), 39–43.CrossRefGoogle Scholar
Bresch, D. & Noble, P. 2007 Mathematical justification of a shallow water model. Meth. Appl. Anal. 14 (2), 87–118.CrossRefGoogle Scholar
Churaev, N.V. & Sobolev, V.D. 1995 Prediction of contact angles on the basis of the Frumkin–Derjaguin approach. Adv. Colloid Interface Sci. 61, 1–16.CrossRefGoogle Scholar
Derjaguin, B.V. 1940 Teoriya kapillyarnoy kondensatsii [Theory of capillary condensation]. Russ. J. Phys. Chem. A 14, 137–147.Google Scholar
Diez, J.A., Kondic, L. & Bertozzi, A. 2000 Global models for moving contact lines. Phys. Rev. E 63, 011208.CrossRefGoogle ScholarPubMed
Ding, H. & Spelt, P.D.M. 2008 Onset of motion of a three-dimensional droplet on a wall in shear flow at moderate Reynolds numbers. J. Fluid Mech. 599, 341–362.CrossRefGoogle Scholar
Dussan, E.B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11 (1), 371–400.CrossRefGoogle Scholar
Espín, L. & Kumar, S. 2017 Droplet wetting transitions on inclined substrates in the presence of external shear and substrate permeability. Phys. Rev. Fluids 2, 014004.CrossRefGoogle Scholar
Fan, J., Wilson, M.C.T. & Kapur, N. 2011 Displacement of liquid droplets on a surface by a shearing air flow. J. Colloid Interface Sci. 356 (1), 286–292.CrossRefGoogle ScholarPubMed
de Gennes, P.G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
de Gennes, P.G., Hua, X. & Levinson, P. 1990 Dynamics of wetting: local contact angles. J. Fluid Mech. 212, 55–63.CrossRefGoogle Scholar
Godunov, S.K. & Bohachevsky, I. 1959 Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Math. Sbornik. 47(89) (3), 271–306.Google Scholar
Gosset, A. 2017 Prediction of rivulet transition in anti-icing applications. In 7th Eur. Conf. Aeronautics and Aerospace Sci., p. 482. EUCASS.Google Scholar
Gottlieb, S., Shu, C.-W. & Tadmor, E. 2001 Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (1), 89–112.CrossRefGoogle Scholar
Haley, P.J. & Miksis, M.J. 1991 The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 57–81.CrossRefGoogle Scholar
Hooshanginejad, A. & Lee, S. 2017 Droplet depinning in a wake. Phys. Rev. Fluids 2, 031601.CrossRefGoogle Scholar
Lallement, J. 2019 Modélisation et simulation numérique d’écoulements de films minces avec effet de mouillage partiel. Theses, Université de Toulouse.Google Scholar
Lallement, J., Trontin, P., Laurent, C. & Villedieu, P. 2018 A shallow water type model to describe the dynamic of thin partially wetting films for the simulation of anti-icing systems. In AIAA Aviation Forum, 2018 Atmospheric and Space Environments Conference, June 25–29, 2018, Atlanta, Georgia, p. 3012. AIAA.CrossRefGoogle Scholar
Lan, H., Friedrich, M., Armaly, B.F. & Drallmeier, J.A. 2008 Simulation and measurement of 3D shear-driven thin liquid film flow in a duct. Intl J. Heat Fluid Flow 29, 449–459.CrossRefGoogle Scholar
Lavalle, G., Vila, J.-P., Lucquiaud, M. & Valluri, P. 2017 Ultraefficient reduced model for countercurrent two-layer flows. Phys. Rev. Fluids 2, 014001.CrossRefGoogle Scholar
Luchini, P. & Charru, F. 2019 On the large difference between Benjamin's and Hanratty's formulations of perturbed flow over uneven terrain. J. Fluid Mech. 871, 534–561.CrossRefGoogle Scholar
Mahé, M., Vignes-Adler, M., Rousseau, A., Jacquin, C.G. & Adler, P.M. 1988 Adhesion of droplets on a solid wall and detachment by a shear flow: I. Pure systems. J. Colloid Interface Sci. 126 (1), 314–328.CrossRefGoogle Scholar
Meredith, K.V., Heather, A., de Vries, J. & Xin, Y. 2011 A numerical model for partially-wetted flow of thin liquid films. In Computational Methods in Multiphase Flows VI 70, 239–250.Google Scholar
Milne, A.J.B. & Amirfazli, A. 2009 Drop shedding by shear flow for hydrophilic to superhydrophobic surfaces. Langmuir 25 (24), 14155–14164.CrossRefGoogle ScholarPubMed
Moghtadernejad, S., Jadidi, M., Esmail, N. & Dolatabadi, A. 2014 Shear driven rivulet dynamics on surfaces with various wettabilities. In Proc. Int. Mech. Engng Congress and Exposition IMECE2014, Montreal, Quebec, Canada, p. 38665. ASME.CrossRefGoogle Scholar
Moghtadernejad, S., Jadidi, M., Esmail, N. & Dolatabadi, A. 2016 Shear-driven droplet coalescence and rivulet formation. Proc. Inst. Mech. Engrs C 230 (5), 793–803.Google Scholar
Noble, P. & Vila, J.P. 2014 Stability theory for difference approximations of Euler–Korteweg equations and application to thin film flows. SIAM J. Numer. Anal. 52 (6), 2770–2791.CrossRefGoogle Scholar
Oron, A., Davis, S.H. & Bankoff, S.G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Popescu, M.N., Oshanin, G., Dietrich, S. & Cazabat, A.-M. 2012 Precursor films in wetting phenomena. J. Phys.: Condens. Matter 24 (24), 243102.Google ScholarPubMed
Pozrikidis, C. 1997 Shear flow over a protuberance on a plane wall. J. Engng Maths 31 (1), 29–42.CrossRefGoogle Scholar
Razzaghi, A. & Amirfazli, A. 2019 Shedding of a pair of sessile droplets. Intl J. Multiphase Flow 110, 59–68.CrossRefGoogle Scholar
Ren, W. & E, W. 2007 Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2), 022101.CrossRefGoogle Scholar
Richard, G.L., Duran, A. & Fabrèges, B. 2019 a A new model of shoaling and breaking waves. Part 2. Run-up and two-dimensional waves. J. Fluid Mech. 867, 146–194.CrossRefGoogle Scholar
Richard, G.L., Gisclon, M., Ruyer-Quil, C. & Vila, J.-P. 2019 b Optimization of consistent two-equation models for thin film flows. Eur. J. Mech. B/Fluids 76, 7–25.CrossRefGoogle Scholar
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21 (9), 092102.CrossRefGoogle Scholar
Schwartz, L.W. & Eley, R.R. 1998 Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interface Sci. 202, 173–188.CrossRefGoogle Scholar
Sellier, M., Grayson, J.W., Renbaum-Wolff, L., Song, M. & Bertram, A.K. 2015 Estimating the viscosity of a highly viscous liquid droplet through the relaxation time of a dry spot. J. Rheol. 59 (3), 733–750.CrossRefGoogle Scholar
Sellier, M., Taylor, J., Bertram, A.K. & Mandin, P. 2019 Models for the bead mobility technique: a droplet-based viscometer. Aerosol Sci. Technol. 53 (7), 749–759.CrossRefGoogle Scholar
Sibley, D.N., Nold, A., Savva, N. & Kalliadasis, S. 2015 A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading. J. Engng Maths 94, 19–41.CrossRefGoogle Scholar
Teshukov, V.M. 2007 Gas-dynamic analogy for vortex free-boundary flows. J. Appl. Mech. Tech. Phys. 48, 303–309.CrossRefGoogle Scholar
Zhang, K., Rothmayer, A.P. & Hu, H. 2016 An experimental invesitgation on the dynamic water runback process over an airfoil surface pertinent to aircraft icing phenomena. In 8th AIAA Atmospheric and Space Environments Conference, 13–17 June Washington D.C., p. 3138. AIAA.CrossRefGoogle Scholar
Zhang, K., Wei, T. & Hu, H. 2015 An experimental investigation on the surface water transport process over an airfoil by using a digital image projection technique. Exp. Fluids 56, 173.CrossRefGoogle Scholar
Zhao, Y. & Marshall, J.S. 2006 Dynamics of driven liquid films on heterogeneous surfaces. J. Fluid Mech. 559, 355–378.CrossRefGoogle Scholar