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A comprehensive model for predicting droplet freezing features on a cold substrate

Published online by Cambridge University Press:  21 November 2018

Moussa Tembely Open the ORCID record for Moussa Tembely [Opens in a new window]  and
Ali Dolatabadi
Moussa Tembely*
Affiliation:
Department of Mechanical, Industrial, and Aerospace Engineering, Concordia University, 1515 Ste-Catherine Street West, H3G 1M8, Montreal, Quebec, Canada
Ali Dolatabadi*
Affiliation:
Department of Mechanical, Industrial, and Aerospace Engineering, Concordia University, 1515 Ste-Catherine Street West, H3G 1M8, Montreal, Quebec, Canada
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

Water droplet freezing affects many aspects of our daily lives, although there is no comprehensive model which retrieves all of the experimentally observed features when a liquid water droplet deposited on a cold substrate turns to ice. In this paper, we present general governing equations to describe water droplet freezing on a solid substrate by accounting for the physical properties of each phase present, namely the liquid and ice, in addition to the solid substrate. The approach, which takes advantage of the full mean curvature expression of both the droplet–air and liquid–ice interfaces, disjoining pressure, the Gibbs–Thomson effect, natural convection and the substrate thermal and physico-chemical properties, enables us to model a more realistic frozen droplet shape, without a prior assumption for the freezing growth angle. In addition to correctly predicting the freezing time, we capture both qualitatively and quantitatively the key experimentally observed features during water droplet freezing such as volume expansion, concave ice front and the cusp singularity. Furthermore, the proposed equation for the tip angle seems to explain its experimentally observed variability.

JFM classification

Drops and Bubbles: Drops Multiphase and Particle-laden Flows: Multiphase flow Phase change: Icing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 859 , 25 January 2019 , pp. 566 - 585
Copyright
© 2018 Cambridge University Press 

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References

Ajaev, V. S. & Davis, S. H. 2003 Boundary-integral simulations of containerless solidification. J. Comput. Phys. 187 (2), 492–503.Google Scholar
Ajaev, V. S. & Davis, S. H. 2004 The effect of tri-junction conditions in droplet solidification. J. Cryst. Growth 264 (13), 452–462.Google Scholar
Alleborn, N. & Raszillier, H. 2004 Spreading and sorption of droplets on layered porous substrates. J. Colloid Interface Sci. 280 (2), 449–464.Google Scholar
Anderson, D. M., Worster, M. G. & Davis, S. H. 1996 The case for a dynamic contact angle in containerless solidification. J. Cryst. Growth 163 (3), 329–338.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.Google Scholar
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids, pp. 282–296. Oxford University Press.Google Scholar
Delplanque, J.-P. & Rangel, R. H. 1997 An improved model for droplet solidification on a flat surface. J. Mater. Sci. 32 (6), 1519–1530.Google Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458–3460.Google Scholar
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–221.Google Scholar
Enríquez, O. R., Marín, Á. G., Winkels, K. G. & Snoeijer, J. H. 2012 Freezing singularities in water drops. Phys. Fluids 24 (9), 091102.Google Scholar
Hao, P., Lv, C. & Zhang, X. 2014 Freezing of sessile water droplets on surfaces with various roughness and wettability. Appl. Phys. Lett. 104 (16), 161609.Google Scholar
Hu., H. & Jin, Z. 2010 An icing physics study by using lifetime-based molecular tagging thermometry technique. Intl J. Multiphase Flow 36 (8), 672–681.Google Scholar
Jung, S., Tiwari, M. K., Doan, N. V. & Poulikakos, D. 2012 Mechanism of supercooled droplet freezing on surfaces. Nature Commun. 3, 615–622.Google Scholar
Madejski, J. 1976 Solidification of droplets on a cold surface. Intl J. Heat Mass Transfer 19 (9), 1009–1013.Google Scholar
Marín, A. G., Enríquez, O. R., Brunet, P., Colinet, P. & Snoeijer, J. H. 2014 Universality of tip singularity formation in freezing water drops. Phys. Rev. Lett. 113, 054301.Google Scholar
Messinger, B. L. 1953 Equilibrium temperature of an unheated icing surface as a function. J. Aeronaut. Sci. 20, 29–41.Google Scholar
Mohammadi, M., Tembely, M. & Dolatabadi, A. 2017 Predictive model of supercooled water droplet pinning/repulsion impacting a superhydrophobic surface: the role of the gas–liquid interface temperature. Langmuir 33 (8), 1816–1825.Google Scholar
Myers, T. G. & Charpin, J. P. F. 2004 A mathematical model for atmospheric ice accretion and water flow on a cold surface. Intl J. Heat Mass Transfer 47 (25), 5483–5500.Google Scholar
Myers, T. G., Charpin, J. P. F. & Chapman, S. J. 2002 The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface. Phys. Fluids 14 (8), 2788–2803.Google Scholar
O’Brien, S. B. G. & Schwartz, L. W. 2002 Theory and modeling of thin film flows. In Encyclopedia of Surface and Colloid Science (ed. Hubbard, A.), pp. 5283–5297. Dekker.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Özgen, S. & Canıbek, M. 2008 Ice accretion simulation on multi-element airfoils using extended messinger model. Heat Mass Transfer 45 (3), 305–322.Google Scholar
Pasandideh-Fard, M., Chandra, S. & Mostaghimi, J. 2002 A three-dimensional model of droplet impact and solidification. Intl J. Heat Mass Transfer 45 (11), 2229–2242; 00218.Google Scholar
Potapczuk, M. 2013 Aircraft icing research at NASA Glenn Research Center. J. Aerosp. Engng 26, 260–276.Google Scholar
Rusche, H.2003 Computational fluid dynamics of dispersed two-phase flows at high phase fractions. PhD thesis, Imperial College London (University of London).Google Scholar
Schremb, M., Roisman, I. V. & Tropea, C. 2018 Normal impact of supercooled water drops onto a smooth ice surface: experiments and modelling. J. Fluid Mech. 835, 1087–1107.Google Scholar
Schultz, W. W., Worster, M. G. & Anderson, D. M. 2001 Solidifying sessile water droplets. In Interactive Dynamics of Convection and Solidification (ed. Ehrhard, D. S. R. P. & Steen, P. H.). Kluwer.Google Scholar
Schwartz, L. W. & Eley, R. R. 1998 Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interface Sci. 202 (1), 173–188.Google Scholar
Sellier, M. & Panda, S. 2010 Beating capillarity in thin film flows. Intl J. Numer. Meth. Fluids 63 (4), 431–448.Google Scholar
Snoeijer, J. H. & Brunet, P. 2012 Pointy ice-drops: how water freezes into a singular shape. Am. J. Phys. 80 (9), 764–771.Google Scholar
Tembely, M., Attarzadeh, R. & Dolatabadi, A. 2018 On the numerical modeling of supercooled micro-droplet impact and freezing on superhydrophobic surfaces. Intl J. Heat Mass Transfer 127, 193–202.Google Scholar
Tembely, M., Vadillo, D., Mackley, M. R. & Soucemarianadin, A. 2012 The matching of a one-dimensional numerical simulation and experiment results for low viscosity Newtonian and non-Newtonian fluids during fast filament stretching and subsequent break-up. J. Rheol. 56 (1), 159–183.Google Scholar
Virozub, A., Rasin, I. G. & Brandon, S. 2008 Revisiting the constant growth angle: estimation and verification via rigorous thermal modeling. J. Cryst. Growth 310 (24), 5416–5422.Google Scholar
Voller, V. & Cross, M. 1983 An explicit numerical method to track a moving phase change front. Intl J. Heat Mass Transfer 26 (1), 147–150.Google Scholar
Voller, V. R. & Prakash, C. 1987 A fixed grid numerical modelling methodology for convection–diffusion mushy region phase-change problems. Intl J. Heat Mass Transfer 30 (8), 1709–1719.Google Scholar
Zadraz̆il, A., Stepanek, F. & Matar, O. K. 2006 Droplet spreading, imbibition and solidification on porous media. J. Fluid Mech. 562 (33), 1–33.Google Scholar
Zhang, H., Wang, X. Y., Zheng, L. L. & Sampath, S. 2004 Numerical simulation of nucleation, solidification, and microstructure formation in thermal spraying. Intl J. Heat Mass Transfer 47 (10–11), 2191–2203.Google Scholar