Abstract
This paper investigates an annular droplet on a horizontal fiber. The static state and the dynamic spreading process of the droplet is analyzed. A full model describing the profile of a static droplet is derived from the energy variation principle. To study the dynamical spreading of the droplet, we derive a lubrication model which is verified by the full model. It indicates that the lubrication model is valid for a thin droplet. Results of the static droplet reveal that, when the fiber radius is very small, the droplet tends to have a spherical shape; if the fiber radius is very large, the droplet approaches to a parabolic profile. Furthermore, the time-evolution study is carried out to investigate the dynamical spreading of the droplet. It is highlighted that when the fiber radius is small, the droplet can breakup into small droplets or contract into a sharp shape. For a large fiber radius, the droplet spreads to a steady profile. In addition, the liquid viscosity is found to retard the deformation of the droplet and the motion of the contact lines.
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References
Alabuzhev, A.A.: Influence of heterogeneous plates on the axisymmetrical oscillations of a cylindrical drop. Microgravity Sci. Technol (2017), https://doi.org/10.1007/s12217-017-9571-8
Benintendi, S.W., Smith, M.K.: The spreading of a non-isothermal liquid droplet. Phys. Fluids. 11, 982–989 (1999)
Berejnov, V., Thorne, R.: Effect of transient pinning on stability of drops sitting on an inclined plane. Phys. Rev. E 75, 066308 (2007)
Bonn, D., Eggers, J., Indekeu, J., Meunier, J., Rolley, E.: Wetting and spreading. Rev. Mod. Phys. 81, 739–805 (2009)
Brochard, F.: Spreading of liquid drops on thin cylinders: the “manchon/droplet” transition. J. Chem. Phys. 84, 4664–4672 (1986)
Chan, K., Borhan, A.: Surfactant-assisted spreading of a liquid drop on a smooth solid surface. J. Colloid Interface Sci. 287, 233–248 (2005)
Chen, X., Zhu, Z.Q., Liu, Q.S., Wang, X.W.: Thermodynamics behaviors of macroscopic liquid droplets evaporation from heated substrates. Microgravity Sci. Technol. 27, 353–360 (2015)
Chen, X., Wang, X., Chen, P.G., Liu, Q.S.: Thermal effects of substrate on Marangoni flow in droplet evaporation: response surface and sensitivity analysis. Int. J. Heat Mass Transfer 113, 354–365 (2017)
Cox, R.: The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194 (1986)
Craster, R., Matar, O.: On viscous beads flowing down a vertical fibre. J. Fluid Mech. 553, 85–105 (2006)
Dimitrakopoulos, P., Higdon, J.: On the gravitational displacement of three-dimensional fluid droplets from inclined solid surfaces. J. Fluid Mech. 395, 181–209 (1999)
Ding, Z., Liu, Q.: Stability of liquid films on a porous vertical cylinder. Phys. Rev. E 84, 046307 (2011)
Ding, Z., Wong, T., Liu, R., Liu, Q.: Viscous liquid films on a porous vertical cylinder: dynamics and stability. Phys. Fluids 25, 105–120 (2013)
Ding, Z., Wong, T.: Three-dimensional dynamics of thin liquid films on vertical cylinders with Marangoni effect. Phys. Fluids 29, 011701 (2017)
Ehrhard, P., Davis, S.: Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365–388 (1990)
Gac, J., Gradoń, L.: Modeling of axial motion of small droplets deposited on smooth and rough fiber surfaces. Colloids Surf. A Physicochem. Eng. Asp. 414, 259–266 (2012)
Gilet, T., Terwagne, D., Vandewalle, N.: Droplets sliding on fibres. Eur. Phys. J. E 31, 253–262 (2010)
Gulati, S., Raghunandan, A., Rasheed, F., McBride, S.A., Hirsa, A.H.: Ring-Sheared Drop (RSD): microgravity module for containerless flow studies. Microgravity Sci. Technol. 29, 81–89 (2017)
Haley, P., Miksis, M.: The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 57–81 (2016)
Hammond, P.: Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363–384 (1983)
Hoffman, R.L.: Discontinuous and dilatant viscosity behavior in concentrated suspensions. II. Theory and experimental tests. J. Colloid Interface Sci. 46, 491–506 (1974)
Hocking, L.M.: Sliding and spreading of thin two-dimensional drops. Q. J. Mech. Appl. Math. 34, 37–55 (1981)
Hocking, L.M.: The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Math. 36, 55–59 (1983)
Karapetsas, G., Crastr, R., Matar, O.: On surfactant-enhanced spreading and super spreading of liquid drops on solid surfaces. J. Fluid Mech. 670, 5–37 (2011)
Karapetsas, G., Sahu, K., Matar, O.: Effect of contact line dynamics on the thermocapillary motion of a droplet on an inclined plate. Langmuir 29, 8892–8906 (2013)
Lister, J., Rallison, J., King, A., Cummings, L., Jensen, O.: Capillary drainage of an annular film: the dynamics of collars and lobes. J. Fluid Mech. 552, 311–343 (2006)
Pratap, V., Moumen, M., Subramanian, R.: Thermocapillary motion of a liquid drop on a horizontal solid surface. Langmuir 24, 5185–5193 (2008)
Reznik, S., Salalha, W., Yarin, A., Zussman, E.: Microscale fibre alignment by a three-dimensional sessile drop on a wettable pad. J. Fluid Mech. 574, 179–207 (2007)
Liu, R., Ding, Z., Zhu, Z.: Themocapillary effect on the absolute and convective instabilities of film flows down a fibre. Int. J. Heat Mass Transf. 112, 918–925 (2017)
Savva, N., Kalliadasis, S.: Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21, 092102 (2009)
Savva, N., Kalliadasis, S., Pavliotis, G.: Two-dimensional droplet spreading over random topographical substrates. Phys. Rev. Lett. 104, 084501 (2010)
Smith, M.K.: Thermocapillary migration of a two-dimensional droplet on a solid surface. J. Fluid Mech. 294, 209–230 (1995)
Wu, X.F., Dzenis, Y. A.: Droplet on a fiber: geometrical shape and contact angle. Acta Mechanica. 185, 215–225 (2006)
Wu, X.F., Bedarkar, A., Akhatov, I.S.: Hydroelastic analysis of an axially loaded compliant fiber wetted with a droplet. J. Appl. Phys. 108, 827 (2010)
Wu, X.F., Yu, M., Zhou, Z., Bedarkar, A., Zhao, Y.: Droplets engulfing on a filament. Appl. Surf. Sci. 294, 49–57 (2014)
Yarin, A., Liu, W., Reneker, D.: Motion of droplets along thin fibers with temperature gradient. J. App. Phys. 91, 4751–4760 (2002)
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This work was financially supported by the National Natural Science Foundation of China (Grants No. 51766002).
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Chen, X., Ding, Z. & Liu, R. Spreading of Annular Droplets on a Horizontal Fiber. Microgravity Sci. Technol. 30, 143–153 (2018). https://doi.org/10.1007/s12217-017-9581-6
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DOI: https://doi.org/10.1007/s12217-017-9581-6