Capillary rise and evaporation of a liquid in a corner between a plane and a cylinder: A model of imbibition into a nanofiber mat coating


Wetting of surfaces with porous coating is relevant for a wide variety of technical applications, such as printing technologies and heat transfer enhancement. Imbibition and evaporation of liquids on surfaces covered with porous layers are responsible for significant improvement of cooling efficiency during drop impact cooling and flow boiling on such surfaces. Up to now, no reliable model exists which is able to predict the kinetics of imbibition coupled with evaporation on surfaces with porous coatings. In this work, we consider one of possible mechanisms of imbibition on a substrate covered by a nanofiber mat. This is the capillary pressure-driven flow in a corner formed between a flat substrate and a fiber attached to it. The shape and the area of the cross-section occupied by the liquid as well as the capillary pressure change along the flow direction. A theoretical/numerical model of simultaneous imbibition and evaporation is developed, in which viscosity, surface tension and evaporation are taken into account. At the beginning of the process the imbibition length is proportional to the square root of time, in agreement with the Lucas-Washburn law. As the influence of evaporation becomes significant, the imbibition rate decreases. The model predictions are compared with experimental data for imbibition of water-ethanol mixtures into nanofiber mat coatings.


  1. 1.

    R. Barhate, S. Ramakrishna, J. Membr. Sci. 296, 1 (2007)

    Article  Google Scholar 

  2. 2.

    X. Deng, L. Mammen, H.-J. Butt, D. Vollmer, Science 335, 67 (2012)

    ADS  Article  Google Scholar 

  3. 3.

    R. Srikar, T. Gambaryan-Roisman, C. Steffes, P. Stephan, C. Tropea, A.L. Yarin, Int. J. Heat Mass Transf. 52, 5814 (2014)

    Article  Google Scholar 

  4. 4.

    E. Alam, S. Yadav, J. Schneider, T. Gambaryan-Roisman, Colloids Surf. A 521, 69 (2017)

    Article  Google Scholar 

  5. 5.

    J. Bico, C. Tordeux, D. Quéré, Europhys. Lett. 55, 214 (2001)

    ADS  Article  Google Scholar 

  6. 6.

    V.M. Starov, S.R. Kostvintsev, V.D. Sobolev, M.G. Velarde, S.A. Zhdanov, J. Colloid Interface Sci. 252, 397 (2002)

    ADS  Article  Google Scholar 

  7. 7.

    T. Gambaryan-Roisman, Curr. Opin. Colloid Interface Sci. 19, 320 (2014)

    Article  Google Scholar 

  8. 8.

    C.K. Wemp, V.P. Carey, Langmuir 33, 14513 (2017)

    Article  Google Scholar 

  9. 9.

    A. Kumar, J. Kleinen, J. Venzmer, A. Trybala, V. Starov, T. Gambaryan-Roisman, Colloids and Interfaces 3, 53 (2019)

    Article  Google Scholar 

  10. 10.

    L. Courbin, J.C. Bird, M. Reyssat, H.A. Stone, J. Phys.: Condens. Matter 21, 464127 (2009)

    ADS  Google Scholar 

  11. 11.

    S. Jun, S. Sinha-Ray, A. Yarin, Int. J. Heat Mass Transf. 62, 99 (2013)

    Article  Google Scholar 

  12. 12.

    Z. Wang, L. Espin, F.S. Bates, S. Kumar, C.W. Macoscko, Chem. Eng. Sci. 146, 104 (2016)

    Article  Google Scholar 

  13. 13.

    M. Freystein, F. Kolberg, L. Spiegel, S. Sinha-Ray, R.P. Sahu, A.L. Yarin, T. Gambaryan-Roisman, P. Stephan, Int. J. Heat Mass Transf. 93, 827 (2016)

    Article  Google Scholar 

  14. 14.

    S. Sinha-Ray, W. Zhang, R.P. Sahu, S. Sinha-Ray, A.L. Yarin, Int. J. Heat Mass Transf. 106, 482 (2017)

    Article  Google Scholar 

  15. 15.

    S. Fischer, R.P. Sahu, S. Sinha-Ray, A.L. Yarin, T. Gambaryan-Roisman, P. Stephan, Int. J. Heat Mass Transf. 108, 2444 (2017)

    Article  Google Scholar 

  16. 16.

    C.M. Weickgenannt, Y. Zhang, A. Lembach, I.V. Roisman, T. Gambaryan-Roisman, A.L. Yarin, C. Tropea, Phys. Rev. E 83, 036305 (2011)

    ADS  Article  Google Scholar 

  17. 17.

    C.M. Weickgenannt, Y. Zhang, S. Sinha-Ray, I.V. Roisman, T. Gambaryan-Roisman, C. Tropea, A.L. Yarin, Phys. Rev. E 84, 036310 (2011)

    ADS  Article  Google Scholar 

  18. 18.

    A.N. Lembach, H.-B. Tan, I.V. Roisman, T. Gambaryan-Roisman, Y. Zhang, C. Tropea, A.L. Yarin, Langmuir 26, 9516 (2010)

    Article  Google Scholar 

  19. 19.

    E.W. Washburn, Phys. Rev. 17, 273 (1921)

    ADS  Article  Google Scholar 

  20. 20.

    S.H. Davis, L.M. Hocking, Phys. Fluids 11, 48 (1999)

    ADS  Article  Google Scholar 

  21. 21.

    N. Alleborn, H. Raszillier, J. Colloid Interface Sci. 280, 449 (2004)

    ADS  Article  Google Scholar 

  22. 22.

    L.A. Romero, F.G. Yost, J. Fluid Mech. 322, 109 (1996)

    ADS  Article  Google Scholar 

  23. 23.

    R.R. Rye, J.A. Mann, F.G. Yost, Langmuir 12, 555 (1996)

    Article  Google Scholar 

  24. 24.

    V. Thammanna Gurumurthy, D. Rettenmaier, I.V. Roisman, C. Tropea, S. Garoff, Colloids Surf. A 544, 118 (2018)

    Article  Google Scholar 

  25. 25.

    V. Thammanna Gurumurthy, I.V. Roisman, C. Tropea, S. Garoff, J. Colloid Interface Sci. 527, 151 (2018)

    ADS  Article  Google Scholar 

  26. 26.

    X. Xu, V.P. Carey, J. Thermophys. Heat Tr. 4, 512 (1990)

    ADS  Article  Google Scholar 

  27. 27.

    M. Markos, V.S. Ajaev, G.M. Homsy, Phys. Fluids 18, 092102 (2006)

    ADS  Article  Google Scholar 

  28. 28.

    L. Mekhitarian, B. Sobac, S. Dehaeck, B. Haut, P. Colinet, Europhys. Lett. 120, 16001 (2017)

    ADS  Article  Google Scholar 

  29. 29.

    P. Kolliopoulos, K.S. Jochem, R.K. Lade, Jr. L.F. Francis, S. Kumar, Langmuir 35, 8131 (2019)

    Article  Google Scholar 

  30. 30.

    T. Gambaryan-Roisman, Interfacial Phenom. Heat Transf. 7, 239 (2019)

    Article  Google Scholar 

  31. 31.

    G. Vázquez, E. Alvarez, J.M. Navaza, J. Chem. Eng. Data 40, 611 (1995)

    Article  Google Scholar 

  32. 32.

    S.J. Spencer, G.T. Andrews, C.G. Deacon, Semicond. Sci. Technol. 28, 055011 (2013)

    ADS  Article  Google Scholar 

  33. 33.

    K. Sefiane, L. Tadrist, M. Douglas, Int. J. Heat Mass Transf. 46, 4527 (2003)

    Article  Google Scholar 

  34. 34.

    C. Diddens, J.G.M. Kuerten, C.W.M. van der Geld, H.M.A. Wijshoff, J. Colloid Interface Sci. 487, 426 (2017)

    ADS  Article  Google Scholar 

  35. 35.

    R. Savino, D. Paterna, Phys. Fluids 18, 118103 (2006)

    ADS  Article  Google Scholar 

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Ghillani, N., Heinz, M. & Gambaryan-Roisman, T. Capillary rise and evaporation of a liquid in a corner between a plane and a cylinder: A model of imbibition into a nanofiber mat coating. Eur. Phys. J. Spec. Top. 229, 1799–1818 (2020).

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