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Quasi-Stationary Evaporation of a Small Liquid Droplet on a Flat Substrate: Analytical Solution in Bipolar Coordinates

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Abstract

The study of a sessile liquid droplet evaporating on a flat surface is of great importance for physicochemical, technical, and medical applications. New analytical expressions have been proposed for the vapor density, evaporation flux density, and total evaporation flux per unit time for a slowly evaporating small axially symmetric droplet placed onto a flat substrate at an arbitrary value of the contact angle ranging from 0 to 180°. When deriving the expressions, the solution of the Laplace equation well-known in electrostatics for a flat wedge has been used. The solution has been transformed by the method of inversion on a sphere into a solution for a lens in bipolar coordinates. The new expressions are mathematically equivalent to previously-proposed equations in toroidal coordinates [Popov, Yu.O., Phys. Rev. E, 2005, vol. 71, p. 036313]; however, in the bipolar coordinates, the evaporation flux density has a simpler form of a single integral of a combination of elementary functions, thus being advantageous from the computational point of view. A new expression has also been proposed for the evaporation flux density in polar coordinates and graphic constructions have been performed for the dependences of the evaporation flux density on the polar angle at different values of the droplet contact angle.

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Funding

The work was supported by the Ministry of Science and Higher Education of the Russian Federation within the framework of the state order to the Federal Scientific Research Center “Crystallography and Photonics” of the Russian Academy of Sciences.

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Authors and Affiliations

  1. Federal Scientific Research Center “Crystallography and Photonics,” Russian Academy of Sciences, 119333, Moscow, Russia

    O. A. Savenko & P. V. Lebedev-Stepanov

  2. National Research Nuclear University, MEPhI, 115409, Moscow, Russia

    P. V. Lebedev-Stepanov

Authors
  1. O. A. Savenko
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  2. P. V. Lebedev-Stepanov
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Contributions

P.V. Lebedev-Stepanov—derivation of the main relations and asymptotics; O.A. Savenko—numerical calculations and construction of the plots.

Corresponding author

Correspondence to P. V. Lebedev-Stepanov.

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Savenko, O.A., Lebedev-Stepanov, P.V. Quasi-Stationary Evaporation of a Small Liquid Droplet on a Flat Substrate: Analytical Solution in Bipolar Coordinates. Colloid J 84, 312–320 (2022). https://doi.org/10.1134/S1061933X22030115

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  • DOI: https://doi.org/10.1134/S1061933X22030115

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