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Timm, Mitchel L.; Alassar, Rajai S. M.; Masoud, Hassan

A second-order-accurate approximation for the shape of a sessile droplet deformed by gravity. (English) Zbl 1526.76012

J. Eng. Math. 142, Paper No. 5, 18 p. (2023).
Summary: We analytically solve the Young- for the shape of a stationary sessile droplet pinned to an inclined substrate, assuming that the droplet’s contact line is circular. In the absence of gravity (or an equivalent external field), a sessile droplet takes the form of a . Here, we calculate deviations from this ideal geometry when gravitational effects are non-negligible. Our calculations are based on a perturbation solution in powers of the Bond number Bo, which is a dimensionless parameter measuring the strength of gravity relative to surface tension. The newly derived solution is second-order accurate and builds on our previous work [the first author et al., Sci. Rep. 9, Paper No. 19803, 13 p. (2019; doi:10.1038/s41598-019-55040-x)], where only the leading-order contributions were calculated. We consider the full range of substrate inclination angle from 0 to \(\pi\) and show that, when the second-order corrections are taken into account, the droplet’s profile is captured more precisely and the volume-conservation error of the solution is reduced considerably, all at a modest computational cost. We also find that our solution accurately approximates the gravity-induced deformation of the droplet for a wide range of droplet volumes and Bond numbers. As an example, we can very well predict the distorted shape of a droplet that is hemispherical at zero gravity up to \(\mathrm{Bo}\, \approx \,4\), \(1.25\), and 2.5 when the substrate is tilted from horizontal by \(0\), \(\pi / 2\), and \(\pi \), respectively. Among other applications, the outcome of our study can serve as the first step toward analyzing the evaporation of sessile droplets deformed by gravity.
© The Author(s), under exclusive licence to Springer Nature B.V. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

MSC:

76B45 Capillarity (surface tension) for incompressible inviscid fluids
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76T10 Liquid-gas two-phase flows, bubbly flows

Keywords:

surface tension; inclined substrate; perturbation theory; Young-Laplace equation; circular contact line; asymptotic solution

Software:

Surface Evolver
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Full Text: DOI

References:

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