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. 2018 Jun 11;8:8910. doi: 10.1038/s41598-018-26121-0

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© The Author(s) 2018

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Streamlines and map of the fluid velocity in the x = 0 symmetry plane of the flow in the thin liquid layer underneath a colloid translating in the positive z direction with velocity U along the film surface in (a) the lab frame in which the colloid is translating, and (b) in the colloid frame in which the liquid is approaching the stationary sphere in the −z direction. (c) Streamlines and map of the fluid velocity in the x = 0 plane of the flow in the liquid layer underneath a sphere rotating counterclockwise around the x axis with velocity Ωa at the interface of the film for (a) the entire film and (d) in the immediate vicinity of the contact line. The thickness of the layer is h/a = 2.14, the contact angle is π/2 and the slip length λ/a is 0.01.