TY - JOUR
T1 - Gravity-driven coatings on curved substrates
T2 - A differential geometry approach
AU - Ledda, Pier Giuseppe
AU - Pezzulla, M.
AU - Jambon-Puillet, E.
AU - Brun, P. T.
AU - Gallaire, F.
N1 - Funding Information:
This work was supported by the Swiss National Science Foundation (grant no. 200021_178971 to P.G.L.).
Publisher Copyright:
©
PY - 2022/10/25
Y1 - 2022/10/25
N2 - Drainage and spreading processes in thin liquid films have received considerable attention in the past decades. Yet, our understanding of three-dimensional cases remains sparse, with only a few studies focusing on flat and axisymmetric substrates. Here, we exploit differential geometry to understand the drainage and spreading of thin films on curved substrates, under the assumption of negligible surface tension and hydrostatic gravity effects. We develop a solution for the drainage on a local maximum of a generic substrate. We then investigate the role of geometry in defining the spatial thickness distribution via an asymptotic expansion in the vicinity of the maximum. Spheroids with a much larger (respectively smaller) height than the equatorial radius are characterized by an increasing (respectively decreasing) coating thickness when moving away from the pole. These thickness variations result from a competition between the variations of the substrate's slope and mean curvature. The coating of a torus presents larger thicknesses and a faster spreading on the inner region than on the outer region, owing to the different curvatures in these two regions. In the case of an ellipsoid with three different axes, spatial modulations in the drainage solution are observed as a consequence of a faster drainage along the short principal axis, faithfully reproduced by a three-dimensional asymptotic solution. Leveraging the conservation of mass, an analytical solution for the average spreading front is obtained. The solutions are in agreement with numerical simulations and experimental measurements obtained from the coating of a curing polymer on diverse substrates.
AB - Drainage and spreading processes in thin liquid films have received considerable attention in the past decades. Yet, our understanding of three-dimensional cases remains sparse, with only a few studies focusing on flat and axisymmetric substrates. Here, we exploit differential geometry to understand the drainage and spreading of thin films on curved substrates, under the assumption of negligible surface tension and hydrostatic gravity effects. We develop a solution for the drainage on a local maximum of a generic substrate. We then investigate the role of geometry in defining the spatial thickness distribution via an asymptotic expansion in the vicinity of the maximum. Spheroids with a much larger (respectively smaller) height than the equatorial radius are characterized by an increasing (respectively decreasing) coating thickness when moving away from the pole. These thickness variations result from a competition between the variations of the substrate's slope and mean curvature. The coating of a torus presents larger thicknesses and a faster spreading on the inner region than on the outer region, owing to the different curvatures in these two regions. In the case of an ellipsoid with three different axes, spatial modulations in the drainage solution are observed as a consequence of a faster drainage along the short principal axis, faithfully reproduced by a three-dimensional asymptotic solution. Leveraging the conservation of mass, an analytical solution for the average spreading front is obtained. The solutions are in agreement with numerical simulations and experimental measurements obtained from the coating of a curing polymer on diverse substrates.
KW - gravity currents
KW - lubrication theory
KW - thin films
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U2 - 10.1017/jfm.2022.758
DO - 10.1017/jfm.2022.758
M3 - Article
AN - SCOPUS:85140727601
SN - 0022-1120
VL - 949
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
M1 - A38
ER -