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Depinning of liquid droplets on substrates by flow of a surrounding immiscible fluid is central to applications such as cross-flow microemulsification, oil recovery and waste cleanup. Surface roughness, either natural or engineered, can cause droplet pinning, so it is of both fundamental and practical interest to determine the flow strength of the surrounding fluid required for droplet depinning on rough substrates. Here, we develop a lubrication-theory-based model for droplet depinning on a substrate with topographical defects by flow of a surrounding immiscible fluid. The droplet and surrounding fluid are in a rectangular channel, a pressure gradient is imposed to drive flow and the defects are modelled as Gaussian-shaped bumps. Using a precursor-film/disjoining-pressure approach to capture contact-line motion, a nonlinear evolution equation is derived describing the droplet thickness as a function of distance along the channel and time. Numerical solutions of the evolution equation are used to investigate how the critical pressure gradient for droplet depinning depends on the viscosity ratio, surface wettability and droplet volume. Simple analytical models are able to account for many of the features observed in the numerical simulations. The influence of defect height is also investigated, and it is found that, when the maximum defect slope is larger than the receding contact angle of the droplet, smaller residual droplets are left behind at the defect after the original droplet depins and slides away. The model presented here yields considerably more information than commonly used models based on simple force balances, and provides a framework that can readily be extended to study more complicated situations involving chemical heterogeneity and three-dimensional effects.
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Film deposition and dynamics of a self-propelled wetting droplet on a conical fibre
We study the dynamic wetting of a self-propelled viscous droplet using the time-dependent lubrication equation on a conical-shaped substrate for different cone radii, cone angles and slip lengths. The droplet velocity is found to increase with the cone angle and the slip length, but decrease with the cone radius. We show that a film is formed at the receding part of the droplet, much like the classical Landau–Levich–Derjaguin film. The film thickness $$h_f$$ is found to decrease with the slip length $$\lambda$$ . By using the approach of matching asymptotic profiles in the film region and the quasi-static droplet, we obtain the same film thickness as the results from the lubrication approach for all slip lengths. We identify two scaling laws for the asymptotic regimes: $$h_fh''_o \sim Ca^{2/3}$$ for $$\lambda \ll h_f$$ and $$h_f h''^{3}_o\sim (Ca/\lambda )^2$$ for $$\lambda \gg h_f$$ ; here, $$1/h''_o$$ is a characteristic length at the receding contact line and $Ca$ is the capillary number. We compare the position and the shape of the droplet predicted from our continuum theory with molecular dynamics simulations, which are in close agreement. Our results show that manipulating the droplet size, the cone angle and the slip length provides different schemes for guiding droplet motion and coating the substrate with a film.
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- Award ID(s):
- 1743794
- PAR ID:
- 10284910
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 907
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2020.834
