The energetics of drop deposition are considered in the capillary-ballistic regime characterized by high Reynolds number and moderate Weber number. Experiments are performed impacting water/glycol drops onto substrates with varying wettability and contact-angle hysteresis. The impacting event is decomposed into three regimes: (i) pre-impact, (ii) inertial spreading and (iii) post contact-line (CL) pinning, conveniently framed using the theory of Dussan & Davis ( J. Fluid Mech. , vol. 173, 1986, pp. 115–130). During fast-time-scale inertial spreading, the only form of dissipation is CL dissipation ( $\mathcal {D}_{CL}$ ). High-speed imaging is used to resolve the stick-slip dynamics of the CL with $\mathcal {D}_{CL}$ measured directly from experiment using the $\Delta \alpha$ - $R$ cyclic diagram of Xia & Steen ( J. Fluid Mech. , vol. 841, 2018, pp. 767–783), representing the contact-angle deviation against the CL radius. Energy loss occurs on slip legs, and this observation is used to derive a closed-form expression for the kinetic K and interfacial $\mathcal{A}$ post-pinning energy $\{K+\mathcal {A}\}_p/\mathcal {A}_o$ independent of viscosity, only depending on the rest angle $\alpha _p$ , equilibrium angle $\bar {\alpha }$ and hysteresis $\Delta \alpha$ , which agrees well with experimental observation over a large range of parameters, and can be used to evaluate contact-line dissipation during inertial spreading. The post-pinning energy is found to be independent of the pre-impact energy, and it is broken into modal components with corresponding energy partitioning approximately constant for low-hysteresis surfaces with fixed pinning angle $\alpha _p$ . During slow-time-scale post-pinning, the liquid/gas ( $lg$ ) interface is found to vibrate with the frequencies and mode shapes predicted by Bostwick & Steen ( J. Fluid Mech. , vol. 760, 2014, pp. 5–38), irrespective of the pre-impact energy. Resonant mode decay rates are determined experimentally from fast Fourier transforms of the interface dynamics.
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Thermodynamically consistent phase-field modelling of contact angle hysteresis
In the phase-field description of moving contact line problems, the two-phase system can be described by free energies, and the constitutive relations can be derived based on the assumption of energy dissipation. In this work we propose a novel boundary condition for contact angle hysteresis by exploring wall energy relaxation, which allows the system to be in non-equilibrium at the contact line. Our method captures pinning, advancing and receding automatically without the explicit knowledge of contact line velocity and contact angle. The microscopic dynamic contact angle is computed as part of the solution instead of being imposed. Furthermore, the formulation satisfies a dissipative energy law, where the dissipation terms all have their physical origin. Based on the energy law, we develop an implicit finite element method that is second order in time. The numerical scheme is proven to be unconditionally energy stable for matched density and zero contact angle hysteresis, and is numerically verified to be energy dissipative for a broader range of parameters. We benchmark our method by computing pinned drops and moving interfaces in the plane Poiseuille flow. When the contact line moves, its dynamics agrees with the Cox theory. In the test case of oscillating drops, the contact line transitions smoothly between pinning, advancing and receding. Our method can be directly applied to three-dimensional problems as demonstrated by the test case of sliding drops on an inclined wall.
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- Award ID(s):
- 1522604
- NSF-PAR ID:
- 10351804
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 899
- 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.2020.465
