Motion of adhering droplets induced by overlapping of gravitational and periodical acceleration
Introduction
Droplets as a consequence of condensation or liquid sprays may move to sensitive regions in technical systems and affect the proper function of components. Also, sensitive sensors can react abruptly to droplets and measurement results can be falsified. Often machine parts undergo mechanical vibrations which influence the droplet motion (Hagemeier et al., 2011). It should be noted that the transport of a droplet into the system may also occur by shear flow (Dussan V., 1987, Dimitrakopoulos and Higdon, 1997, Barwari et al., 2019) or gravity (Extrand and Kumagai, 1995, Quéré et al., 1998, Lv et al., 2010).
The analysis of the droplet oscillation can be traced back to the 19th century. In 1863 (Lord Kelvin 1863) Lord Kelvin investigated oscillations of incompressible liquid spheres of different densities under their own gravitational influence but neglected capillary effects. In 1873 (Plateau, 1873) Plateau further investigated the dynamics of a vibrating droplet in floating conditions. Only six years later (Lord Rayleigh 1879) Lord Rayleigh developed a mathematical formula to determine the natural frequency of free vibrations of liquid droplets. The theory is based on the assumption that the liquid droplet has a spherical shape and that the oscillation amplitude of the droplet surface is small. Shortly afterwards in 1881 Lamb (Lamb, 1881) extended Rayleigh's derivation by analyzing the influence of droplet viscosity on the natural frequency. In recent research (Shin and Lim, 2014) it has been approved that the theoretical natural frequencies derived by Lamb and Rayleigh are different to the natural frequencies of adhering droplets on substrates. Several factors are responsible for the difference of up to 20 % (Sharp et al., 2011) including friction between substrate and droplet as well as moving/pinned contact line.
Investigations (Daniel and Chaudhury, 2002, Daniel et al., 2004) of adhering droplets on gradient surfaces show that the contact angle hysteresis can be overcome by induced horizontal asymmetric surface vibration. Furthermore, it is proved that the velocity of the sliding droplet increases linearly with the amplitude of the surface vibration and nonlinearly with the frequency. The maximum velocity of the droplet is detected at the first and second natural frequency.
In general, vertical surface vibrations also favor the motion of the droplet. Investigations (Noblin et al., 2004, Noblin et al., 2009) on vertical excitation with low accelerations up to 0.4 g confirm an increased motion of the adhering droplet that happens around the first natural frequency. Two modes of oscillation become clear: at low amplitudes the contact line is pinned and a linear oscillation behavior is detected. At higher amplitudes the contact angle hysteresis is overcome and the droplet radius expands whereas a nonlinear oscillation is observed: the contact line of the droplet is pinned for a short time, then it changes its radius, stops for a while and the radius moves in the opposite direction and the process repeats. This motion behavior is called "stick-slip" motion (Noblin et al., 2004). However, there is no complete linear transversal motion of the contact line in one direction. In recent investigations (Brunet et al., 2009, Brunet et al., 2013), the overlapping of gravity and surface vibration (experiments on inclined substrate) enables transversal motion. Two phenomena can be detected for a moving droplet: climbing against gravity and sliding with gravity down the surface. Experiments (Sartori et al., 2015) confirm that hydrophilic substrates favor sliding, while droplets climb on highly hydrophobic surfaces. However, there exist no well-defined transitions between gliding and climbing depending on droplet volume and tilt angle.
The phenomenon of separation of adhering droplets on superhydrophobic substrates was investigated by (Boreyko and Chen, 2009, Mettu and Chaudhury, 2011). A complete detachment of the droplet by overcoming the adhesion between droplet and surface can be observed for both horizontal and vertical vibration. By using less hydrophobic materials, a larger part of the droplet is always pinned to the substrate and therefore a complete jump-off is not possible. However, in this case a part of the droplet separates forming a satellite droplet. The phenomenon of the droplet jump-off from the surface has not been observed in literature so far for moderate static contact angles.
Still the phenomena of wetting and the associated static droplet oscillation, transversal motion and separation regimes are not fully described and understood. The parameter selection is limited to a minimum and a clear characteristic of the droplet behavior is still missing, i.e. specific droplet phenomena are only described to a limited extent. A detailed subdivision and description of these phenomena is initially published in the present study by closing the gap with a systematic investigation of the hydrodynamic instabilities of adhering droplets, taking into account a wide range of parameters. The investigation of the overlapping of surface vibration (horizontal and vertical excitation) and gravity effect is performed for moderate static contact angles. A definition of the droplet motion is given and a defined stability threshold for the beginning of the droplet motion is found.
Section snippets
Experiments and Method
In Fig. 1 the test rig is shown schematically. An electromagnetic shaker (LDS V455) excites the droplet either vertically or horizontally. The substrates used are acrylic glass (abbr. PMMA) with a moderate hydrophilic behavior and the hydrophobic coated silicon wafer (abbr. cSW). The static contact angle for PMMA is 74.4° ± 0.3° and for the cSW it is 104.8° ± 0.7°. For reasons of technical applications like the motion of droplets on visors of motorcycle helmets, standard PMMA was used (
Droplet contour deformation
The effect of frequency and acceleration on droplet deformation is described for the example of the vertical surface vibration. A metal grid with a mesh size of 0.25 mm is placed under the transparent PMMA substrate. This grid highlights elevations and depressions in order to better visualize the droplet deformation of the top view. Figs. 3 and 4 show the influence of the vertical acceleration (25, 100 and 150 m/s2) on a 10 µl droplet for the first and second natural frequency. The experiments
Horizontal surface vibration
The results with horizontal surface vibration (HV) are discussed in the upcoming part. Two droplet volumes (5 and 20 µl) are analyzed (Fig. 11). For all following diagrams the acceleration a is normalized with respect to the acceleration of gravity (g = 9,81 m/s²). The frequency f is normalized with the first naturally frequency f0. All points shown in the diagrams stand for individual measurements. Accordingly, there is no smooth transition, i.e. the points only represent discrete states. The
Conclusions
The analysis reveals that a droplet motion mainly takes place around the first natural frequency. The reason for this phenomenon is the asymmetric oscillation behavior which is characterized by a contour deformation in the horizontal plane causing a deflection of the center of mass. With increasing the acceleration of the excitation, the surface energy of the droplet is inferior to the external energy and the cohesion of the droplet cannot be guaranteed. The dominance of the contour surface
CRediT authorship contribution statement
B. Barwari: Conceptualization, Methodology, Formal analysis, Visualization, Investigation, Writing - original draft, Writing - review & editing, Validation. M. Rohde: Conceptualization, Methodology, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Validation. O. Wladarz: Investigation, Formal analysis. S. Burgmann: Writing - review & editing. U. Janoske: Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
Authors acknowledge the support of the German Research Foundation (DFG) with the project number 398314989.
References (26)
Practice of vehicle soiling investigations: A review
Int. J. Multiph. Flow
(2011)- et al.
Liquid Drops on an Inclined Plane: The Relation between Contact Angles, Drop Shape, and Retentive Force
J. Colloid Interface Sci.
(1995) - et al.
Displacement of fluid droplets from solid surfaces in low-Reynolds-number shear flows
J. Fluid Mech.
(1997) - et al.
Hydrodynamic Instabilities of Adhering Droplets Due to a Shear Flow in a Rectangular Channel
Chem. Ing. Tech.
(2019) On the ability of drops to stick to surfaces of solids. Part 3. The influences of the motion of the surrounding fluid on dislodging drops
J. Fluid Mech.
(1987)- et al.
Drops at rest on a tilted plane
Langmuir
(1998) Sliding of water droplets on microstructured hydrophobic surfaces
Langmuir
(2010)Dynamical problems regarding elastic spherodial shells and speroids of incompressible liquid
Phil. Trans. R. Soc. Lond
(1863)Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires
(1873)- Lord Rayleigh, On the Stability, or Instability, of certain Fluid Motions, Proceedings of the London Mathematical...
Shape oscillation and detachment conditions for a droplet on a vibrating flat surface
Eur. Phys. J. E
Contact Angle Dependence of the Resonant Frequency of Sessile Water Droplets
Langmuir
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