Modeling Internal Flow Patterns of Sessile Droplets on Horizontally Vibrating Substrates
Abstract
:1. Introduction
2. Mathematical Model
2.1. Physical Description and Assumptions
- (1)
- The flow inside the droplet is incompressible and non-axisymmetric under the horizontal vibration, and the contact line of the droplet remains pinned.
- (2)
- Since the oscillation period of droplets is much smaller than the evaporation time of droplets, evaporation is not considered during vibrations.
- (3)
- The radius of the droplet is less than the capillary length of the droplet (, about 2.73 mm for water, where is the acceleration of gravity, is the surface tension, and is the density of the liquid), the influence of gravity on the shape of the droplet is negligible compared to the surface tension, and the droplets on the substrate are assumed to be spherical caps.
2.2. VOF-CSF Model
2.3. Dynamic Contact Angle Model
2.4. Boundary Conditions
3. Results and Discussion
3.1. Model Validation
3.2. The Evolution of Droplet Oscillation Modes
3.3. Internal Flow Patterns Inside Oscillating Droplets
3.4. Variations in Contact Angle of the Droplet during Vibrations
3.5. Variations in Average Velocity within the Droplet during Vibrations
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
amplitude, | substrate velocity | ||
capillary number | velocity vector | ||
frequency | velocity of x direction | ||
surface tension force | volume fraction | ||
acceleration of gravity | contact angle | ||
capillary length | initial contact angle | ||
droplet mass | dynamic contact angle | ||
order of modes | contact angles difference | ||
outward unit normal vector | curvature of interface | ||
unit normal vector of the substrate | viscosity | ||
time | density | ||
tangent vector of the substrate | surface tension | ||
fluid-phase velocity along the x, y, z direction | resonant frequency of droplet |
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Shan, Y.; Yin, T. Modeling Internal Flow Patterns of Sessile Droplets on Horizontally Vibrating Substrates. Processes 2024, 12, 667. https://doi.org/10.3390/pr12040667
Shan Y, Yin T. Modeling Internal Flow Patterns of Sessile Droplets on Horizontally Vibrating Substrates. Processes. 2024; 12(4):667. https://doi.org/10.3390/pr12040667
Chicago/Turabian StyleShan, Yanguang, and Tianyi Yin. 2024. "Modeling Internal Flow Patterns of Sessile Droplets on Horizontally Vibrating Substrates" Processes 12, no. 4: 667. https://doi.org/10.3390/pr12040667