Identities for droplets with circular footprint on tilted surfaces

Area of Science:

• Physics
• Fluid Dynamics
• Surface Science

Background:

• Droplet behavior on surfaces is governed by surface tension and gravity.
• The Young-Laplace equation describes the pressure difference across a curved interface.
• Understanding droplet parameters on tilted surfaces is crucial for various applications.

Purpose of the Study:

• Derive exact mathematical identities for droplet parameters on flat tilted surfaces.
• Validate these identities using analytical approximations and numerical simulations.
• Demonstrate the utility of these identities in solving the Young-Laplace equation and error estimation.

Main Methods:

• Derivation of identities from force and torque balance principles.
• Testing identities against small Bond number linear response approximations.
• Utilizing Surface Evolver software for numerical solutions and analysis of angle-averages.

Main Results:

• Established three exact mathematical identities relating droplet parameters on tilted surfaces.
• Confirmed the validity of identities through comparisons with analytical and numerical solutions.
• Identified and discussed numerical subtleties in obtaining angle-averages.

Conclusions:

• The derived identities offer a novel approach to parameter substitution in the Young-Laplace equation.
• These identities provide a method for error estimation in approximate solutions without needing exact solutions.
• The findings are applicable to sessile, pendant, and wall-adhering droplets across all tilt angles.