Identities for Droplets with Circular Footprint on Tilted Surfaces

TL;DR

This paper derives and tests exact mathematical identities relating droplet parameters on tilted surfaces, applicable across all inclinations, and demonstrates their usefulness in simplifying equations and estimating solution errors.

Contribution

It introduces new identities based on force and torque balances for droplets on tilted surfaces, valid for all inclinations including vertical walls.

Findings

Identities are validated using numerical and approximate solutions.

Identities help replace unknowns in the Young-Laplace equation.

They can estimate errors in approximate solutions without exact references.

Abstract

Exact mathematical identities are presented between the relevant parameters of droplets displaying circular contact boundary based on flat tilted surfaces. Two of the identities are derived from the force balance, and one from the torque balance. The tilt surfaces cover the full range of inclinations for sessile or pendant drops, including the intermediate case of droplets on a wall (vertical surface). The identities are put under test both by the available solutions of a linear response approximation at small Bond numbers as well as the ones obtained from numerical solutions, making use of the Surface Evolver software. The subtleties to obtain certain angle-averages appearing in identities by the numerical solutions are discussed in detail. It is argued how the identities are useful in two respects. First is to replace some unknown values in the Young-Laplace equation by their expressions obtained from the identities. Second is to use the identities to estimate the error for approximate analytical or numerical solutions without any reference to an exact solution.