On the gravitational displacement of three-dimensional fluid droplets from inclined solid surfaces

Abstract

The yield conditions for the gravitational displacement of three-dimensional fluid droplets from inclined solid surfaces are studied through a series of numerical computations. The study considers both sessile and pendant droplets and includes interfacial forces with constant surface tension. An extensive study is conducted, covering a wide range of Bond numbers B_d, angles of inclination β and advancing and receding contact angles, θ_A and θ_R. This study seeks the optimal shape of the contact line which yields the maximum displacing force (or B_T ≡ B_d sin β) for which a droplet can adhere to the surface. The yield conditions B_T are presented as functions of (B_d or β, θ_A, Δθ) where Δθ = θ_A − θ_R is the contact angle hysteresis. The solution of the optimization problem provides an upper bound for the yield condition for droplets on inclined solid surfaces. Additional contraints based on experimental observations are considered, and their effect on the yield condition is determined. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The numerical results are compared with asymptotic theories (Dussan V. & Chow 1983; Dussan V. 1985) and the useful range of these theories is identified. The normal component of the gravitational force B_N ≡ B_d cos β was found to have a weak effect on the displacement of sessile droplets and a strong effect on the displacement of pendant droplets, with qualitatively different results for sessile and pendant droplets.