On the displacement of three-dimensional fluid droplets adhering to a plane wall in viscous pressure-driven flows

Abstract

The yield conditions for the displacement of three-dimensional fluid droplets adhering to a plane solid boundary in pressure-driven flows are studied through a series of numerical computations. The study considers low-Reynolds-number flows between two parallel plates and includes interfacial forces with constant surface tension. A comprehensive study is conducted, covering a wide range of viscosity ratio λ, capillary number Ca, advancing and receding contact angles, θA and θR, and dimensionless plate separation H/h (where H is the plate spacing and h is the unperturbed droplet height). This study seeks the optimal shape of the contact line which yields the maximum flow rate (or Ca) for which a droplet can adhere to the surface. The critical shear rates are presented as functions Ca(λ, H/h, θA, Δθ) where Δθ = θA − θR is the contact angle hysteresis. The numerical solutions are based on an efficient, three-dimensional Newton 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 critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets. As the viscosity of a droplet increases, the critical flow rate decreases, facilitating the displacement. This is consistent with our previous results for shear flows (Dimitrakopoulos & Higdon 1997, 1998), which represent the limit of infinite plate spacing. As the plate spacing is reduced, the critical flow rate increases until a maximum value is reached. Further reduction in the plate spacing decreases the critical flow rate. The effects of both viscosity ratio and plate separation are much more pronounced for high contact angles. Inviscid droplets (or bubbles) show behaviour dramatically different from that of viscous droplets. For these droplets, a significantly higher flow rate is required for drop displacement, but this critical flow rate decreases monotonically as the distance between the plates decreases. In the Appendix, we clarify the necessary conditions for low-Reynolds-number flows past low viscosity droplets or bubbles.