Displacement of fluid droplets from solid surfaces in low-Reynolds-number shear flows

The yield conditions for the displacement of fluid droplets from solid boundaries are studied through a series of numerical computations. The study includes gravitational and interfacial forces, but is restricted to two-dimensional droplets and low-Reynolds-number flow. A comprehensive study is conducted, covering a wide range of viscosity ratio λ, Bond number B_d, capillary number Ca and contact angles θ_A and θ_R. The yield conditions for drop displacement are calculated and the critical shear rates are presented as functions Ca(θ, B_d, θ_A, Δθ) where Δθ = θ_A - θ_R is the contact angle hysteresis. 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 surface profiles. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While excellent agreement is found in the joint asymptotic limits Δθ ≪ θ_A ≪ 1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets. Gravitational forces normal to the solid boundary have a significant effect on the displacement process, reducing the critical shear rate for viscous drops and increasing the rate for inviscid droplets. The low-viscosity limit λ → 0 is shown to be a singular limit in the lubrication theory, and the proper scaling for Ca at small λ is identified.