Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows

The dynamic behaviour and stability of a two-dimensional immiscible droplet subject to shear or pressure-driven flow between parallel plates is studied under conditions of negligible inertial and gravitational forces. The droplet is attached to the lower plate and forms two contact lines that are either fixed or mobile. The boundary-integral method is used to numerically determine the flow along and dynamics of the free surface. For surfactant-free interfaces with fixed contact lines, the deformation of the interface is determined for a range of capillary numbers, droplet to displacing fluid viscosity ratios, droplet sizes and flow type. It is shown that as the capillary number or viscosity ratio or size of the droplet increases, the deformation of the interface increases and above critical values of the capillary number no steady shape exists. For small droplets, and at low capillary numbers, shear and pressure-driven flows are shown to yield similar steady droplet shapes. The effect of surfactants is studied assuming a fixed amount of surfactant that is subject to convective-diffusive transport along the interface and no transport to or from the bulk fluids. Increasing the surface Peclet number, the ratio of convective to diffusive transport, leads to an accumulation of surfactant at the downstream end of the droplet and creates Marangoni stresses that immobilize the interface and reduce deformation. The no-slip boundary condition is then relaxed and an integral form of the Navier-slip model is used to examine the effects of allowing the droplet to slip along the solid surface in a pressure-driven flow. For contact angles less than or equal to 90 degrees, a stable droplet spreads along the wall until a steady shape is reached, when the droplet translates across the wall at a constant velocity. The critical capillary number is larger for these droplets compared to those with pinned contact lines. For contact angles greater than 90 degrees, the wetted area between a stable droplet and the wall decreases until a steady shape is reached. The critical capillary number for these droplets is less than that for pinned droplets. Above the critical capillary number the droplet completely detaches for a contact angle of 120 degrees, or part of it is pinched off leaving behind a smaller attached droplet for contact angles less than or equal to 90 degrees.