Displacement of a two-dimensional immiscible droplet adhering to a wall inshear 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 studie d under conditions of negligible inertial and gravitational forces. The dro plet is attached to the lower plate and forms two contact lines that are ei ther fixed or mobile. The boundary-integral method is used to numerically d etermine the flow along and dynamics of the free surface. For surfactant-fr ee interfaces with fixed contact lines, the deformation of the interface is determined for a range of capillary numbers, droplet to displacing fluid v iscosity ratios, droplet sizes and flow type. It is shown that as the capil lary number or viscosity ratio or size of the droplet increases, the deform ation of the interface increases and above critical values of the capillary number no steady shape exists. For small droplets, and at low capillary nu mbers, shear and pressure-driven flows are shown to yield similar steady dr oplet shapes. The effect of surfactants is studied assuming a fixed amount of surfactant that is subject to convective-diffusive transport along the i nterface and no transport to or from the bulk fluids. Increasing the surfac e Peclet number, the ratio of convective to diffusive transport, leads to a n accumulation of surfactant at the downstream end of the droplet and creat es 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 t o slip along the solid surface in a pressure-driven flow. For contact angle s 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 wa ll at a constant velocity. The critical capillary number is larger for thes e droplets compared to those with pinned contact lines. For contact angles greater than 90 degrees, the wetted area between a stable droplet and the w all decreases until a steady shape is reached. The critical capillary numbe r for these droplets is less than that for pinned droplets. Above the criti cal capillary number the droplet completely detaches for a contact angle of 120 degrees, or part of it is pinched off leaving behind a smaller attache d droplet for contact angles less than or equal to 90 degrees.