Displacement of a two-dimensional immiscible droplet in a channel

We used the lattice Boltzmann method to study the displacement of a two-dimensional immiscible droplet subject to gravitational forces in a channel. The dynamic behavior of the droplet is shown, and the effects of the contact angle, Bond number (the ratio of gravitational to surface forces), droplet size, and density and viscosity ratios of the droplet to the displacing fluid are investigated. For the case of a contact angle less than or equal to 90°, at a very small Bond number, the wet length between the droplet and the wall decreases with time until a steady shape is reached. When the Bond number is large enough, the droplet first spreads and then shrinks along the wall before it reaches steady state. Whether the steady-state value of the wet length is greater or less than the static value depends on the Bond number. When the Bond number exceeds a critical value, a small portion of the droplet pinches off from the rest of the droplet for a contact angle less than 90°; a larger portion of the droplet is entrained into the bulk for a contact angle equal to 90°. For the nonwetting case, however, for any Bond number less than a critical value, the droplet shrinks along the wall from its static state until reaching the steady state. For any Bond number above the critical value, the droplet completely detaches from the wall. Either increasing the contact angle or viscosity ratio or decreasing the density ratio decreases the critical Bond number. Increasing the droplet size increases the critical Bond number while it decreases the critical capillary number.