Wetting boundary condition for three-dimensional curved geometries in lattice Boltzmann color-gradient model

A wetting boundary condition for handling contact line dynamics on three-dimensional curved geometries is developed in the lattice Boltzmann color-gradient framework. By combining the geometrical formation and the prediction-correction wetting scheme, the present wetting boundary condition is able to avoid the necessity to select an appropriate interface normal vector from its multiple solutions in the previous prediction-correction method. The effectiveness and accuracy of the wetting boundary condition are first validated by several benchmark cases, namely a droplet resting on a flat surface and on a solid sphere, and the spontaneous imbibition into a cylindrical tube. We then use the color-gradient model equipped with the developed wetting boundary condition to study the trapping behavior of a confined droplet in a microchannel with a cylindrical hole on the top surface, in which the effects of the hole radius and the droplet radius are identified for varying capillary numbers. Results show that the simulated critical capillary numbers, below which the droplet would be anchored by the hole, and the steady-state shapes of the anchored droplet generally match well with their theoretical solutions. The critical capillary number is found to decrease by either decreasing the hole radius or increasing the droplet radius, which is attributed to the weakened anchoring surface energy gradient and the enhanced driving force from outer flow, respectively. In addition, we show that the previous theoretical solutions are valid only when the initial droplet radius is greater than twice the height of the channel.