Projection method for droplet dynamics on groove-textured surface with
merging and splitting
We study the full dynamics of droplets placed on an inclined groove-textured
surface with merging and splitting. The motion of droplets can be determined by
the contact line dynamics and motion by mean curvature, which are driven by the
competition between surfaces tensions of three phases and gravitational effect.
We reformulate the dynamics as a gradient flow on a Hilbert manifold with
boundary, which can be further reduced to a parabolic variational inequality
under some differentiable assumptions. To efficiently solve the parabolic
variational inequality, the convergence and stability of projection method for
obstacle problem in Hilbert space is revisited using Trotter-Kato's product
formula. Based on this, we proposed a projection scheme for the droplets
dynamics, which incorporates both the obstacle information and the phase
transition information when merging and splitting happen. Several challenging
examples including splitting and merging of droplets are demonstrated.