A variational approach to a quasi-static droplet model

Abstract

We consider a quasi-static droplet motion based on contact angle dynamics on a planar surface. We derive a natural time-discretization and prove the existence of a weak global-in-time solution in the continuum limit. The time discrete interface motion is described in comparison with barrier functions, which are classical sub-and super-solutions in a local neighborhood. This barrier property is different from standard viscosity solutions since there is no comparison principle for our problem. In the continuum limit the barrier properties still hold in a modified sense.

Liquid drops sliding down an inclined plane

“…In one dimension, one can rigorously show that the above heuristics is valid, using the discretetime (JKO) scheme introduced in [6] (see also [8]…”

Liquid Drops on a Rough Surface

“…As for a dynamic description of evolving liquid drops, many different models are available: see, for instance, [2,15,16,20,22,23]. In general, regularity near the contact line, or even the topology of the drop, is largely unknown for drops that are not global minimizers except drops with strong geometric properties (for example, see Feldman and Kim [18]).…”

Dynamic Stability of Equilibrium Capillary Drops

“…Barrier Properties for the Gradient Flow. In [17] a discrete gradient flow is constructed for (P-V) without restrictions on the star-shapedness of the domain or the maximum speed of the free boundary. There it is proven that the discrete solutions satisfy a barrier property with respect to smooth strict sub and super-solutions of (P-λ).…”

“…The cost of the restriction to nicer sets in (4.5) is that the class of barriers for which the sub and super-solution properties hold is reduced. The necessary conditions on admissible barriers can be deduced by inspecting the proofs in Section 3 of [17]. We give modified statement of the super-solution barrier property applicable to our situation below.…”