Droplet numerics

A non-physical1 droplet evolution model using a discrete Morse flow discretization of a wave equation with a volume constraint. In essence the surface is described by a graph of a function \(u(x,t)\) that is formally a solution of the wave equation with a volume constraint and a positivity constraint.

The time discretization is formulated as the minimization of a functional \[ J_n(v) := \int_\Omega \frac{|v - 2u_{n-1} + u_{n-2}|^2}{2\tau^2} + \frac{|Dv|^2}{2} + Q(x) \chi_{v > 0} + \rho \left[(x g_1 + y g_2)v + \frac12 g_3 v^2\right] \;dx \] for \(n = 2, 3, \ldots\) to find the shape of the surface given by \(u_n\) at time \(t_n = n \tau\). The minimization is performed in the set \[ K_n := \left\{v \in H^1(\Omega): v \geq 0, \int_\Omega v = V(t_n)\right\}, \] where \(V = V(t)\) is the prescribed volume. \(Q = Q(x)\) characterizes the contact angle as \(\theta = \sqrt{2 Q}\).

Pinning of the contact line

However, the model illustrates some important properties of the contact angle inhomogeneity. For instance, in the following video, the droplet sliding on a vertical surface (right is down) stops unless its size is large enough.

On the other hand, if the contact angle is smooth, nothing prevents the droplet from sliding.

Contact line facets

A periodic inhomogeneous contact angle influences the large-scale features of the contact line of an advancing droplet. For instance, in this case the contact line develops macroscopic flat parts (“facets”) parallel to the axes.

  1. Non-physical in the sense that this has very little to do with how an actual fluid in a droplet behaves, but it produces cool videos and is easy to implement.