# 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.