# Droplet numerics

A non-physical^{1} **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.

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.↩