zero. The quantitative results we will present in the next section use a resolution of $128^3$, confirmed by simulations with $256^3$. Grids with $64^3$ are used to plot the visualisations of the drop.

The physical parameters are as given in section \[sec:phys\_par\]. We study either gravity or capillary waves, i.e. a Bond number $\rho g R^2/\sigma$ of either infinity or zero. Figure \[fig:BLUE\_tongues\] locates the parameter values that we have used for our simulations within the instability tongues for the gravitational or capillary cases, where

$$\begin{aligned}
\omega_g^2 \equiv \frac{g}{R} && \qquad\omega_c^2 \equiv \frac{\sigma}{\rho R^3}\\
a_g \equiv R\omega_g^2 = g && \qquad a_c \equiv R\omega_c^2 =\frac{\sigma}{\rho R^2}\end{aligned}$$

\[eq:nondim\]

See [@Ali1] for more details on this non-dimensionalization. Because our viscosity is low, the tongues are very close to the inviscid ones; we plot the inviscid tongues for simplicity.

Frequencies were chosen to induce instabilities from $\ell = 1$ to $\ell = 6$, as predicted from linear Floquet theory. In each case, the value of $\ell$ from the full three-dimensional nonlinear numerical simulations agreed with the theoretical value.

Results
=======

Case ${\ell}= 1$ {#sec:l=1}
----------------

We begin by presenting the ${\ell}=1$ case in the purely gravitational regime, i.e. in the presence of a 