  In this paper, we present port-Hamiltonian formulations of the incompressible
Euler equations with a free surface governed by surface tension and gravity
forces, modelling e.g. capillary and gravity waves and the evolution of
droplets in air. Three sets of variables are considered, namely $(v,\Sigma)$,
$(\eta,\phi_{\partial},\Sigma)$ and $(\omega,\phi_{\partial},\Sigma)$, with $v$
the velocity, $\eta$ the solenoidal velocity, $\phi_{\partial}$ a potential,
$\omega$ the vorticity, and $\Sigma$ the free surface, resulting in the
incompressible Euler equations in primitive variables and the vorticity
equation. First, the Hamiltonian formulation for the incompressible Euler
equations in a domain with a free surface combined with a fixed boundary
surface with a homogeneous boundary condition will be derived in the proper
Sobolev spaces of differential forms. Next, these results will be extended to
port-Hamiltonian formulations allowing inhomogeneous boundary conditions and a
non-zero energy flow through the boundaries. Our main results are the
construction and proof of Dirac structures in suitable Sobolev spaces of
differential forms for each variable set, which provides the core of any
port-Hamiltonian formulation. Finally, it is proven that the state dependent
Dirac structures are related to Poisson brackets that are linear,
skew-symmetric and satisfy the Jacobi identity.
