  We propose and analyze unfitted finite element approximations for the
two-phase incompressible Navier--Stokes flow in an axisymmetric setting. The
discretized schemes are based on an Eulerian weak formulation for the
Navier--Stokes equation in the 2d-meridian halfplane, together with a
parametric formulation for the generating curve of the evolving interface. We
use the lowest order Taylor--Hood and piecewise linear elements for
discretizing the Navier--Stokes formulation in the bulk and the moving
interface, respectively. We discuss a variety of schemes, amongst which is a
linear scheme that enjoys an equidistribution property on the discrete
interface and good volume conservation. An alternative scheme can be shown to
be unconditionally stable and to conserve the volume of the two phases exactly.
Numerical results are presented to show the robustness and accuracy of the
introduced methods for simulating both rising bubble and oscillating droplet
experiments.
