  We analyze numerical approximations for axisymmetric two-phase flow in the
arbitrary Lagrangian-Eulerian (ALE) framework. We consider a parametric
formulation for the evolving fluid interface in terms of a one-dimensional
generating curve. For the two-phase Navier-Stokes equations, we introduce both
conservative and nonconservative ALE weak formulations in the 2d meridian
half-plane. Piecewise linear parametric elements are employed for discretizing
the moving interface, which is then coupled to a moving finite element
approximation of the bulk equations. This leads to a variety of ALE methods,
which enjoy either an equidistribution property or unconditional stability.
Furthermore, we adapt these introduced methods with the help of suitable
time-weighted discrete normals, so that the volume of the two phases is exactly
preserved on the discrete level. Numerical results for rising bubbles and
oscillating droplets are presented to show the efficiency and accuracy of these
introduced methods.
