  The numerical modelling of convection dominated high density ratio two-phase
flow poses several challenges, amongst which is resolving the relatively thin
shear layer at the interface. To this end we propose a sharp discretisation of
the two-velocity model of the two-phase Navier-Stokes (NS) equations. This
results in the ability to model the shear layer, rather than resolving it, by
allowing for a velocity discontinuity in the direction(s) tangential to the
interface.
  In this paper we focus our attention on the transport of mass and momentum in
the presence of such a velocity discontinuity. We propose a generalisation of
the dimensionally unsplit geometric volume of fluid (VOF) method for the
advection of the interface in the two-velocity formulation. Sufficient
conditions on the construction of donating regions are derived that ensure
boundedness of the volume fraction for dimensionally unsplit advection methods.
We propose to interpolate the mass fluxes resulting from the dimensionally
unsplit geometric VOF method for the advection of the staggered momentum field,
resulting in semi-discrete energy conservation. Division of the momentum by the
respective mass, to obtain the velocity, is not always well-defined for nearly
empty control volumes and therefore care is taken in the construction of the
momentum flux interpolant: our proposed flux interpolant guarantees that this
division is always well-defined without being unnecessarily dissipative.
  Besides the newly proposed two-velocity model we also detail our exactly
conservative (mass per phase and total linear momentum) implementation of the
one-velocity formulation of the two-phase NS equations, which will be used for
comparison.
  The discretisation methods are validated using classical time-reversible flow
fields, where in this paper the advection is uncoupled from the NS solver,
which will be developed in a later paper.
