  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 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 a previous paper (Remmerswaal and Veldman (2022), arXiv:2209.14934) we
have discussed the transport of mass and momentum, where the two fluids were
not yet coupled. In this paper an implicit coupling of the two fluids is
proposed, which imposes continuity of the velocity field in the interface
normal direction. The coupling is included in the pressure Poisson problem, and
is discretised using a multidimensional generalisation of the ghost fluid
method. Furthermore, a discretisation of the diffusive forces is proposed,
which leads to recovering the continuous one-velocity solution as the interface
shear layer is resolved.
  The proposed two-velocity formulation is validated and compared to our
one-velocity formulation, where we consider a multitude of two-phase flow
problems. It is demonstrated that the proposed two-velocity model is able to
consistently, and sharply, approximate solutions to the inviscid Euler
equations, where velocity discontinuities appear analytically as well.
Furthermore, the proposed two-velocity model is shown to accurately model the
interface shear layer in viscous problems, and it is successfully applied to
the simulation of breaking waves where the model was used to sharply capture
free surface instabilities.
