Performing Direct Numerical Simulation of atomization processes is a challenge due to the presence of different length scales appearing as a consequence of interfaces. This is why it is difficult to capture numerically vorticity production or energy dissipated during the atomization of a liquid jet. In this work, we discuss the various sources of vorticity in a general multiphase flow problem showing that surface tension can play a major role. We use numerical simulations of simplified flows in order to clarify the influence of surface tension, viscosity and grid resolution to capture the mechanisms responsible for vorticity fluxes at interfaces. We show that correctly capturing vorticity fluxes during liquid breakup is critical to correctly predict the vorticity generated in the overall atomization process. The DNS of the breakup of a liquid ligament by the Rayleigh-Plateau instability is used to shed light into the conditions that must be satisfied to correctly resolve all the scales of the flow in multiphase flow problems. Under some conditions, poorly resolved simulations tend to underestimate the total dissipation. However we also observe that in some other configurations where it is not possible to resolve all flow scales numerical discretization errors can lead to a large unphysical vorticity production overestimating the energy dissipation and significantly polluting the quality of the solution obtained.