Atomization of a liquid jet is a challenge for Direct Numerical Simulation due to a wide range of length scales produced by the dynamics of interfaces. In this paper, we present the circulation production at an interface and its importance for energy dissipation. We point out the major role played by surface tension at interfaces on both production and dissipation. Evaluating numerically circulation production during liquid breakup is indeed critical to predict the vorticity generated in the overall atomization process. In order to shed light on the necessary conditions to resolve all scales in multiphase problems, we study the influence of surface tension, viscosity and grid resolution on two interface problems : the breakup of a liquid ligament by Rayleigh-Plateau instability and the liquid jet injection. Poorly resolved simulations generally tend to underestimate the total dissipation but in other cases, numerical discretization errors can lead to a large unphysical vorticity production overestimating the energy dissipation and deteriorating the quality of the numerical solution.