  We propose a kinetic framework for single-component non-ideal isothermal
flows. Starting from a kinetic model for a non-ideal fluid, we show that under
conventional scaling the Navier-Stokes equations with a non-ideal equation of
state are recovered in the hydrodynamic limit. A scaling based on the smallness
of velocity increments is then introduced, which recovers the full
Navier-Stokes-Korteweg equations. The proposed model is realized on a standard
lattice and validated on a variety of benchmarks. Through a detailed study of
thermodynamic properties including co-existence densities, surface tension,
Tolman length and sound speed, we show thermodynamic consistency,
well-posedness and convergence of the proposed model. Furthermore, hydrodynamic
consistency is demonstrated by verification of Galilean invariance of the
dissipation rate of shear and normal modes and the study of visco-capillary
coupling effects. Finally, the model is validated on dynamic test cases in
three dimensions with complex geometries and large density ratios such as drop
impact on textured surfaces and mercury drops coalescence.
