Van der Waals normal form for a one-dimensional hydrodynamic model

Phase separation in a fluidized granular system is studied. We consider a one-dimensional hydrodynamic model that mimics a two-dimensional fluidized granular system with a vibrating wall and without gravity, which exhibits a phase separation. Close to the critical point, by means of an adiabatic elimination of the temperature, we deduce the van der Waals normal form, which is the equation that describes the slow dynamics of the system and predicts the qualitative behavior in different regions of parameters. This allows us to understand the origin of the effective viscosity and the spatial saturation at the onset of the bifurcation. The hydrodynamic model and van der Waals normal form exhibit a behavior similar to the one observed in molecular dynamics simulations.