A one-dimensional hydrodynamic model that mimicked a two-dimensional fluidized granular system with a vibrating and without gravity, and exhibited a phase separation, was investigated. The objective was to achieve an explicit derivation of the van der Waals normal form (VDWNF) from the hydrodynamic model. The VDWNF was deduced close to the critical point by means of an adiabatic elimination the temperature. The VDWNF was the equation that described the slow dynamics of the system and predicted the qualitative behavior in different regions of parameters. The origin of the effective viscosity and the spatial saturation at the onset of the bifurcation were also discussed.
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The authors thank P. Cordero, S. Rica, and E. Tirapegui for fruitful discussions. The simulation software developed at INLN, France, has been used for the simulations of model . The simulations of the two-dimensional hydrodynamic model were done in the parallel cluster of CIMAT. M.G.C. and R.S. acknowledge the support of FONDAP Grant No. 11980002. M.G.C. and R.S. are very grateful for the support of FONDECYT Projects No. 1020782 and No. 1030993, respectively. C.C. acknowledges the support of a DIPUC grant.