Résumé
n this note, we justify rigorously the formal method proposed in [M. HILLAIRET, J. Math. Fluid Mech. 2007] to derive viscous and compressible multi-component flow equations. We present here a simpler proof than in [D. BRESCH, X. HUANG, Arch. Ration. Mech. Anal. 2011] to show that the homogenized system may be reduced to a viscous and compressible multi-component flow system (with one velocity-field) getting rid of the no-crossing assumption on the partial densities. We also discuss formally why our multi-component system may be seen as a physically-relevant relaxed system for the well-known bi-fluid system with algebraic closure (pressure equilibrium) in the isothermal case.