Binary fluids with long range segregating interaction. I: Derivation of kinetic and hydrodynamic equations

We study the evolution of a two component fluid consisting of "blue" and '' red" particles which interact via strong short range (hard core) and weak l ong range pair potentials. At low temperatures the equilibrium state of the system is one in which there are two coexisting phases. Under suitable cho ices of space-time scalings and system parameters we first obtain (formally ) a mesoscopic kinetic Vlasov-Boltzmann equation for the one particle posit ion and velocity distribution functions, appropriate for a description of t he phase segregation kinetics in this system, Further scalings then yield V lasov-Euler and incompressible Vlasov-Navier-Stokes equations. We also obta in, via the usual truncation of the Chapman-Enskog expansion, compressible Vlasov-Navier-Stokes equations.