Binary Fluids with Long Range Segregating Interaction I: Derivation of Kinetic and Hydrodynamic Equations

TL;DR

This paper derives kinetic and hydrodynamic equations for a binary fluid with long-range interactions, describing phase segregation and fluid dynamics through systematic scalings and expansions.

Contribution

It introduces a formal derivation of Vlasov-Boltzmann, Vlasov-Euler, and Vlasov-Navier-Stokes equations for a binary fluid with long-range interactions, linking microscopic interactions to macroscopic behavior.

Findings

Derivation of mesoscopic Vlasov-Boltzmann equation for phase segregation.

Obtaining Vlasov-Euler and Vlasov-Navier-Stokes equations via scalings.

Extension to compressible Vlasov-Navier-Stokes equations through Chapman-Enskog expansion.

Abstract

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 long range pair potentials. At low temperatures the equilibrium state of the system is one in which there are two coexisting phases. Under suitable choices of space-time scalings and system parameters we first obtain (formally) a mesoscopic kinetic Vlasov-Boltzmann equation for the one particle position and velocity distribution functions, appropriate for a description of the phase segregation kinetics in this system. Further scalings then yield Vlasov-Euler and incompressible Vlasov-Navier-Stokes equations. We also obtain, via the usual truncation of the Chapman-Enskog expansion, compressible Vlasov-Navier-Stokes equations.