Exact Hydrodynamics of the Lattice BGK Equation

TL;DR

This paper introduces an exact, non-Markovian approach using projection operators to derive hydrodynamic equations from the lattice BGK model, surpassing traditional methods in accuracy and applicability to complex fluids.

Contribution

It presents a novel projection operator method that yields exact hydrodynamic difference equations from the lattice BGK model, avoiding Chapman-Enskog limitations.

Findings

Produces exact hydrodynamic difference equations accurate to all orders in Knudsen number.

Can be Taylor expanded to recover standard hydrodynamic equations.

Applicable to complex fluids with sharp gradients where Chapman-Enskog fails.

Abstract

We apply the projection operator formalism to the problem of determining the asymptotic behavior of the lattice BGK equation in the hydrodynamic limit. As an alternative to the more usual Chapman-Enskog expansion, this approach offers many benefits. Most remarkably, it produces absolutely exact, though non-Markovian, hydrodynamic difference equations as an intermediate step. These are accurate to all orders in Knudsen number and hence contain all of the physics of the Burnett equations and beyond. If appropriate, these equations may then be Taylor expanded to second order in Knudsen number to obtain the usual hydrodynamic equations that result from the Chapman-Enskog analysis. The method offers the potential to derive hydrodynamic difference equations for complex fluids with sharp gradients, such as immiscible and amphiphilic flow, for which the assumptions underlying the Chapman-Enskog approach are generally invalid.