Rigorous Hydrodynamics from Linear Boltzmann Equations and Viscosity-Capillarity Balance

TL;DR

This paper rigorously derives hydrodynamic equations from the linear Boltzmann equation using spectral theory and slow manifold techniques, revealing a viscosity-capillarity balance and applying it to the Knudsen minimum paradox.

Contribution

It introduces a novel, exact closure for hydrodynamics from kinetic theory, incorporating a non-local viscosity-capillarity balance with a rigorous mathematical foundation.

Findings

Derived a unique hydrodynamic closure from the linear Boltzmann equation.

Identified a non-local viscosity-capillarity balance consistent with entropy principles.

Applied the theory to explain the Knudsen minimum paradox in channel flow.

Abstract

An exact closure for hydrodynamic variables is rigorously derived from the linear Boltzmann kinetic equation. Our approach, based on spectral theory, structural properties of eigenvectors and the theory of slow manifolds, allows us to define a unique, optimal reduction in phase space close to equilibrium. The hydrodynamically constrained system induces a modification of entropy that ensures pure viscous dissipation on the hydrodynamic manifold, which is interpreted as a non-local variant of Korteweg's theory of viscosity-capillarity balance. The rigorous hydrodynamic equations are exemplified on the Knudsen minimum paradox in a channel flow.