Rigorous hydrodynamics from linear Boltzmann equations and viscosity-capillarity balance

Area of Science:

• Fluid dynamics
• Statistical mechanics
• Kinetic theory

Background:

• Deriving accurate hydrodynamic equations from kinetic theory is challenging.
• Existing models often lack rigorous justification or are limited in scope.
• Understanding dissipation and non-local effects is crucial for complex fluid behaviors.

Purpose of the Study:

• To rigorously derive exact closure for hydrodynamic variables from the linear Boltzmann equation.
• To develop a unique, optimal reduction in phase space near equilibrium.
• To investigate the implications for entropy modification and dissipation in hydrodynamic systems.

Main Methods:

• Spectral theory and analysis of eigenvector properties.
• Theory of slow manifolds for phase space reduction.
• Modification of entropy within a hydrodynamically constrained system.

Main Results:

• A unique, optimal reduction in phase space near equilibrium was defined.
• A modified entropy ensuring pure dissipation on the hydrodynamic manifold was established.
• The derived equations were exemplified using the Knudsen minimum paradox in channel flow.

Conclusions:

• The study provides a rigorous foundation for hydrodynamic equations derived from kinetic theory.
• The findings offer a nonlocal variant of Korteweg's theory, linking viscosity and capillarity.
• The approach successfully explains complex phenomena like the Knudsen minimum paradox.