A Theoretical Investigation of the Maxwellian Velocity Distribution with Connection to Continuum Transport Phenomena

TL;DR

This paper provides a theoretical framework connecting Maxwellian velocity distribution to continuum transport phenomena, deriving fluid equations from statistical mechanics and establishing benchmarks for simple flow models.

Contribution

It introduces a modified statistical mechanical approach to derive Euler and Navier-Stokes equations, linking microscopic distributions to macroscopic continuum transport.

Findings

Derived fluid equations from Maxwellian velocity distribution

Established benchmark problems for simple flow cases

Provided insights into the mathematical structure of transport equations

Abstract

The Euler and Navier-Stokes fluid mechanics equations are derived using a modified statistical mechanical approach using theory taken from the Chapman-Enskog perturbation analysis used to support the lattice Boltzmann method. Additional distributions such as the velocity vector and total scalar energy distributions are established. A complete discretization in velocity space is provided. Benchmark problems are established for simple cases modeling isothermal compressible inviscid flows. Thermal viscous flows will be the focus of future work. Overall, a more fundamental description of mass, momentum, and energy transport is uncovered and provides insights into the mathematical nature of the continuum transport equations such as the incorporation of viscosity and thermal conductivity into space and time dependence.