# First-order Approximation to the Boltzmann-Curtiss Equations for Flows with Local Spin

## TL;DR

This paper derives a first-order approximation to the Boltzmann-Curtiss equation, modeling particle gyration as an independent variable, and connects it to continuum theories like MCT and Navier-Stokes, providing new physical insights.

## Contribution

It introduces a novel first-order distribution function for particle gyration, linking kinetic theory with continuum mechanics and extending the understanding of local rotation effects in fluid flows.

## Key findings

- Distribution function treats rotation as an independent variable.
- Governing equations match morphing continuum theory and Navier-Stokes.
- New physical parameters and relaxation times are derived for better modeling.

## Abstract

The first-order approximation to the solution of the Boltzmann--Curtiss transport equation is derived. The resulting distribution function treats the rotation or gyration of spherical particles as an independent classical variable, deviating from the quantum mechanical treatment of molecular rotation found in the Wang Chang-Uhlenbeck equation. The Boltzmann-Curtiss equation, therefore, does not treat different rotational motions as separate molecular species. The first-order distribution function yields momentum equations for the translational velocity and gyration that match the form of the governing equations of morphing continuum theory (MCT), a theory derived from the approach of rational continuum thermomechanics. The contribution of the local rotation to the Cauchy stress and the viscous diffusion are found to be proportional to an identical expression based off the relaxation time, number density, and equilibrium temperature of the fluid. When gyration is equated to the macroscopic angular velocity, the kinetic description reduces to the first-order approximation for a classical monatomic gas, and the governing equations match the form of the Navier--Stokes equations. The relaxation time used for this approximation is shown to be more complex due to the additional variable of local rotation. The approach of De Groot and Mazur is invoked to give an initial approximation for the relaxation of the gyration. The incorporation of this relaxation time, and other physical parameters, into the coefficients of the governing equations provides a more in-depth physical treatment of the new terms in the MCT equations, allowing experimenters to test these expressions and get a better understanding of new coefficients in MCT.