Kinetic theory of dilute granular gases with hard-core and inverse power-law potentials under uniform shear flow: comparison with simplified model

TL;DR

This work develops a first-principles kinetic theory for a dilute charged granular gas under uniform shear, incorporating both hard-core and inverse-power-law repulsive interactions via a velocity- and distance-dependent restitution framework. Using Boltzmann dynamics with Grad's moment expansion, it derives closed evolution equations for the temperature, temperature anisotropy, and shear stress, and introduces a practical fitting form for the collision integrals to capture temperature dependence across shear regimes. The theory predicts Bagnold-like scaling at high shear, reveals suppression of inelastic dissipation at intermediate and low shear due to the repulsive potential, and shows excellent agreement with DSMC simulations for key rheological quantities; the velocity distribution remains nearly Maxwellian with modest non-Gaussian features. Overall, the framework provides a tractable, quantitatively accurate description of dilute charged granular rheology and offers a path to extensions to denser systems and more complex charge dynamics.

Abstract

We develop a kinetic-theory framework to investigate the steady rheology of a dilute gas interacting via a repulsive potential under uniform shear flow. Starting from the Boltzmann equation with a restitution coefficient that depends on the impact velocity and potential strength, we derive evolution equations for the stress tensor based on Grad's moment expansion. The resulting expressions for the collisional rates and transport coefficients are fitted with simple analytical functions that capture their temperature dependence over a wide range of shear rates. Comparison with direct simulation Monte Carlo (DSMC) results shows excellent quantitative agreement for the shear stress, temperature anisotropy, and steady viscosity. We also analyze the velocity distribution functions, revealing that the system remains nearly Maxwellian even under strong shear.

Figures (9)

• Figure 1: Schematic of our system. Monodisperse particles are randomly distributed in a cubic box. A shear is applied with the shear rate $\dot\gamma$.
• Figure 2: The interparticle potential used in this paper. The shaded and non-shaded region represents whether a collision is elastic or inelastic.
• Figure 3: A schematic of a scattering process. Two particles are colliding with each other with the impact parameter $b$ and the relative speed $v$.
• Figure 4: Dependence of the collision angle $\theta_\alpha$ on the relative speed $v$ and impact parameter $b$ for $\alpha=4$. The color scale represents the magnitude of the angle. The dashed line represents Eq. \ref{['eq:b_boundary']}.
• Figure 5: Temperature dependence of $\Omega_{\alpha, n}^{(i)}$ for various sets of $(i, \alpha, e)$ with $\alpha=4$, where $\Omega_{\infty,n}^{(i)}$ represents the hard-core limit of $\Omega_{\alpha, n}^{(i)}$.