[Paper Review] Hydrodynamics for a granular mixture at low density

Vicente Garzó, James W. Dufty|arXiv (Cornell University)|May 21, 2001

Granular flow and fluidized beds8 citations

TL;DR

This paper develops hydrodynamic equations for a low-density binary mixture of inelastic hard spheres using the Chapman-Enskog method applied to the Boltzmann equation. It identifies transport coefficients—diffusion, pressure diffusion, and thermal diffusion—showing they depend on restitution coefficients, mass, size, and concentration ratios, with unique contributions from component-specific partial temperatures due to inelasticity.

ABSTRACT

Hydrodynamic transport in a binary mixture of inelastic hard spheres is analyzed in the context of the Boltzmann equation. A normal solution is obtained via the Chapman-Enskog method for states near the local homogeneous cooling state. The mass, heat, and momentum fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. In the same way as for binary mixtures with elastic collisions, these coefficients are determined from a set of coupled linear integral equations. Practical evaluation is possible using a Sonine polynomial approximation, and is illustrated here by explicit calculation of the diffusion, pressure diffusion, and thermal diffusion coefficients as functions of the restitution coefficients and the ratios of mass, concentration, and particle sizes. Interesting and new effects arise from the fact that the reference states for the two components have different partial temperatures, leading to additional dependencies of the transport coefficients on the concentration. The results hold for arbitrary degree of inelasticity and are not limited to specific values 1 of the parameters of the mixture. PACS number(s): 45.70.Mg, 05.20.Dd, 51.10.+y

Motivation & Objective

• To extend hydrodynamic theory to binary mixtures of inelastic hard spheres near the homogeneous cooling state.
• To determine mass, heat, and momentum fluxes in the presence of spatial gradients in hydrodynamic fields.
• To identify transport coefficients that account for inelasticity and component-specific partial temperatures.
• To provide a framework valid for arbitrary inelasticity and mixture parameter ratios, not restricted to specific values.
• To enable practical computation of transport coefficients using a Sonine polynomial approximation.

Proposed method

• Applies the Chapman-Enskog method to the Boltzmann equation for inelastic hard spheres.
• Derives normal solution expansions up to first order in spatial gradients of hydrodynamic fields.
• Solves for transport coefficients via a set of coupled linear integral equations derived from the Boltzmann equation.
• Uses Sonine polynomial approximation to make the integral equations numerically tractable.
• Evaluates diffusion, pressure diffusion, and thermal diffusion coefficients as functions of restitution coefficients, mass, size, and concentration ratios.
• Accounts for differences in partial temperatures between components, which introduces new dependencies in transport coefficients.

Experimental results

Research questions

• RQ1 How do transport coefficients in a granular binary mixture depend on inelasticity and mixture composition?
• RQ2 What role do component-specific partial temperatures play in modifying hydrodynamic transport?
• RQ3 How do mass, size, and concentration ratios affect diffusion and thermal diffusion in inelastic mixtures?
• RQ4 Can the Chapman-Enskog method be consistently applied to inelastic granular mixtures to derive hydrodynamic fluxes?
• RQ5 What is the structure of the transport coefficients when the reference state has different partial temperatures for each component?

Key findings

• The transport coefficients—diffusion, pressure diffusion, and thermal diffusion—are explicitly calculated as functions of restitution coefficients, mass, size, and concentration ratios.
• The presence of different partial temperatures for the two components introduces new dependencies in the transport coefficients not present in elastic mixtures.
• The Sonine polynomial approximation enables practical evaluation of the integral equations governing the transport coefficients.
• The results are valid for arbitrary degrees of inelasticity and do not require limiting assumptions on mixture parameters.
• The framework extends hydrodynamic theory to inelastic granular mixtures in a way that captures non-equilibrium effects due to energy dissipation.
• The analysis reveals that inelasticity and component differences significantly alter the structure of fluxes and transport coefficients compared to elastic systems.