A kinetic interpretation of thermomechanical restrictions of continua

TL;DR

This paper links thermodynamic continuum mechanics with kinetic theory, proposing a hybrid approach that uses Chapman--Enskog expansion and the Rajagopal--Srinivasa principle to derive constitutive laws.

Contribution

It establishes a kinetic interpretation of the maximal entropy production principle and introduces a hybrid method combining Chapman--Enskog and thermodynamic principles.

Findings

Equivalent minimal relaxation-time principle for Bhatnagar--Gross--Krook approximation.

Standard Euler and Navier--Stokes--Fourier laws recovered for gases.

Alternative selection procedures can improve modeling of liquid crystals.

Abstract

Rajagopal and Srinivasa's thermodynamic framework derives constitutive relations in continuum mechanics from two scalar functions describing energy storage and entropy production via a constrained optimization principle. In parallel, kinetic theory obtains constitutive laws through moment closure, most notably via the Chapman--Enskog expansion. This work has three objectives. First, we establish a connection between these approaches by providing a kinetic interpretation of the Rajagopal--Srinivasa principle of maximal entropy production, under appropriate albeit restrictive hypotheses. For a Bhatnagar--Gross--Krook-type approximation, we show that the Rajagopal--Srinivasa principle is equivalent to a minimal relaxation-time principle, selecting among admissible constitutive responses the one with the fastest compatible relaxation toward equilibrium. Second, we review the classical kinetic description of continua in a manner accessible to those familiar with continuum thermodynamics. Third, we propose a hybrid Chapman--Enskog--Rajagopal--Srinivasa approach which computes the thermodynamic relations and entropy production from the Chapman--Enskog expansion, and then invokes the Rajagopal--Srinivasa principle to determine the other constitutive relations. This recovers the standard Euler and Navier--Stokes--Fourier constitutive laws for monatomic gases. We also demonstrate how different choices of selection procedure can be more informative than the classical Chapman--Enskog closure in the context of an inviscid compressible Leslie--Ericksen model arising in liquid crystals.