A kinetic interpretation of thermomechanical restrictions of continua

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.