Coupled constitutive relations: a second law based higher order closure for hydrodynamics

TL;DR

This paper introduces coupled constitutive relations based on the second law of thermodynamics to extend the validity of hydrodynamic equations, enabling the modeling of rarefaction effects beyond classical Navier-Stokes-Fourier limits.

Contribution

It develops a nonlinear, coupled thermodynamic framework that captures rarefaction phenomena, extending classical hydrodynamics with second law consistent boundary conditions.

Findings

Successfully models Knudsen paradox and transpiration flows.

Predicts heat flux without temperature gradients.

Demonstrates applicability through benchmark problems.

Abstract

In the classical framework, the Navier-Stokes-Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic description is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier-Stokes-Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which can not be predicted by the classical Navier-Stokes-Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.