Entropy Analysis for a Nonlinear Fluid with a Nonlinear Heat Flux Vector

Hyunjin Yang,Mehrdad Massoudi,A D Kirwan

doi:10.3390/e19120689

Hyunjin Yang, Mehrdad Massoudi + Show 1 more

Open Access

https://doi.org/10.3390/e19120689

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Abstract

Flowing media in both industrial and natural processes are often characterized as assemblages of densely packed granular materials. Typically, the constitutive relations for the stress tensor and heat flux vector are fundamentally nonlinear. Moreover, these equations are coupled through the Clausius–Duhem inequality. However, the consequences of this coupling are rarely studied. Here we address this issue by obtaining constraints imposed by the Clausius–Duhem inequality on the constitutive relations for both the stress tensor and the heat flux vector in which the volume fraction gradient plays an important role. A crucial result of the analysis is the restriction on the dependency of phenomenological coefficients appearing in the constitutive equations on the model objective functions.

Highlights

Complex or nonlinear fluids such as coal slurries, polymers, and drilling fluids are fundamental to many industrial processes
Based on the restrictions imposed by the Clausius–Duhem inequality, the principle of material frame-indifference, incompressibility of the grains, and the assumption that the constitutive parameters depend on the temperature gradient, velocity gradient, and gradient of the volume fraction, in a linear fashion, Goodman and Cowin derived a constitutive equation for the stress tensor
The basic issue is whether a single constitutive relation for T and a single constitutive relation q are sufficient to describe the behavior of granular materials and situations encountered in many engineering applications

Summary

Introduction

Complex or nonlinear fluids such as coal slurries, polymers, and drilling fluids are fundamental to many industrial processes. In many engineering applications, it is possible to treat these fluids as suspensions and model them as nonlinear—and sometimes non-homogeneous—single-component fluids This avoids the necessity of specifying numerous phenomenological coefficients to characterize constituent interactions. The theory of Goodman and Cowin is applicable to situations where the stress levels are smaller than 10 psi They assumed that the pneumatic effects can be neglected; that is, the effects of the interstitial gas were ignored. Based on the restrictions imposed by the Clausius–Duhem inequality, the principle of material frame-indifference, incompressibility of the grains, and the assumption that the constitutive parameters depend on the temperature gradient, velocity gradient, and gradient of the volume fraction, in a linear fashion, Goodman and Cowin derived a constitutive equation for the stress tensor. The report concludes with a summary of our results and a commentary on further work in this area

The Governing Equations

The Constitutive Equations

Entropy Analysis

Discussion