Thermodynamically Consistent Evolution Equations in Continuum Mechanics

Abstract

This paper addresses the modelling of material behaviour in terms of differential (or rate) equations. To comply with the objectivity principle, recourse is made to invariant fields in the Lagrangian description or to objective time derivatives in the Eulerian description. The thermodynamic consistency is investigated in terms of the Clausius–Duhem inequality with two unusual features. Firstly, the (non-negative) entropy production is viewed as a constitutive function per se. Secondly, the inequality is viewed as a constraint on the pertinent fields and it is solved by using a representation formula, which allows for the the admissibility of a class of models. For definiteness, models of heat conduction are established, within Lagrangian descriptions, while models of the Navier–Stokes–Voigt fluid are investigated within Eulerian descriptions. In connection with thermo-viscous fluids, evolution equations are investigated within the Eulerian description. It is shown that the thermodynamic consistency is compatible with both objective and non-objective evolution equations.

1. Introduction

The constitutive properties of materials are often described through evolution equations whereby the time derivative of suitable fields is eventually related to appropriate fields and their derivatives of various order. The dependence on the space and time derivatives is a way of accounting for non-local properties in space and time. Relative to the description of response of the material in terms of space and time integrals, differential evolution equations show the advantage of greater flexibility.

Starting from rheological models [1,2], examples of differential equations are settled in the mechanical domain. For example, in connection with viscoelastic materials a simple equation is that associated with the Maxwell model. In the three-dimensional setting, we can write the stress–strain relation in the form,

$$\dot{\mathbf{T}}=E\dot{\mathbf{ϵ}}-\lambda \mathbf{T},$$

(1)

where $\mathbf{T}$ is the stress tensor and $\mathbf{ϵ}$ is the (infinitesimal) strain tensor. Likewise, the Kelvin–Voigt model of fluids is expressed by

$$\dot{\mathbf{D}}=a_1\mathbf{D}+a_2\mathcal{T},$$

(2)

where $\mathcal{T}$ is the viscous stress and $\mathbf{D}$ is the stretching tensor. Here, $E,\lambda$ and $a_1,a_2$ are constants; we might view these coefficients as functions of temperature and mass density. Other models may involve higher-order time differentiation, as seen in the Burgers material, which involves second-order time derivatives (see, e.g., [3], ch. 4).

As for thermal models, it is worth mentioning the Maxwell–Cattaneo equation [4,5,6,7]

$$\dot{\mathbf{q}}+\lambda \nabla \theta +\nu \mathbf{q}=\mathbf{0}$$

(3)

for the temperature $\theta$ and the heat flux vector $\mathbf{q}$. In nanoscale systems continuum models might be applicable provided non-local and/or rate properties are properly described. The evolution is often modelled by the Guyer–Krumhansl [8,9] equation, namely

$$\dot{\mathbf{q}}+\lambda \nabla \theta +\nu \mathbf{q}-\zeta (\Delta \mathbf{q}+\nabla (\nabla ·\mathbf{q}))=\mathbf{0},$$

(4)

where $\mathbf{q}$ is the heat flux vector, ∇ is the gradient operator, $\Delta$ is the Laplacian, and the superposed dot denotes the material time derivative. Equation (4) is then a generalization of (18) with the dependence of $\dot{\mathbf{q}}$ on the higher-order spatial derivatives $\Delta \mathbf{q}$ and $\nabla (\nabla ·\mathbf{q})$.

Within continuum mechanics, the constitutive properties are required to be objective, meaning they remain form-invariant under the set of Euclidean transformations, SO(3). However, Equations (1)–(4) are not objective, merely because the time derivative is not objective. Hence, these equations deserve an objective form. Furthermore, the physical admissibility of any constitutive equation, like, e.g., (1)–(4), is required to be consistent with the second law of thermodynamics. Consequently, we need a procedure to ascertain the physical admissibility of constitutive equations.

The physical admissibility is by now a well-known topic; however, it is performed in various ways. A formal statement about the thermodynamic consistency traces back to Coleman and Noll [10]. Next, Müller [11] generalized the statement by letting the entropy flux be a constitutive quantity to be determined. Lately, a further generalization has been accomplished by regarding also the entropy production as a constitutive quantity per se ([2], ch. 3).

The purpose of this paper is to establish a systematic procedure to ascertain whether given constitutive equations are thermodynamically consistent. While this is classical in the literature, albeit with technical variants, this paper shows how the outcome is in general a set of constitutive equations that possibly includes a selected equation, such as Equations (1)–(4). Mathematically, the procedure is made operative by using a representation formula that determines a wide class of equations once a set of variables is chosen depending on the given particular equation.

This paper revisits the thermodynamic approach and emphasizes the operative character of the second law in connection with physically admissible models. Next, some significant examples of evolution equations are investigated, both in the Lagrangian and in the Eulerian description.

2. Notation and Balance Equations

We denote by $\Omega$, the time-dependent, three-dimensional, region occupied by the body under consideration. The position vector of a point in $\Omega$ is denoted by $\mathbf{x}$. Hence, $\rho (\mathbf{x},t)$ and $\mathbf{v}(\mathbf{x},t)$ are the mass density and the velocity fields at $\mathbf{x}$, at time $t\in \mathbb{R}$. The symbol ∇ denotes the gradient, with respect to $\mathbf{x}$, while $\nabla ·$ is the divergence operator. For any pair of vectors $\mathbf{u},\mathbf{w}$, or tensors $\mathbf{A},\mathbf{B}$, the notations $\mathbf{u}·\mathbf{w}$ and $\mathbf{A}·\mathbf{B}$ denote the inner product. Cartesian coordinates are used and then, in the suffix notation, $\mathbf{u}·\mathbf{w}=u_i{w}_i$, $\mathbf{A}·\mathbf{B}=A_{ij}{B}_{ij}$, the summation over repeated indices being understood. Also, $\mathrm{sym}\mathbf{A}$ and $\mathrm{skw}\mathbf{A}$ denote the symmetric and skew-symmetric parts of $\mathbf{A}$, while $\mathrm{Sym}$ is the space of symmetric tensors. A superposed dot denotes the total time derivative and hence, for any function $f(\mathbf{x},t)$ on $\Omega \times \mathbb{R}$, we have $\dot{f}={\partial }_{t}f+(\mathbf{v}·\nabla )f$. The symbol $\mathbf{L}$ denotes the velocity gradient, $L_{ij}={\partial }_{x_j}{v}_i$, while $\mathbf{D}=\mathrm{sym}\mathbf{L}$ and $\mathbf{W}=\mathrm{skw}\mathbf{L}$. Further, $\mathbf{T}$ is the Cauchy stress tensor, $\mathbf{b}$ is the specific body force, and ⊗ denotes the dyadic product.

2.1. Balance Equations

The conservation of mass is expressed locally by the continuity equation

$$\dot{\rho }+\rho \nabla ·\mathbf{v}=0.$$

The balance of linear momentum leads to the equation of motion

$$\rho \dot{\mathbf{v}}=\nabla ·\mathbf{T}+\rho \mathbf{b},$$

(5)

where ${(\nabla ·\mathbf{T})}_i={\partial }_{x_j}{T}_{ij}$. The balance of angular momentum implies that $\mathbf{T}={\mathbf{T}}^T$. To obtain the balance of energy, we consider the specific energy density ${\displaystyle \frac{1}{2}}{\mathbf{v}}^2+\epsilon$ (per unit mass) and the power $(\mathbf{T}\mathbf{n})·\mathbf{v}-\mathbf{q}·\mathbf{n}$ per unit area, with unit external normal $\mathbf{n}$, and $\mathbf{b}·\mathbf{v}+r$ per unit mass; $\epsilon ,\mathbf{q},r$ are quantities of a non-mechanical character. In light of the equation of motion (5), it follows that the internal energy $\epsilon$ is subject to the local equation

$$\rho \dot{\epsilon }=\mathbf{T}·\mathbf{D}-\nabla ·\mathbf{q}+\rho r.$$

(6)

Let $\eta$ be the specific entropy density, $\mathbf{j}$ the entropy flux, and $r/\theta$ the entropy supply. The balance of entropy leads to

$$\rho \dot{\eta }+\nabla ·\mathbf{j}-{\displaystyle \frac{\rho r}{\theta }}=\rho \gamma ,$$

where $\gamma$ is the (rate of) entropy production. It is assumed that $\gamma \ge 0$, which implies the increase of entropy for isolated systems ($\mathbf{j}=\mathbf{0},r=0$).

We assume $\mathbf{b}$ and r are arbitrarily given time-dependent fields on $\Omega \times \mathrm{R}$. A process is the set $\mathcal{P}=(\rho ,\mathbf{v},\mathbf{T},\epsilon ,\eta ,\theta ,\mathbf{q},\mathbf{j},\gamma )$ entering the balance equations. Since the number of components in a process is larger than the number of balance equations, then a corresponding set of constitutive equations are needed to make the two numbers equal. Hence, the balance equations provide $\dot{\rho },\dot{\mathbf{v}},\dot{\epsilon }$; the assumed arbitrariness of $\mathbf{b}(\mathbf{x},t)$ and $r(\mathbf{x},t)$ allows us to view $\dot{\mathbf{v}}$, $\dot{\epsilon }$, and their derivatives as arbitrary.

2.2. Second Law of Thermodynamics

The second law of thermodynamics is assumed as follows.

• Postulate.For every process $\mathcal{P}$ admissible in a body, the inequality

  $$\rho \dot{\eta }+\nabla ·\mathbf{j}-{\displaystyle \frac{\rho r}{\theta }}=\rho \gamma \ge 0$$

  (7)

  is valid at any internal point.

Henceforth, we refer to Equation (7) as the Clausius–Duhem (CD) inequality. It is convenient to let

$$\mathbf{j}={\displaystyle \frac{\mathbf{q}}{\theta }}+\mathbf{k},\psi =\epsilon -\theta \eta ;$$

$\mathbf{k}$ is said to be the extra-entropy flux and $\psi$ the Helmholtz free energy. Substitution of $\nabla ·\mathbf{q}-\rho r$ from (6) makes (7) in the form

$$-\rho (\dot{\psi }+\eta \dot{\theta })+\mathbf{T}·\mathbf{D}+\theta \nabla ·\mathbf{k}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma \ge 0.$$

(8)

This is the thermodynamic scheme for fields in the Eulerian description, where $\Omega \times \mathbb{R}$ is the space–time domain of the process. For later use, we now establish the corresponding relations in the space–time domain $\mathrm{R}\times \mathbb{R}$, where $\mathrm{R}$ is the region $\Omega$ at some convenient time. To fix ideas, let $\mathrm{R}={\Omega }_{t=0}$.

2.3. Lagrangian Description

Let $\mathbf{X}$ be the position vector of a point in $\mathrm{R}$ and hence, $\mathbf{x}(\mathbf{X},t)$ is the motion. We denote by $\nabla R={\partial }_{\mathbf{X}}$ the gradient operator in $\mathrm{R}$. We let

$$\mathbf{F}={\nabla }_R\mathbf{x},F_{iK}={\partial }_{X_K}{x}_i,$$

be the deformation gradient and hence, for any differentiable function $f(\mathbf{x},t)=f(\mathbf{x}(\mathbf{X},t),t)$, we have

$$\nabla Rf=(\nabla f)\mathbf{F}.$$

Define

$$J=det\mathbf{F}>0,$$

and let

$$\mathbf{E}=\frac{1}{2}({\mathbf{F}}^T\mathbf{F}-\mathbf{1})$$

be the Green–Lagrange strain tensor. Hence,

$$\dot{\mathbf{F}}=\mathbf{L}\mathbf{F},\dot{\mathbf{E}}={\mathbf{F}}^T\mathbf{D}\mathbf{F}.$$

Let

$${\mathbf{T}}_{RR}=J{\mathbf{F}}^{-1}\mathbf{T}{\mathbf{F}}^{-T},{\mathbf{q}}_R=J\mathbf{q}{\mathbf{F}}^{-T};$$

${\mathbf{T}}_{RR}$ is the second Piola (or Piola–Kirchhoff) stress while ${\mathbf{q}}_R$ is the heat flux in $\mathrm{R}$. We can prove that ([2], ch. 1; [12], §25)

$$J\mathbf{T}·\mathbf{D}={\mathbf{T}}_{RR}·\dot{\mathbf{E}},J\mathbf{q}·\nabla \theta ={\mathbf{q}}_R·\nabla R\theta ,$$

Furthermore, letting ${\mathbf{q}}_R=J\mathbf{q}{\mathbf{F}}^{-T}$m, we have

$$J\nabla ·\mathbf{k}=\nabla R·{\mathbf{k}}_R.$$

Since $J\rho ={\rho }_R$ is the mass density in the reference configuration $\mathrm{R}$, then J times the CD inequality (8) yields

$$-{\rho }_R(\dot{\psi }+\eta \dot{\theta })+{\mathbf{T}}_{RR}·\dot{\mathbf{E}}+\theta \nabla R·{\mathbf{k}}_R-{\displaystyle \frac{1}{\theta }}{\mathbf{q}}_R·\nabla R\theta ={\rho }_R\theta \gamma \ge 0.$$

(9)

Two points characterize the present approach. Firstly, the entropy production $\gamma$ is given by a constitutive equation per se as is the case for the flux ${\mathbf{k}}_R$. Secondly, the constitutive equation for ${\dot{\mathbf{q}}}_R$ is not assumed a priori but is derived as a consequence of the other constitutive equations and the CD inequality. In particular, this shows how ${\mathbf{k}}_R$ and $\gamma$ influence the equation for ${\mathbf{q}}_R$.

2.4. Representation Formula

Depending on the set of variables, the CD inequality (8) or (9) has the form

$$\mathbf{w}·\mathbf{z}=f,$$

(10)

in the unknown $\mathbf{z}$ with $\mathbf{w},\mathbf{z}$ as vectors or tensors. For example, if $\psi$ depends on ${\mathbf{q}}_R$, then (9) eventually has the form (10) for the unknown $\mathbf{z}={\dot{\mathbf{q}}}_R$, with

$$\mathbf{w}={\rho }_R{\partial }_{{\mathbf{q}}_R}\psi ,f=-{\rho }_R\theta \gamma +\theta \nabla ·{\mathbf{k}}_R-{\displaystyle \frac{1}{\theta }}{\mathbf{q}}_R·\nabla R\theta .$$

Notice that, for any unit vector $\mathbf{n}$, a vector $\mathbf{z}$ can be represented in the form

$$\mathbf{z}=(\mathbf{z}·\mathbf{n})\mathbf{n}+{\mathbf{z}}_{\perp },$$

where $(\mathbf{z}·\mathbf{n})\mathbf{n}$ is the longitudinal part (in the direction $\mathbf{n}$) while ${\mathbf{z}}_{\perp }$ is the transverse part, ${\mathbf{z}}_{\perp }·\mathbf{n}=0$. If only $\mathbf{z}·\mathbf{n}$ is known, say $\mathbf{z}·\mathbf{n}=g$, then

$$\mathbf{z}=g\mathbf{n}+{\mathbf{z}}_{\perp }.$$

Any vector ${\mathbf{z}}_{\perp }$ perpendicular to $\mathbf{n}$ can be represented in the form

$${\mathbf{z}}_{\perp }=(\mathbf{1}-\mathbf{n}\otimes \mathbf{n})\mathbf{\xi },$$

for any vector $\mathbf{\xi }$, where $\mathbf{1}$ is the identity tensor. This is so in that

$$[(\mathbf{1}-\mathbf{n}\otimes \mathbf{n})\mathbf{\xi }]·\mathbf{n}=\mathbf{\xi }·\mathbf{n}-(\mathbf{\xi }·\mathbf{n})\mathbf{n}·\mathbf{n}=0.$$

Back to (10), we have

$$\mathbf{z}={\displaystyle \frac{\mathbf{z}·\mathbf{w}}{{\left|\mathbf{w}\right|}^2}}\mathbf{w}+(\mathbf{1}-{\displaystyle \frac{\mathbf{w}\otimes \mathbf{w}}{{\left|\mathbf{w}\right|}^2}})\mathbf{\xi }.$$

(11)

If $\mathbf{w}$ and $\mathbf{z}$ are tensors, say

$$\mathbf{U}·\mathbf{Z}=f,$$

then we repeat the formal steps to conclude that

$$\mathbf{Z}={\displaystyle \frac{\mathbf{Z}·\mathbf{U}}{{\left|\mathbf{U}\right|}^2}}\mathbf{U}+\left(\mathsf{I}-{\displaystyle \frac{\mathbf{U}\otimes \mathbf{U}}{{\left|\mathbf{U}\right|}^2}}\right)\Xi ,$$

(12)

where $\mathsf{I}$ is the fourth-order identity tensor and $\Xi$ is any second-order tensor.

3. Thermodynamic Restrictions

Our purpose is to determine evolution equations of ${\mathbf{q}}_R$ consistent with thermodynamics and hence, we look for a constitutive equation of ${\dot{\mathbf{q}}}_R$. Possible nonlocal effects are described by a suitable dependence on the spatial derivatives. For definiteness, we direct our attention to a dependence on $\nabla R{\mathbf{q}}_R$ and $\nabla R\nabla R{\mathbf{q}}_R$. A dependence on the strain $\mathbf{E}$ is allowed to account for the elastic properties of the solid. Hence, we let

$$\Gamma =(\theta ,\mathbf{E},{\mathbf{q}}_R,\nabla R\theta ,\nabla R{\mathbf{q}}_R,\nabla R\nabla R{\mathbf{q}}_R)$$

be the set of independent variables and assume $\eta ,{\mathbf{T}}_{RR},{\mathbf{k}}_R,\gamma$, and ${\dot{\mathbf{q}}}_R$ are continuous functions of $\Gamma$, while $\psi$ is continuously differentiable.

Compute $\dot{\psi }$ and substitute in (9) to obtain

$$\begin{array}{c}\hfill -{\rho }_R({\partial }_{\theta }\psi +\eta )\dot{\theta }+({\mathbf{T}}_{RR}-{\rho }_R{\partial }_{\mathbf{E}}\psi )·\dot{\mathbf{E}}-{\rho }_R{\partial }_{\nabla R\theta }\psi ·\nabla R\dot{\theta }-{\rho }_R{\partial }_{\nabla R{\mathbf{q}}_R}\psi ·\nabla R{\dot{\mathbf{q}}}_R\\ \hfill -{\rho }_R{\partial }_{\nabla R\nabla R{\mathbf{q}}_R}\psi ·\nabla R\nabla R{\dot{\mathbf{q}}}_R-{\rho }_R{\partial }_{{\mathbf{q}}_R}\psi ·{\dot{\mathbf{q}}}_R+\theta {\nabla }_R·{\mathbf{k}}_R-{\displaystyle \frac{1}{\theta }}{\mathbf{q}}_R·\nabla R\theta ={\rho }_R\theta \gamma \ge 0.\end{array}$$

(13)

The linearity and arbitrariness of $\nabla R\dot{\theta },\nabla R{\dot{\mathbf{q}}}_R,\nabla R\nabla R{\dot{\mathbf{q}}}_R,\dot{\theta },\dot{\mathbf{E}}$, imply that

$${\partial }_{\nabla R\theta }\psi =\mathbf{0},{\partial }_{\nabla R{\mathbf{q}}_R}\psi =\mathbf{0},{\partial }_{\nabla R\nabla R{\mathbf{q}}_R}\psi =\mathbf{0},$$

$$\eta =-{\partial }_{\theta }\psi ,{\mathbf{T}}_{RR}={\partial }_{\mathbf{E}}\psi .$$

(14)

Hence, (13) simplifies to

$$-{\rho }_R{\partial }_{{\mathbf{q}}_R}\psi ·{\dot{\mathbf{q}}}_R+\theta {\nabla }_R·{\mathbf{k}}_R-{\displaystyle \frac{1}{\theta }}{\mathbf{q}}_R·\nabla R\theta ={\rho }_R\theta \gamma \ge 0.$$

(15)

Inequality (15) requires the compatibility among the constitutive functions $\psi ,{\mathbf{K}}_R,\gamma$ along with the non-negative valuedness of $\gamma$. For the present purpose, we now use Equation (15) as an equation in the unknown ${\dot{\mathbf{q}}}_R$.

The representation formula (11) is now applied to obtain ${\dot{\mathbf{q}}}_R$ from (10). Assume that ${\partial }_{{\mathbf{q}}_R}\psi \ne \mathbf{0}$ and let $\mathbf{n}={\partial }_{{\mathbf{q}}_R}\psi /\left|{\partial }_{{\mathbf{q}}_R}\psi \right|$. By (11), it follows that

$${\dot{\mathbf{q}}}_R={\displaystyle \frac{-{\rho }_R\theta \gamma +\theta \nabla R·{\mathbf{k}}_R-(1/\theta ){\mathbf{q}}_R·\nabla R\theta }{{\rho }_R{\left|{\partial }_{{\mathbf{q}}_R}\psi \right|}^2}}{\partial }_{{\mathbf{q}}_R}\psi +(\mathbf{1}-{\displaystyle \frac{{\partial }_{{\mathbf{q}}_R}\psi \otimes {\partial }_{{\mathbf{q}}_R}\psi }{|{\partial }_{{\mathbf{q}}_R}{\psi |}^2}})\mathbf{\xi }.$$

(16)

Equation (16) is the general thermodynamic requirement on ${\dot{\mathbf{q}}}_R$ under the assumption ${\partial }_{{\mathbf{q}}_R}\psi \ne \mathbf{0}$ with variables $\Gamma$.

The representation (16) and the arbitrariness of the field $\mathbf{\xi }$ show that any transverse term relative to ${\partial }_{{\mathbf{q}}_R}\psi$ is allowed, irrespective of its connection with the entropy production $\gamma$ or the extra-entropy flux ${\mathbf{k}}_R$. This property is apparent from the CD inequality (9), where ${\dot{\mathbf{q}}}_R$ occurs through the component along ${\partial }_{{\mathbf{q}}_R}\psi$. Physical examples are given in the next section.

4. Remarkable Models of Heat Conduction

Models are now derived by taking different assumptions on $\gamma ,{\mathbf{k}}_R,\mathbf{\xi }$.

4.1. Non-Dissipative Heat Conduction

Let $\gamma =0,{\mathbf{k}}_R=\mathbf{0},\mathbf{\xi }=\mathbf{0}$. Hence, Equation (16) simplifies to

$${\dot{\mathbf{q}}}_R=-{\displaystyle \frac{{\mathbf{q}}_R·\nabla R\theta }{{\rho }_R\theta {\left|{\partial }_{{\mathbf{q}}_R}\psi \right|}^2}}{\partial }_{{\mathbf{q}}_R}\psi .$$

The non-dissipative character is characterized by the assumption $\gamma =0$. Now, as $\theta$ and $\mathbf{E}$ are constants, we have

$$\dot{\psi }={\partial }_{{\mathbf{q}}_R}\psi ·{\dot{\mathbf{q}}}_R=-{\displaystyle \frac{1}{\theta {\rho }_R}}{\mathbf{q}}_R·\nabla R\theta .$$

The classical entropy production $-1/{\rho }_R{\theta }^2$ times ${\mathbf{q}}_R·\nabla R\theta$, which enters the heat conduction inequality, produces a corresponding rate of the free energy in this model.

4.2. Fourier Model

Let ${\mathbf{k}}_R=\mathbf{0}$ and $\mathbf{\xi }=\mathbf{0}$. If, further, we let ${\rho }_R{\theta }^2\gamma =-{\mathbf{q}}_R·\nabla R\theta$, then we find ${\dot{\mathbf{q}}}_R=\mathbf{0}$, which means that ${\mathbf{q}}_R$ is stationary. Now, if we let

$$\gamma =\nabla R\theta ·\mathbf{K}(\theta ,\mathbf{E})\nabla R\theta ,$$

where $\mathbf{K}$ is positive definite, then it follows that

$${\mathbf{q}}_R=-{\rho }_R{\theta }^2\mathbf{K}\nabla R\theta .$$

Thus, the Fourier law follows with heat conductivity tensor ${\rho }_R{\theta }^2\mathbf{K}$.

4.3. Some Evolutionary Models

We now look for some equations involving ${\dot{\mathbf{q}}}_R$ and relate them to models that appear in the literature. Hence, for definiteness, hereafter, we let

$$\psi ={\psi }_0(\theta ,\mathbf{E})+\frac{1}{2}\alpha {\left|{\mathbf{q}}_R\right|}^2,$$

where $\alpha$ is possibly dependent on $\theta$ and $\mathbf{E}$. Equation (16) then is written in the form

$${\dot{\mathbf{q}}}_R={\displaystyle \frac{-{\rho }_R\theta \gamma +\theta \nabla ·{\mathbf{k}}_R-(1/\theta ){\mathbf{q}}_R·\nabla R\theta }{{\rho }_R\alpha {\left|{\mathbf{q}}_R\right|}^2}}{\mathbf{q}}_R+(\mathbf{1}-{\displaystyle \frac{{\partial }_{{\mathbf{q}}_R}\psi \otimes {\partial }_{{\mathbf{q}}_R}\psi }{|{\partial }_{{\mathbf{q}}_R}{\psi |}^2}})\mathbf{\xi }.$$

(17)

Maxwell–Cattaneo Equation

Let ${\mathbf{k}}_R=\mathbf{0}$. Hence, Equation (17) simplifies to

$${\dot{\mathbf{q}}}_R={\displaystyle \frac{-{\rho }_R\theta \gamma -(1/\theta ){\mathbf{q}}_R·\nabla R\theta }{{\rho }_R\alpha {\left|{\mathbf{q}}_R\right|}^2}}{\mathbf{q}}_R+(\mathbf{1}-{\displaystyle \frac{{\mathbf{q}}_R\otimes {\mathbf{q}}_R}{|{\mathbf{q}}_R{|}^2}})\mathbf{\xi }.$$

Assume $\mathbf{\xi }=\beta \nabla R\theta$. If

$$\beta =-{\displaystyle \frac{1}{{\rho }_R\theta \alpha }},$$

it follows

$${\dot{\mathbf{q}}}_R+{\displaystyle \frac{\theta \gamma }{\alpha |{\mathbf{q}}_R{|}^2}}{\mathbf{q}}_R=-{\displaystyle \frac{1}{{\rho }_R\theta \alpha }}\nabla R\theta .$$

(18)

Equation (18) is just the Maxwell–Cattaneo (MC for short) equation with relaxation time $\tau$ and heat conductivity $\kappa$, given by

$$\tau ={\displaystyle \frac{\alpha |{\mathbf{q}}_R{|}^2}{\theta \gamma }},\kappa ={\displaystyle \frac{|{\mathbf{q}}_R{|}^2}{{\rho }_R{\theta }^2\gamma }}.$$

The positive value of the entropy dissipation $\gamma$ implies the positive value of $\kappa$. Instead, the sign of $\alpha$ is left undetermined by thermodynamics; it is customary to assume $\tau >0$, which follows from the assumption $\alpha >0$.

4.4. Cross-Couplings between Deformation and Heat Conduction

Cross-coupling terms are modelled by the joint dependence on $\mathbf{E}$ and ${\mathbf{q}}_R$. For simplicity, let

$$\mathbf{\xi }=\zeta \mathbf{E}{\mathbf{q}}_R,$$

where $\zeta$ is any scalar function of $\Gamma$. Hence, Equation (16) takes the form

$${\dot{\mathbf{q}}}_R={\displaystyle \frac{-{\rho }_R\theta \gamma +\theta \nabla ·{\mathbf{k}}_R-(1/\theta ){\mathbf{q}}_R·\nabla R\theta -\zeta {\rho }_R\alpha {\mathbf{q}}_R·\mathbf{E}{\mathbf{q}}_R}{{\rho }_R\alpha {\left|{\mathbf{q}}_R\right|}^2}}{\mathbf{q}}_R+\zeta \mathbf{E}{\mathbf{q}}_R.$$

(19)

The selection of appropriate functions $\gamma \ge 0$ and ${\mathbf{k}}_R$ completes the model.

4.5. Second-Order Non-Local Models

Non-local models are established in terms of the dependence on the spatial derivatives of ${\mathbf{q}}_R$. A model often considered in the literature is named after Guyer and Krumhansl and involves second-order derivatives [8,9]. Indeed, we consider the generalized (Lagrangian) form

$${\dot{\mathbf{q}}}_R+\lambda \nabla R\theta +\nu {\mathbf{q}}_R=c_1{\Delta }_R{\mathbf{q}}_R+c_2\nabla R(\nabla R·{\mathbf{q}}_R);$$

(20)

the coefficients $\lambda ,\nu ,c_1,c_2$ are expected to be positive-valued functions of temperature.

With the purpose of obtaining an evolution equation like (20) and recalling the assumption $\mathbf{\xi }=\beta \nabla R\theta$ made in connection with the MC Equation (18), we let

$$\mathbf{\xi }=\beta \nabla R\theta +c_1{\Delta }_R{\mathbf{q}}_R+c_2\nabla R(\nabla R·{\mathbf{q}}_R),$$

where $\beta ,c_1,c_2$ are functions of $\theta$ and $\mathbf{E}$. Letting $\beta =-1/{\rho }_R\theta \alpha$, we can write (17) in the form

$$\begin{array}{c}\hfill {\dot{\mathbf{q}}}_R={\displaystyle \frac{-{\rho }_R\theta \gamma +\theta \nabla ·{\mathbf{k}}_R}{{\rho }_R\alpha {\left|{\mathbf{q}}_R\right|}^2}}{\mathbf{q}}_R-{\displaystyle \frac{1}{{\rho }_R\theta \alpha }}\nabla R\theta +c_1{\Delta }_R{\mathbf{q}}_R+c_2\nabla R(\nabla R·{\mathbf{q}}_R)\\ \hfill -{\displaystyle \frac{c_1}{|{\mathbf{q}}_R{|}^2}}({\mathbf{q}}_R·{\Delta }_R{\mathbf{q}}_R){\mathbf{q}}_R-{\displaystyle \frac{c_2}{|{\mathbf{q}}_R{|}^2}}\left[({\mathbf{q}}_R·\nabla R){\nabla }_R{\mathbf{q}}_R\right]{\mathbf{q}}_R.\end{array}$$

We can get Equation (20) by the identifications $\lambda =1/{\rho }_R\theta \alpha$ and

$$\nu ={\displaystyle \frac{1}{{\rho }_R\alpha {\left|{\mathbf{q}}_R\right|}^2}}\{{\rho }_R\theta \gamma -\theta \nabla ·{\mathbf{k}}_R+{\rho }_R\alpha c_1\mathbf{q}·{\Delta }_R{\mathbf{q}}_R+{\rho }_R\alpha c_2({\mathbf{q}}_R·\nabla R)\nabla R·{\mathbf{q}}_R\}.$$

(21)

The form (20) is then obtained, albeit with $\nu$ as a function of the second-order derivatives $\nabla R\nabla R{\mathbf{q}}_R$. Owing to the occurrence of $\gamma$ and ${\mathbf{k}}_R$, we ask whether we can eliminate the dependence of $\nu$ on the derivatives of $\theta$ and ${\mathbf{q}}_R$.

Let

$$h={\displaystyle \frac{{\rho }_R\alpha c_1}{\theta }},l={\displaystyle \frac{{\rho }_R\alpha c_2}{\theta }}.$$

We can then write Equation (21) in the form

$${\displaystyle \frac{{\rho }_R\alpha \nu }{\theta }}{\left|{\mathbf{q}}_R\right|}^2={\rho }_R\gamma -\nabla ·{\mathbf{k}}_R+h\mathbf{q}·{\Delta }_R{\mathbf{q}}_R+l({\mathbf{q}}_R·\nabla R)\nabla R·{\mathbf{q}}_R.$$

Notice that

$$h{\mathbf{q}}_R·{\Delta }_R{\mathbf{q}}_R=\nabla R·(h\nabla R\frac{1}{2}|{\mathbf{q}}_R{|}^2)-[\nabla R(h{\mathbf{q}}_R)]·[\nabla R{\mathbf{q}}_R],$$

$$l({\mathbf{q}}_R·\nabla R)\nabla R·{\mathbf{q}}_R=\nabla R·(l{\mathbf{q}}_R\nabla R·{\mathbf{q}}_R)-[\nabla R·(l{\mathbf{q}}_R)]\nabla R·{\mathbf{q}}_R.$$

Hence, we select

$${\mathbf{k}}_R=h\nabla R\frac{1}{2}{\left|{\mathbf{q}}_R\right|}^2-l{\mathbf{q}}_R\nabla R·{\mathbf{q}}_R$$

(22)

to find

$${\displaystyle \frac{{\rho }_R\alpha {\left|{\mathbf{q}}_R\right|}^2\nu }{\theta }}+[\nabla R(h{\mathbf{q}}_R)]·[\nabla R{\mathbf{q}}_R]+[\nabla R·(l{\mathbf{q}}_R)]\nabla R·{\mathbf{q}}_R={\rho }_R\gamma .$$

If h and l are constants, then

$$[\nabla R(h{\mathbf{q}}_R)]·[\nabla R{\mathbf{q}}_R]+[\nabla R·(l{\mathbf{q}}_R)]\nabla R·{\mathbf{q}}_R=h|\nabla R{\mathbf{q}}_R{|}^2+l{(\nabla R·{\mathbf{q}}_R)}^2\ge 0,$$

provided only that $h,l>0$ and hence, $\alpha c_1,\alpha c_2>0$.

The generalized form (20) is thermodynamically consistent if h and l are positive constants. In that case the extra-entropy flux has the form (22), while the entropy production is

$$\gamma ={\displaystyle \frac{\alpha |{\mathbf{q}}_R{|}^2\nu }{\theta }}+{\displaystyle \frac{\alpha c_1}{\theta }}(\nabla R{\mathbf{q}}_R)·(\nabla R{\mathbf{q}}_R)+{\displaystyle \frac{\alpha c_2}{\theta }}{(\nabla R·{\mathbf{q}}_R)}^2.$$

5. Higher-Order Differential Equations

Experiments on heat propagation in graphene [13] show that the results are best fitted with the differential equation

$${\displaystyle \frac{1}{\alpha }}{\partial }_t\theta +{\displaystyle \frac{{\tau }_q}{\alpha }}{\partial }_t^2\theta =\Delta \theta +{\tau }_{\theta }{\partial }_t\Delta \theta ,$$

(23)

for the temperature $\theta (\mathbf{x},t)$. There are various physical models leading to (23). The simplest one is to assume the conductor is rigid and let $c={\partial }_{\theta }\epsilon$ be the specific heat so that the balance of energy simplifies to

$$\rho c{\partial }_t\theta =-\nabla ·\mathbf{q},$$

with a zero energy supply. The evolution of the heat flux is assumed to be governed by the MC equation with a higher-order correction so that

$${\tau }_q\dot{\mathbf{q}}+\mathbf{q}=-\lambda \nabla \theta -\lambda {\tau }_{\theta }{\partial }_t\nabla \theta .$$

(24)

Assuming c is constant, upon some rearrangements, we find Equation (23) with $\alpha =1/\rho c$. Owing to the formal occurrence of two times ${\tau }_q,{\tau }_{\theta }$, Equation (23) is also referred to as a heat equation with two phase lags. Other physical motivations of (23) are given in [14].

We now examine the thermodynamic consistency of heat equations of the form (24). For generality, we let the body be deformable and hence, we follow again the Lagrangian description. We then consider equations of the form

$${\tau }_q{\dot{\mathbf{q}}}_R+{\mathbf{q}}_R=-\lambda \nabla R\theta -\lambda {\tau }_{\theta }\nabla R\dot{\theta }.$$

(25)

Owing to the term $\nabla R\dot{\theta }$ in Equation (25), we let both $\nabla R\theta$ and $\dot{\theta }$ be among the variables. Hence, we let

$$\Gamma =(\theta ,\mathbf{E},{\mathbf{q}}_R,\nabla R\theta ,\dot{\theta })$$

be the set of variables and $\psi ,\eta ,{\mathbf{T}}_{RR},{\mathbf{k}}_R,\gamma$ and ${\dot{\mathbf{q}}}_R$ be given by constitutive functions. Computation of $\dot{\psi }$ and substitution in the CD inequality yields

$$-{\rho }_R({\partial }_{\theta }\psi +\eta )\dot{\theta }-{\rho }_R{\partial }_{{\mathbf{q}}_R}\psi ·{\dot{\mathbf{q}}}_R-{\rho }_R{\partial }_{\nabla R\theta }\psi ·\nabla R\dot{\theta }-{\rho }_R{\partial }_{\dot{\theta }}\psi \ddot{\theta }+\theta \nabla R·{\mathbf{k}}_R-{\displaystyle \frac{1}{\theta }}{\mathbf{q}}_R·\nabla R\theta ={\rho }_R\theta \gamma .$$

(26)

The linearity and arbitrariness of $\ddot{\theta }$ implies that

$${\partial }_{\dot{\theta }}\psi =0.$$

The occurrence of $\nabla R\dot{\theta }$ affects the properties of the entropy $\eta$. Hence, we define

$${\widehat{\mathbf{q}}}_R={\dot{\mathbf{q}}}_R+d(\theta ,\mathbf{E})\nabla R\dot{\theta }.$$

In (25), it is $d=\lambda {\tau }_{\theta }/{\tau }_q$; consistent with (25), we assume ${\dot{\mathbf{q}}}_R$, and hence, ${\widehat{\mathbf{q}}}_R$, are independent of $\dot{\theta }$. Substitution of ${\dot{\mathbf{q}}}_R$ in (26) and division by $\theta$ result in

$$-{\displaystyle \frac{{\rho }_R}{\theta }}({\partial }_{\theta }\psi +\eta )\dot{\theta }-{\displaystyle \frac{{\rho }_R}{\theta }}{\partial }_{{\mathbf{q}}_R}\psi ·{\widehat{\mathbf{q}}}_R-{\displaystyle \frac{{\rho }_R}{\theta }}\mathbf{y}·\nabla R\dot{\theta }+\nabla R·{\mathbf{k}}_R-{\displaystyle \frac{1}{{\theta }^2}}{\mathbf{q}}_R·\nabla R\theta ={\rho }_R\gamma ,$$

(27)

where

$$\mathbf{y}=d{\partial }_{{\mathbf{q}}_R}\psi +{\partial }_{\nabla R\theta }\psi .$$

Now, observe that

$$-{\displaystyle \frac{{\rho }_R}{\theta }}\mathbf{y}·\nabla R\dot{\theta }+\nabla R·{\mathbf{k}}_R=\nabla R·({\mathbf{k}}_R-{\displaystyle \frac{{\rho }_R}{\theta }}\mathbf{y}\dot{\theta })+[\nabla R·({\displaystyle \frac{{\rho }_R}{\theta }}\mathbf{y}]\dot{\theta }.$$

Hence, we let

$${\mathbf{k}}_R={\displaystyle \frac{{\rho }_R}{\theta }}\mathbf{y}\dot{\theta }.$$

Consequently, we can write Equation (27) in the form

$$-{\displaystyle \frac{{\rho }_R}{\theta }}({\delta }_{\theta }\psi +\eta )\dot{\theta }-{\displaystyle \frac{{\rho }_R}{\theta }}{\partial }_{{\mathbf{q}}_R}\psi ·{\widehat{\mathbf{q}}}_R-{\displaystyle \frac{1}{{\theta }^2}}{\mathbf{q}}_R·\nabla R\theta ={\rho }_R\gamma ,$$

(28)

where

$${\delta }_{\theta }\psi :={\partial }_{\theta }\psi -{\displaystyle \frac{\theta }{{\rho }_R}}\nabla R·\left[{\displaystyle \frac{{\rho }_R}{\theta }}(d{\partial }_{{\mathbf{q}}_R}\psi +{\partial }_{\nabla R\theta }\psi )\right].$$

If ${\rho }_R$ and $\theta$ are uniform, that is $\nabla R{\rho }_R=\mathbf{0},\nabla R\theta =\mathbf{0}$, then

$${\delta }_{\theta }\psi :={\partial }_{\theta }\psi -\nabla R·(d{\partial }_{{\mathbf{q}}_R}\psi +{\partial }_{\nabla R\theta }\psi ).$$

If, further, $d=0$, then ${\delta }_{\theta }\psi$ equals the variational derivative of $\psi$ with respect to $\theta$. Accordingly, we can say that ${\delta }_{\theta }\psi$ is a generalized variational derivative of $\psi$.

While $\widehat{\mathbf{q}}$ is independent of $\dot{\theta }$, both $\eta$ and $\gamma$ can depend on $\dot{\theta }$. If we let

$$\eta ={\eta }_0(\theta ,\mathbf{E},{\mathbf{q}}_R,\nabla R\theta )+{\eta }_1(\Gamma )\dot{\theta },$$

then we find that

$${\eta }_0=-{\delta }_{\theta }\psi ,{\eta }_1\le 0,{\rho }_R\gamma ={\rho }_R{\gamma }_0+{\rho }_R{\gamma }_1,{\gamma }_1=-{\displaystyle \frac{1}{\theta }}{\eta }_1{\dot{\theta }}^2,$$

where ${\gamma }_0\ge 0$ is the value of the entropy production $\gamma$ at $\dot{\theta }=0$.

Equation (28) reduces to

$$-{\rho }_R{\partial }_{{\mathbf{q}}_R}\psi ·{\widehat{\mathbf{q}}}_R-{\displaystyle \frac{1}{\theta }}{\mathbf{q}}_R·\nabla R\theta ={\rho }_R\theta {\gamma }_0.$$

(29)

Hence, using the representation (11) with $\mathbf{n}={\partial }_{{\mathbf{q}}_R}\psi /\left|{\partial }_{{\mathbf{q}}_R}\psi \right|$, we can determine ${\widehat{\mathbf{q}}}_R$ in the form

$${\widehat{\mathbf{q}}}_R=-{\displaystyle \frac{(1/\theta ){\mathbf{q}}_R·\nabla R\theta +{\rho }_R\theta {\gamma }_0}{{\rho }_R{\left|{\partial }_{{\mathbf{q}}_R}\psi \right|}^2}}{\partial }_{{\mathbf{q}}_R}\psi +(\mathbf{1}-{\displaystyle \frac{{\partial }_{{\mathbf{q}}_R}\psi \otimes {\partial }_{{\mathbf{q}}_R}\psi }{|{\partial }_{{\mathbf{q}}_R}{\psi |}^2}})\mathbf{\xi }.$$

We assume

$${\partial }_{{\mathbf{q}}_R}\psi =\alpha (\theta ,\mathbf{E},\nabla R\theta ){\mathbf{q}}_R,$$

whence

$$\psi ={\psi }_0(\theta ,\mathbf{E},\nabla R\theta )+\frac{1}{2}\alpha (\theta ,\mathbf{E},\nabla R\theta ){\left|{\mathbf{q}}_R\right|}^2.$$

The selection

$$\mathbf{\xi }=\beta \nabla R\theta$$

leads to

$${\widehat{\mathbf{q}}}_R=-{\displaystyle \frac{{\rho }_R\theta {\gamma }_0}{{\rho }_R\alpha {\left|{\partial }_{{\mathbf{q}}_R}\right|}^2}}{\mathbf{q}}_R-{\displaystyle \frac{{\mathbf{q}}_R·\nabla R\theta }{\theta {\rho }_R\alpha {\left|{\mathbf{q}}_R\right|}^2}}{\mathbf{q}}_R+\beta \nabla R\theta -\beta {\displaystyle \frac{{\mathbf{q}}_R·\nabla R\theta }{|{\mathbf{q}}_R{|}^2}}{\mathbf{q}}_R.$$

The particular case

$$\beta =-{\displaystyle \frac{1}{\theta {\rho }_R\alpha }}$$

results in

$${\widehat{\mathbf{q}}}_R=-{\displaystyle \frac{\theta {\gamma }_0}{\alpha |{\mathbf{q}}_R{|}^2}}{\mathbf{q}}_R-{\displaystyle \frac{1}{\theta {\rho }_R\alpha }}\nabla R\theta .$$

This in turn leads to the rate equation for ${\mathbf{q}}_R$ in the form

$${\dot{\mathbf{q}}}_R+{\displaystyle \frac{\theta {\gamma }_0}{\alpha |{\mathbf{q}}_R{|}^2}}{\mathbf{q}}_R=-{\displaystyle \frac{1}{\theta {\rho }_R\alpha }}\nabla R\theta -d\nabla R\dot{\theta },$$

whence

$$\lambda ={\displaystyle \frac{|{\mathbf{q}}_R{|}^2}{{\theta }^2{\gamma }_0{\rho }_R}}>0,{\tau }_q={\displaystyle \frac{\alpha |{\mathbf{q}}_R{|}^2}{\theta {\gamma }_0}},{\tau }_{\theta }=\theta {\rho }_R\alpha d.$$

In stationary conditions, we have

$${\mathbf{q}}_R=-{\displaystyle \frac{|{\mathbf{q}}_R{|}^2}{{\theta }^2{\rho }_R{\gamma }_0}}\nabla R\theta ,$$

which ascribes to

$$K={\displaystyle \frac{|{\mathbf{q}}_R{|}^2}{{\theta }^2{\rho }_R{\gamma }_0}}$$

the meaning of heat conductivity. Since ${\gamma }_0>0$ the conductivity K proves to be positive, as expected.

6. Rate Equations in the Eulerian Description

In accordance with the objectivity principle ([15], p. 35; [16], §41), the constitutive equations must be form-invariant under Euclidean transformations, namely under the map $\mathbf{x}\mapsto {\mathbf{x}}^{\ast }$ such that

$${\mathbf{x}}^{\ast }=\mathbf{c}+\mathbf{Q}\mathbf{x},$$

where the vector $\mathbf{c}$ and the tensor $\mathbf{Q}$ are time-dependent and $\mathbf{Q}$ is orthogonal, ${\mathbf{Q}}^T\mathbf{Q}=\mathbf{1}$. Consequently, if a constitutive equation is expressed by a rate equation then the time derivatives must be objective in that the objective time derivative of a vector transforms as a vector, and the same is true for tensors.

A natural question arises since there are infinitely many objective derivatives and the non-uniqueness remains, even though we require the thermodynamic consistency to hold. We now examine rate-type models. For formal simplicity, we consider separately the occurrence of rate effects for the stress and the heat flux.

6.1. The Navier–Stokes–Voigt Fluid

Roughly, the NSV (Navier–Stokes–Voigt) fluid model is the counterpart of the Kelvin–Voigt solid, whereby the stress is a linear combination of the strain and the strain-rate. Consistently, the NSV system of equations modifies the Navier–Stokes equations by addition of a term of form $\nu \Delta {\partial }_t\mathbf{v}$ to the acceleration $\dot{\mathbf{v}}$. Really, this is the approximation induced by replacing the Navier–Stokes stress $\mu \mathbf{D}+\lambda (\mathrm{tr}\mathbf{D})\mathbf{1}$ with the linear combination $\mu \mathbf{D}+\lambda (\mathrm{tr}\mathbf{D})\mathbf{1}+\widehat{\mu }\dot{\mathbf{D}}+\widehat{\lambda }(\mathrm{tr}\dot{\mathbf{D}})\mathbf{1}$. However, the total time derivative, along with the partial time derivative, is not objective and hence, the idea of the linear combination has to be revisited.

Here, for generality, we let the fluid be compressible, though usually in the literature the fluid is incompressible (see, e.g., [17,18,19]). Hence, we let

$$\mathbf{T}=-p(\theta ,\rho )\mathbf{1}+\mathcal{T},$$

where p is the thermodynamic pressure, assumed to be a function of the temperature $\theta$ and the mass density $\rho$. The stress $\mathcal{T}$ is expected to be related to $\mathbf{D}$ and the rate of $\mathbf{D}$. Furthermore, we keep modelling heat conduction through a rate equation. Hence, we let

$$\Lambda =(\theta ,\rho ,\mathcal{T},\mathbf{D},\mathbf{q},\dot{\theta },\nabla \theta )$$

be the set of variables for the constitutive equations of $\psi ,\eta$, and the rates of $\mathbf{D}$ and $\mathbf{q}$. Indeed, the rate equations for $\mathbf{D}$ and $\mathbf{q}$ have to be established via objective time derivatives.

Compute the time derivative of $\psi (\Lambda )$ and substitute in the CD inequality (8) to obtain

$$\begin{array}{c}\hfill -\rho ({\partial }_{\theta }\psi +\eta )\dot{\theta }+({\rho }^2{\partial }_{\rho }\psi -p)\nabla ·\mathbf{v}-\rho {\partial }_{\mathbf{D}}\psi ·\dot{\mathbf{D}}-\rho {\partial }_{\mathbf{q}}\psi ·\dot{\mathbf{q}}-\rho {\partial }_{\nabla \theta }\psi ·\dot{(\nabla \theta )}\\ \hfill -\rho {\partial }_{\dot{\theta }}\psi \ddot{\theta }+\mathcal{T}·\mathbf{D}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta +\theta \nabla ·\mathbf{k}=\rho \theta \gamma .\end{array}$$

(30)

The linearity and arbitrariness of $\ddot{\theta }$ imply that

$${\partial }_{\dot{\theta }}\psi =0.$$

By the thermodynamic character of the pressure, we conclude that

$$p={\rho }^2{\partial }_{\rho }\psi .$$

Thus, Equation (30) simplifies to

$$\begin{array}{c}\hfill -{\displaystyle \frac{\rho }{\theta }}({\partial }_{\theta }\psi +\eta )\dot{\theta }-{\displaystyle \frac{\rho }{\theta }}{\partial }_{\mathbf{D}}\psi ·\dot{\mathbf{D}}-{\displaystyle \frac{\rho }{\theta }}{\partial }_{\mathbf{q}}\psi ·\dot{\mathbf{q}}-{\displaystyle \frac{\rho }{\theta }}{\partial }_{\nabla \theta }\psi ·\dot{(\nabla \theta )}\\ \hfill +{\displaystyle \frac{1}{\theta }}\mathcal{T}·\mathbf{D}-{\displaystyle \frac{1}{{\theta }^2}}\mathbf{q}·\nabla \theta +\nabla ·\mathbf{k}=\rho \gamma ,\end{array}$$

(31)

where, for technical purposes, we have divided throughout by $\theta$.

Notice that

$$\dot{(\nabla \theta )}=\nabla \dot{\theta }-{\mathbf{L}}^T\nabla \theta$$

and then

$$-{\displaystyle \frac{\rho }{\theta }}{\partial }_{\nabla \theta }\psi ·\dot{(\nabla \theta )}+\nabla ·\mathbf{k}=\nabla ·(\mathbf{k}-{\displaystyle \frac{\rho }{\theta }}{\partial }_{\nabla \theta }\psi \dot{\theta })+\{\nabla ·({\displaystyle \frac{\rho }{\theta }}{\partial }_{\nabla \theta }\psi )\}\dot{\theta }.$$

Consequently, we let

$$\mathbf{k}={\displaystyle \frac{\rho }{\theta }}{\partial }_{\nabla \theta }\psi \dot{\theta }.$$

Hence, $\theta$ times the remaining part of (31) results in

$$-\rho ({\delta }_{\theta }\psi +\eta )\dot{\theta }-\rho {\partial }_{\mathbf{D}}\psi ·\dot{\mathbf{D}}-\rho {\partial }_{\mathbf{q}}\psi ·\dot{\mathbf{q}}+\mathcal{T}·\mathbf{D}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma ,$$

(32)

where ${\delta }_{\theta }\psi$ denotes the variational derivative,

$${\delta }_{\theta }\psi ={\partial }_{\theta }\psi -{\displaystyle \frac{\theta }{\rho }}\nabla ·({\displaystyle \frac{\rho }{\theta }}{\partial }_{\nabla \theta }\psi ).$$

By Equation (32), the set $\Lambda$ of variables allows for a rich variety of properties. Formally, one of the rates, e.g., $\dot{\theta }$, is a function of $\dot{\mathbf{D}},\dot{\mathbf{q}},\gamma$ and the remaining quantities. For definiteness, we look for separate effects or, mathematically, sufficient conditions to satisfy Equation (32).

Firstly, let $\eta$, as well as $\dot{\mathbf{D}},\dot{\mathbf{q}},\gamma$ be independent of $\dot{\theta }$. Hence, Equation (32) holds only if

$$\eta =-{\delta }_{\theta }\psi ,$$

and

$$-\rho {\partial }_{\mathbf{D}}\psi ·\dot{\mathbf{D}}-\rho {\partial }_{\mathbf{q}}\psi ·\dot{\mathbf{q}}+\mathcal{T}·\mathbf{D}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .$$

(33)

To proceed with the analysis of (33), it is worth observing that a rate equation of the form

$$\dot{\mathbf{D}}=\mathcal{F}(\theta ,\rho ,\mathcal{T},\mathbf{D},\mathbf{q},\nabla \theta )$$

does not comply with the objectivity principle in that $\dot{\mathbf{D}}$ is not form invariant. We then look for a recourse to objective derivatives. To fix ideas, consider the objective derivative

$$\stackrel{\diamond }{\mathbf{D}}=\dot{\mathbf{D}}-\mathbf{W}\mathbf{D}-\mathbf{D}{\mathbf{W}}^T-2c_1\mathbf{D}\mathbf{D}+c_2(\nabla ·\mathbf{v})\mathbf{D},\stackrel{\diamond }{\mathbf{q}}=\dot{\mathbf{q}}-\mathbf{W}\mathbf{Q}-c_1\mathbf{D}\mathbf{q}+c_2(\nabla ·\mathbf{v})\mathbf{q}.$$

If $c_1=1,c_2=1$, then $\stackrel{\diamond }{\mathbf{D}},\stackrel{\diamond }{\mathbf{q}}$ are the Truesdell derivatives of $\mathbf{D}$ and $\mathbf{q}$. If, instead, $c_1=0,c_2=0$, then $\stackrel{\diamond }{\mathbf{D}}$ and $\stackrel{\diamond }{\mathbf{q}}$ coincide with the corotational derivatives,

$$\stackrel{\circ }{\mathbf{D}}=\dot{\mathbf{D}}-\mathbf{W}\mathbf{D}-\mathbf{D}{\mathbf{W}}^T,\stackrel{\circ }{\mathbf{q}}=\dot{\mathbf{q}}-\mathbf{W}\mathbf{q}.$$

The other objective derivatives have as common part the corotational derivative [2]. Replacing

$$\dot{\mathbf{D}}=\stackrel{\diamond }{\mathbf{D}}+\mathbf{W}\mathbf{D}+\mathbf{D}{\mathbf{W}}^T+2\mathbf{D}\mathbf{D}-(\nabla ·\mathbf{v})\mathbf{D},\dot{\mathbf{q}}=\stackrel{\diamond }{\mathbf{q}}+\mathbf{W}\mathbf{Q}+\mathbf{D}\mathbf{q}-(\nabla ·\mathbf{v})\mathbf{q},$$

we can write Equation (33) in the form

$$\begin{array}{c}\hfill -\rho {\partial }_{\mathbf{D}}\psi ·[\stackrel{\diamond }{\mathbf{D}}+\mathbf{W}\mathbf{D}+\mathbf{D}{\mathbf{W}}^T+2c_1\mathbf{D}\mathbf{D}-c_2(\nabla ·\mathbf{v})\mathbf{D}]\\ \hfill -\rho {\partial }_{\mathbf{q}}\psi ·[\stackrel{\diamond }{\mathbf{q}}+\mathbf{W}\mathbf{q}+c_1\mathbf{D}\mathbf{q}-c_2(\nabla ·\mathbf{v})\mathbf{q}]+\mathcal{T}·\mathbf{D}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .\end{array}$$

(34)

The linearity and arbitrariness of the spin $\mathbf{W}$ in (34) imply that

$${\partial }_{\mathbf{D}}\psi \mathbf{D}\in \mathrm{Sym},{\partial }_{\mathbf{q}}\psi \otimes \mathbf{q}\in \mathrm{Sym},$$

whence

$${\partial }_{\mathbf{D}}\psi ={\alpha }_1\mathbf{D},{\partial }_{\mathbf{q}}\psi ={\alpha }_2\mathbf{q}.$$

(35)

Hence, (34) simplifies to

$$\begin{array}{c}\hfill -\rho {\alpha }_1\mathbf{D}·[\stackrel{\diamond }{\mathbf{D}}+2c_1\mathbf{D}\mathbf{D}-c_2(\nabla ·\mathbf{v})\mathbf{D}]+\mathcal{T}·\mathbf{D}\\ \hfill -\rho {\alpha }_2\mathbf{q}·[\stackrel{\diamond }{\mathbf{q}}+c_1\mathbf{D}\mathbf{q}-c_2(\nabla ·\mathbf{v})\mathbf{q}]-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .\end{array}$$

(36)

Furthermore, in light of (35), we have

$$\psi ={\psi }_0(\theta ,\rho ,\mathcal{T},\nabla \theta )+\frac{1}{2}{\alpha }_1{\left|\mathbf{D}\right|}^2+\frac{1}{2}{\alpha }_2{\left|\mathbf{q}\right|}^2.$$

(37)

Two significant instances are taken from the literature.

6.2. Non-Heat Conducting NSV Fluid

This approximation traces back to Oskolkov, though in the particular case of incompressible fluids [20]. We let $\mathbf{q}=\mathbf{0}$ and notice that

$$\mathbf{D}·\mathbf{D}={\left|\mathbf{D}\right|}^2,\mathbf{D}·(\mathbf{D}\mathbf{D})=\mathrm{tr}(\mathbf{D}\mathbf{D}\mathbf{D}).$$

Consequently, Equation (36) simplifies to

$$-\rho \alpha \mathbf{D}·\stackrel{\diamond }{\mathbf{D}}-2\rho \alpha c_1\mathrm{tr}(\mathbf{D}\mathbf{D}\mathbf{D})+\rho \alpha c_2{\left|\mathbf{D}\right|}^2\nabla ·\mathbf{v}+\mathcal{T}·\mathbf{D}=\rho \theta \gamma .$$

(38)

We now apply the representation formula (11) for the unknown $\stackrel{\diamond }{\mathbf{D}}$ with

$$\mathbf{N}={\displaystyle \frac{\mathbf{D}}{\left|\mathbf{D}\right|}}.$$

Hence,

$$\begin{array}{cc}\stackrel{\diamond }{\mathbf{D}}\hfill & ={\displaystyle \frac{\rho \alpha \mathbf{D}·\stackrel{\diamond }{\mathbf{D}}}{{\rho \alpha \left|\mathbf{D}\right|}^2}}\mathbf{D}+[\mathsf{I}-{\displaystyle \frac{\mathbf{D}\otimes \mathbf{D}}{{\left|\mathbf{D}\right|}^2}}]\Xi \hfill \\ & =-{\displaystyle \frac{\theta \gamma }{{\alpha \left|\mathbf{D}\right|}^2}}\mathbf{D}+{\displaystyle \frac{\mathcal{T}·\mathbf{D}}{{\rho \alpha \left|\mathbf{D}\right|}^2}}\mathbf{D}-{\displaystyle \frac{2c_1\mathrm{tr}(\mathbf{D}\mathbf{D}\mathbf{D})}{{\left|\mathbf{D}\right|}^2}}\mathbf{D}+c_2(\nabla ·\mathbf{v})\mathbf{D}+[\mathsf{I}-{\displaystyle \frac{\mathbf{D}\otimes \mathbf{D}}{{\left|\mathbf{D}\right|}^2}}]\Xi ,\hfill \end{array}$$

where $\Xi$ is any tensor function of $\theta ,\rho ,\mathcal{T},\mathbf{D},\nabla \theta$. If $c_1=0,c_2=0$, then $\stackrel{\diamond }{\mathbf{D}}$ reduces to the corotational derivative; thus, we have

$$\stackrel{\circ }{\mathbf{D}}=-{\displaystyle \frac{\theta \gamma }{{\alpha \left|\mathbf{D}\right|}^2}}\mathbf{D}+{\displaystyle \frac{\mathcal{T}·\mathbf{D}}{{\rho \alpha \left|\mathbf{D}\right|}^2}}\mathbf{D}+[\mathsf{I}-{\displaystyle \frac{\mathbf{D}\otimes \mathbf{D}}{{\left|\mathbf{D}\right|}^2}}]\Xi .$$

If $\Xi =\beta \mathcal{T}$, then

$$\stackrel{\circ }{\mathbf{D}}=-{\displaystyle \frac{\theta \gamma }{{\alpha \left|\mathbf{D}\right|}^2}}\mathbf{D}+{\displaystyle \frac{\mathcal{T}·\mathbf{D}}{{\rho \alpha \left|\mathbf{D}\right|}^2}}\mathbf{D}+\beta \mathcal{T}-\beta {\displaystyle \frac{\mathcal{T}·\mathbf{D}}{{\left|\mathbf{D}\right|}^2}}\mathbf{D}.$$

The choice $\beta =1/\rho \alpha$ results in

$$\mathcal{T}={\displaystyle \frac{\rho \theta \gamma }{{\left|\mathbf{D}\right|}^2}}\mathbf{D}+\rho \alpha \stackrel{\circ }{\mathbf{D}}.$$

Borrowing from the Navier–Stokes theory we identify the coefficient of $\mathbf{D}$ with the shear viscosity $\mu$ and hence, we can write

$$\rho \theta \gamma ={\mu \left|\mathbf{D}\right|}^2,$$

which gives

$$\mathcal{T}=\mu (\mathbf{D}+\tau \stackrel{\circ }{\mathbf{D}}),$$

where $\tau =\rho \alpha /\mu$ represents the relaxation time.

In light of (35), the free energy has the form

$$\psi ={\psi }_0(\theta ,\rho ,\mathcal{T},\nabla \theta )+\frac{1}{2}\alpha {\left|\mathbf{D}\right|}^2.$$

Hence, also $\eta$ and $\epsilon =\psi +\eta \theta$ have additive terms proportional to ${\left|\mathbf{D}\right|}^2$.

Remark 1.

Two theories of NSV fluids have been developed with different schemes; however, both of them use a kinetic energy density proportional to ${|\nabla \mathbf{v}|}^2$ [21,22]. We then see a similarity with the free energy ψ, comprising a term proportional to ${\left|\mathbf{D}\right|}^2$.

6.3. NSV Fluid with Non-Fourier Conduction

Back to Equation (34), we now focus our attention on the corotational derivative as the objective derivative (and hence, we let $c_1=0,c_2=0$). Furthermore, the model of heat conduction is sought by looking for a rate-type equation; thus, we obtain, in particular, a Maxwell-Cattaneo equation. In this way, a generalization is established of the model applied by Straughan [23].

By (36) and (37), it follows

$$-\rho {\alpha }_1\mathbf{D}·\stackrel{\circ }{\mathbf{D}}-\rho {\alpha }_2\mathbf{q}·\stackrel{\circ }{\mathbf{q}}+\mathcal{T}·\mathbf{D}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .$$

(39)

For simplicity, cross couplings (e.g., $\mathbf{D}\mathbf{q}$ terms) are set aside and hence, the entropy production is assumed in the separate form

$$\gamma ={\gamma }_{\mathcal{T}}(\theta ,\rho ,\mathcal{T},\mathbf{D})+{\gamma }_{\mathbf{q}}(\theta ,\rho ,\mathbf{q},\nabla \theta ).$$

Both ${\gamma }_{\mathcal{T}}$ and ${\gamma }_{\mathbf{q}}$ are non-negative, in that ${\gamma }_{\mathcal{T}}$ is the value of $\gamma$ when $\mathcal{T}=\mathbf{0},\mathbf{D}=\mathbf{0}$, and ${\gamma }_{\mathbf{q}}$ is the value of $\gamma$ when $\mathbf{q}=\mathbf{0}$. Hence, (39) splits into two equations,

$$-\rho {\alpha }_1\mathbf{D}·\stackrel{\circ }{\mathbf{D}}+\mathcal{T}·\mathbf{D}=\rho \theta \gamma ,$$

(40)

$$-\rho {\alpha }_2\mathbf{q}·\stackrel{\circ }{\mathbf{q}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta {\gamma }_q.$$

(41)

Equation (40) has just been considered in the previous subsection. As for (41), we apply the representation formula with $\mathbf{n}=\mathbf{q}/\left|\mathbf{q}\right|$ to obtain

$$\stackrel{\circ }{\mathbf{q}}={\displaystyle \frac{\rho \theta {\gamma }_q+(1/\theta )\mathbf{q}·\nabla \theta }{-\rho {\alpha }_2{\left|\mathbf{q}\right|}^2}}\mathbf{q}+\left[\mathbf{1}-{\displaystyle \frac{\mathbf{q}\otimes \mathbf{q}}{{\left|\mathbf{q}\right|}^2}}\right]\mathbf{\xi }.$$

(42)

If we assume $\mathbf{\xi }=\mathbf{0}$, then it follows

$$\stackrel{\circ }{\mathbf{q}}={\displaystyle \frac{\rho \theta {\gamma }_q+(1/\theta )\mathbf{q}·\nabla \theta }{-\rho {\alpha }_2{\left|\mathbf{q}\right|}^2}}\mathbf{q}.$$

If no action on the system happens ($\mathbf{\xi }=\mathbf{0}$), then $\stackrel{\circ }{\mathbf{q}}$ is directed along $\mathbf{q}$. Furthermore, in stationary conditions, $\stackrel{\circ }{\mathbf{q}}=\mathbf{0}$, it follows

$$-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta {\gamma }_q\ge 0;$$

that is, the standard thermodynamic condition on the heat flux when only the temperature gradient is considered to produce the heat flux.

If we assume $\mathbf{\xi }=\beta \nabla \theta$, then it follows

$$\stackrel{\circ }{\mathbf{q}}={\displaystyle \frac{\rho \theta {\gamma }_q+(1/\theta )\mathbf{q}·\nabla \theta }{-\rho {\alpha }_2{\left|\mathbf{q}\right|}^2}}\mathbf{q}+\beta \nabla \theta -\beta {\displaystyle \frac{\mathbf{q}·\nabla \theta }{{\left|\mathbf{q}\right|}^2}}\mathbf{q}.$$

Hence, letting $\beta =-1/\rho \theta {\alpha }_2$, we have

$$\stackrel{\circ }{\mathbf{q}}+{\displaystyle \frac{\theta {\gamma }_q}{{\alpha }_2{\left|\mathbf{q}\right|}^2}}\mathbf{q}=-{\displaystyle \frac{1}{\rho \theta {\alpha }_2}}\nabla \theta .$$

(43)

Equation (43) has the form of the MC equation where the time derivative is in the corotational (and hence objective) form. Indeed, we can view

$$\tau ={\displaystyle \frac{{\alpha }_2{\left|\mathbf{q}\right|}^2}{\theta {\gamma }_q}},\kappa ={\displaystyle \frac{{\left|\mathbf{q}\right|}^2}{\rho {\theta }^2{\gamma }_q}},$$

as the relaxation time and the heat conductivity. The positive value of the entropy production ${\gamma }_q$ yields $\kappa >0$. Any assumption ${\gamma }_q=g(\theta ,\rho ){\left|\mathbf{q}\right|}^2$ makes $\kappa$ a function of $\theta$ and $\rho$. The assumption ${\alpha }_2>0$ implies $\tau >0$ and hence, guarantees the boundedness of the solution $\mathbf{q}$ in time. This procedure in turn shows how the MC equation is merely a particular case of the family of thermodynamically consistent evolution equations $\stackrel{\circ }{\mathbf{q}}=\widehat{\mathbf{q}}(\theta ,\rho ,\mathbf{q},\nabla \theta )$ with $\mathbf{\xi }=\beta \nabla \theta$.

7. Evolution Equations, Second Law, and Objective Derivatives

By analogy with the MC equation for the heat flux, in fluid-dynamics, evolution equations for the stress tensor are considered in the Eulerian description. In this context, we contemplate the selection of objective derivatives and the restrictions placed by the second law, as well as the objectivity of thermodynamically consistent evolution equations.

The stress $\mathbf{T}$ is represented in the form $\mathbf{T}=-p\mathbf{1}+\mathcal{T}$, with p representing the thermodynamic pressure. Describe a thermo-viscous fluid through the variables

$$\theta ,\rho ,\mathcal{T},\mathbf{q},\mathbf{D},\nabla \theta ,$$

for the constitutive functions $\psi ,\eta ,p,\dot{\mathbf{T}},\dot{\mathbf{q}},\gamma$. The Clausius–Duhem inequality (with $\mathbf{k}=\mathbf{0}$) reads

$$\begin{array}{c}\hfill -\rho ({\partial }_{\theta }\psi +\eta )\dot{\theta }-\rho {\partial }_{\rho }\psi \dot{\rho }-\rho {\partial }_{\mathcal{T}}\psi ·\dot{\mathcal{T}}-\rho {\partial }_{\mathbf{q}}\psi ·\dot{\mathbf{q}}-\rho {\partial }_{\mathbf{D}}\psi ·\dot{\mathbf{D}}-\rho {\partial }_{\nabla \theta }\psi ·\dot{(\nabla \theta )}\\ \hfill -p\nabla ·\mathbf{v}+\mathcal{T}·\mathbf{D}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .\end{array}$$

The linearity and arbitrariness of $\dot{\mathbf{D}},\dot{(\nabla \theta )},$ and $\dot{\theta }$ imply

$${\partial }_{\mathbf{D}}\psi =\mathbf{0},{\partial }_{\nabla \theta }=\mathbf{0},\eta =-{\partial }_{\theta }\psi .$$

Substitute $\dot{\rho }$ with $-\rho \nabla ·\mathbf{v}$ and define the thermodynamic pressure by

$$p={\rho }^2{\partial }_{\rho }\psi .$$

The remaining condition is

$$-\rho {\partial }_{\mathcal{T}}\psi ·\dot{\mathcal{T}}-\rho {\partial }_{\mathbf{q}}\psi ·\dot{\mathbf{q}}+\mathbf{T}·\mathbf{D}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta \gamma .$$

For simplicity, we neglect cross-coupling terms (i.e., $\dot{\mathbf{q}}$ is independent of $\mathbf{D}$ and $\mathcal{T}$, while $\dot{\mathcal{T}}$ is independent of $\mathbf{q}$ and $\nabla \theta$) and

$$\gamma ={\gamma }_{\mathcal{T}}(\theta ,\rho ,\mathcal{T},\mathbf{D})+{\gamma }_{\mathbf{q}}(\theta ,\rho ,\mathbf{q},\nabla \theta );$$

both ${\gamma }_{\mathcal{T}}$ and ${\gamma }_{\mathbf{q}}$ are non-negative. Hence, we have

$$-\rho {\partial }_{\mathcal{T}}\psi ·\dot{\mathcal{T}}+\mathcal{T}·\mathbf{D}=\rho \theta {\gamma }_{\mathcal{T}}.$$

(44)

$$-\rho {\partial }_{\mathbf{q}}\psi ·\dot{\mathbf{q}}-{\displaystyle \frac{1}{\theta }}\mathbf{q}·\nabla \theta =\rho \theta {\gamma }_{\mathbf{q}}.$$

(45)

By applying the representation formula to (44), it follows

$$\dot{\mathcal{T}}={\displaystyle \frac{\mathcal{T}·\mathbf{D}-\rho \theta {\gamma }_{\mathcal{T}}}{\rho |{\partial }_{\mathcal{T}}{\psi |}^2}}{\partial }_{\mathcal{T}}\psi +[\mathsf{I}-{\displaystyle \frac{{\partial }_{\mathcal{T}}\psi \otimes {\partial }_{\mathcal{T}}\psi }{|{\partial }_{\mathcal{T}}{\psi |}^2}}]\Xi ,$$

(46)

where $\Xi$ is any tensor function of $\theta ,\rho ,\mathcal{T},\mathbf{D}$. Equation (46) is not objective because $\dot{\mathcal{T}}$ is not. Thus, the second law per se does not provide objective evolution equations. Yet, we show that thermodynamic consistency and objectivity are compatible.

Consider the objective time derivative

$$\stackrel{\diamond }{\mathcal{T}}=\dot{\mathcal{T}}-\mathbf{W}\mathcal{T}-\mathcal{T}{\mathbf{W}}^T+c_1(\mathbf{D}\mathcal{T}+\mathcal{T}\mathbf{D})+c_2(\nabla ·\mathbf{v})\mathcal{T},$$

with generic coefficients $c_1,c_2$, and examine the thermodynamic consistency of the sought constitutive function $\stackrel{\diamond }{\mathcal{T}}(\theta ,\rho ,\mathcal{T},\mathbf{D})$. Substitution of $\dot{\mathcal{T}}$ in (44) results in

$$-\rho {\partial }_{\mathcal{T}}\psi ·[\stackrel{\diamond }{\mathcal{T}}+\mathbf{W}\mathcal{T}+\mathcal{T}{\mathbf{W}}^T-c_1(\mathbf{D}\mathcal{T}+\mathcal{T}\mathbf{D})-c_2(\nabla ·\mathbf{v})\mathcal{T}]+\mathcal{T}·\mathbf{D}=\rho \theta {\gamma }_{\mathcal{T}}.$$

(47)

Since

$${\partial }_{\mathcal{T}}\psi ·[\mathbf{W}\mathcal{T}+\mathcal{T}{\mathbf{W}}^T]=\mathbf{W}·[{\partial }_{\mathcal{T}}\psi {\mathcal{T}}^T+{({\partial }_{\mathcal{T}}\psi )}^T\mathcal{T}],$$

then the linearity and arbitrariness of $\mathbf{W}$ in (47) imply

$${\partial }_{\mathcal{T}}\psi {\mathcal{T}}^T+{({\partial }_{\mathcal{T}}\psi )}^T\mathcal{T}\in \mathrm{Sym};$$

(48)

the condition (48) holds if $\psi$ depends on $\mathcal{T}$ through the invariants $\mathrm{tr}\mathcal{T}$ or $\left|\mathcal{T}\right|$. Hence, Equation (47) reduces to

$$-\rho {\partial }_{\mathcal{T}}\psi ·[\stackrel{\diamond }{\mathcal{T}}-c_1(\mathbf{D}\mathcal{T}+\mathcal{T}\mathbf{D})-c_2(\nabla ·\mathbf{v})\mathcal{T}]+\mathcal{T}·\mathbf{D}=\rho \theta {\gamma }_{\mathcal{T}}.$$

For definiteness, let

$$\psi ={\psi }_0(\theta ,\rho )+\frac{1}{2}\lambda {\left|\mathcal{T}\right|}^2.$$

The representation formula for $\stackrel{\diamond }{\mathcal{T}}$ yields

$$\stackrel{\diamond }{\mathcal{T}}=-{\displaystyle \frac{\theta {\gamma }_{\mathcal{T}}}{{\lambda \left|\mathcal{T}\right|}^2}}\mathcal{T}-{\displaystyle \frac{c_1(\mathbf{D}\mathcal{T}+\mathcal{T}\mathbf{D})·\mathcal{T}+c_2(\nabla ·\mathbf{v}){\left|\mathcal{T}\right|}^2}{{\left|\mathcal{T}\right|}^2}}\mathcal{T}+{\displaystyle \frac{\mathcal{T}·\mathbf{D}}{{\rho \lambda \left|\mathcal{T}\right|}^2}}\mathcal{T}+[\mathsf{I}-{\displaystyle \frac{\mathcal{T}\otimes \mathcal{T}}{{\left|\mathcal{T}\right|}^2}}]\Xi .$$

(49)

Equation (49) shows various terms for the objective derivative $\stackrel{\diamond }{\mathcal{T}}$, namely the term induced by the entropy production, the terms due to the chosen derivative, the term induced by the power $\mathcal{T}·\mathbf{D}$ and the term associated with the transverse tensor $[\mathsf{I}-\mathcal{T}\otimes \mathcal{T}/|\mathcal{T}{|}^2]\Xi$. Among the particular cases, we may consider the tensor $\Xi =\beta \mathbf{D}$. Letting $\beta =1/\rho \lambda$, we find that

$$\stackrel{\diamond }{\mathcal{T}}+{\displaystyle \frac{\theta {\gamma }_{\mathcal{T}}}{{\lambda \left|\mathcal{T}\right|}^2}}\mathcal{T}=-{\displaystyle \frac{c_1(\mathbf{D}\mathcal{T}+\mathcal{T}\mathbf{D})·\mathcal{T}+c_2(\nabla ·\mathbf{v}){\left|\mathcal{T}\right|}^2}{{\left|\mathcal{T}\right|}^2}}\mathcal{T}+{\displaystyle \frac{1}{\lambda \rho }}\mathbf{D}.$$

If, further, $c_1=0,c_2=0$, then we have

$$\stackrel{\diamond }{\mathcal{T}}+{\displaystyle \frac{\theta {\gamma }_{\mathcal{T}}}{{\lambda \left|\mathcal{T}\right|}^2}}\mathcal{T}={\displaystyle \frac{1}{\lambda \rho }}\mathbf{D}.$$

(50)

Equation (50) can be viewed as the tensor form of the Maxwell–Wiechert fluid model ([2], sec. 6.1.4). Accordingly,

$$\tau ={\displaystyle \frac{{\lambda \left|\mathcal{T}\right|}^2}{\theta {\gamma }_{\mathcal{T}}}},$$

can be viewed as the relaxation time. In stationary conditions ($\stackrel{\diamond }{\mathcal{T}}=\mathbf{0}$), we have

$$\mathcal{T}=2\mu \mathbf{D},\mu ={\displaystyle \frac{{\left|\mathcal{T}\right|}^2}{2\theta {\gamma }_{\mathcal{T}}\rho }},$$

thus ascribing to $\mu$ the role of viscosity coefficient. The assumption ${\gamma }_{\mathcal{T}}>0$ implies that $\mu >0$.

Remarks about Rate Equations and Thermodynamic Restrictions

Depending on the set of variables, it may happen that the CD inequality has the form (as, e.g., in (44))

$$f(\theta ,\rho ,\mathcal{T},\mathbf{D},\dot{\mathcal{T}})=0.$$

If ${\partial }_{\dot{\mathcal{T}}}f\ne \mathbf{0}$, then by the representation formula, we can derive $\dot{\mathcal{T}}$, as in (46), but this rate equation would be non-objective. Hence, objectivity is an additional requirement. In fact, this requirement is realized by replacing the time derivative with an objective derivative, say $\stackrel{\diamond }{\mathcal{T}}$, and then deriving the (objective) equation for $\stackrel{\diamond }{\mathcal{T}}$. Owing to the non-uniqueness of objective derivatives, this shows that thermodynamics and objectivity lead to a class of rate-type models rather than to a single equation.

The need of objective constitutive equations and the non-uniqueness of the objective variables are well established in the literature. For instance, in connection with fluids of differential type [24,25], the dependence on the variables $\mathbf{L},\dot{\mathbf{L}},\ddot{\mathbf{L}},...$ is replaced from the start with that on the Rivlin–Ericksen tensors ${\mathbf{A}}_1,{\mathbf{A}}_2,{\mathbf{A}}_3,...$, e.g.,

$$\widehat{w}(\mathbf{L},\dot{\mathbf{L}},\ddot{\mathbf{L}},...)=\tilde{w}({\mathbf{A}}_1,{\mathbf{A}}_2,{\mathbf{A}}_3,...).$$

The literature also shows another way of modelling through objective derivatives. The starting assertion in [26] is that a scalar subjected to an objective rate is equal to its conventional time rate. Hence, one might expect the validity of the equation

$$\dot{G}=\dot{\mathbf{a}}·\mathbf{b}+\mathbf{a}·\dot{\mathbf{b}}=\stackrel{\diamond }{\mathbf{a}}·\mathbf{b}+\mathbf{a}·\stackrel{\diamond }{\mathbf{b}}=\stackrel{\diamond }{G}$$

for the two vector functions $\mathbf{a},\mathbf{b}$ and $G=\mathbf{a}·\mathbf{b}$; in [26] $\stackrel{\diamond }{G}$ is the logarithmic derivative. Yet, this equation in general is not true. For simplicity, let

$$\stackrel{\diamond }{\mathbf{a}}=\dot{\mathbf{a}}-\mathbf{W}\mathbf{a}+\mathbf{D}\mathbf{a}.$$

We have

$$\stackrel{\diamond }{\mathbf{a}}·\mathbf{b}+\mathbf{a}·\stackrel{\diamond }{\mathbf{b}}=\dot{\mathbf{a}}·\mathbf{b}+\mathbf{a}·\dot{\mathbf{b}}-(\mathbf{b}\otimes \mathbf{a}+\mathbf{a}\otimes \mathbf{b})·\mathbf{W}+(\mathbf{b}\otimes \mathbf{a}+\mathbf{a}\otimes \mathbf{b})·\mathbf{D}.$$

Since $\mathbf{b}\otimes \mathbf{a}+\mathbf{a}\otimes \mathbf{b}\in \mathrm{Sym}$, then $(\mathbf{b}\otimes \mathbf{a}+\mathbf{a}\otimes \mathbf{b})·\mathbf{W}=0$, but $(\mathbf{b}\otimes \mathbf{a}+\mathbf{a}\otimes \mathbf{b})·\mathbf{D}$ need not be zero. Accordingly, in general, the most convenient procedure is to start from the time derivatives and then to replace them through the selected objective derivatives.

8. Conclusions

This paper addresses the modelling of material behaviour in continuous bodies. In general, non-locality effects, in space and time, have to be described by involving time and space dependence of the pertinent fields. In this connection, the paper looks at the modelling of constitutive properties in terms of differential equations as relations among space and time derivatives of various orders.

The occurrence of time derivatives has to comply with the objectivity principle. Objective equations are obtained in two ways. In the Lagrangian description some pertinent fields (e.g., ${\mathbf{T}}_{RR},{\mathbf{q}}_R,\mathbf{E}$, as well as the densities $\psi ,\epsilon$) are invariant under the set of Euclidean transformations. Hence, their time derivatives are invariant and objective. Furthermore, the recourse to the Lagrangian description naturally allows models with finite deformations. Instead, in the Eulerian description, the standard fields (e.g., $\mathbf{T},\mathbf{q},\mathbf{D}$) are not invariant and their derivatives are not objective. Objectivity is obtained by using objective time derivatives; the corotational derivative is the simplest one and seemingly the most appropriate.

Physically admissible models are required to comply with the second law of thermodynamics. Here, the postulate of the second law is eventually expressed in terms of the Clausius–Duhem inequality. Two features have to be emphasized with respect to the tradition. Firstly, the (non-negative) entropy production is viewed as a constitutive function per se and not merely deduced from the other constitutive properties. Secondly, the Clausius–Duhem inequality is viewed as a constraint on the pertinent fields and it is solved with respect to the interested vector or tensor by using a representation formula (Section 2; see also [2]). This allows for qualitatively different constitutive equations, as is emphasized in Section 4, Section 5 and Section 6. Rather than confining to the admissibility of a single equation, the procedure leads to the admissibility of a class of models characterized by the entropy production $\gamma$ and the transverse vector or tensor (determined by $\mathbf{\xi }$ or $\Xi$).

Section 3, Section 4 and Section 5 are devoted to models of heat conduction, within Lagrangian descriptions, while Section 6 develops some models of the Navier–Stokes–Voigt fluid, within Eulerian descriptions. In connection with thermo-viscous fluids, Section 7 gives a systematic approach to the modelling of evolution equations in the Eulerian description. It is shown that the thermodynamic consistency is compatible with both objective and non-objective evolution equations, which emphasizes that objectivity is an independent principle of continuum mechanics.