1 Introduction

Compressible fluids are usually described by a function \(p(\rho , \theta )\) specifying the pressure p in terms of the mass density \(\rho\) and the (absolute) temperature \(\theta\). In practice we should account also for relaxation phenomena where the connection between p and \(\rho\), or \(\theta\), depends on the rate at which the phenomenon occurs. A dependence on rates is likely to determine hysteresis and visco-elastic effects. The thermodynamic consistency should determine operative schemes where these effects are modelled. Lately we have developed thermodynamic analyses where rate dependences result in hysteretic effects via the constitutive properties of the entropy production.

The idea that the entropy production \(\sigma\) be given by a constitutive equation traces back to Green and Naghdi [9]. Conceptually our approach extends this idea assuming from the outset that all constitutive functions depend on the common set of (physical) variables. However, unlike the Green-Naghdi theories, here \(\sigma\) is independent of the constitutive prescription of \(\psi\) and is required to be non-negative in all processes. We mention that a similar approach was developed in [4, 5] for deformable ferroelectrics, in [6, 7] for elastic-plastic materials and in [8] for viscoelastic and viscoplastic materials.

It is a purpose of this paper to establish models of fluids where rate-type dependences play a significant role and to cast them in a rational thermodynamic scheme. Recently, rheological hysteresis has been studied systematically in a wide range of complex fluids [2, 20]. Also in light of late observations of hysteresis effects in fluids [11, 24, 25], in Sec. 3 we show that a constitutive scheme of fluids is determined where hysteretic properties, of the dependence of pressure on the mass density, are modelled by a set of functions of mass density parameterized by the temperature.

The Navier–Stokes–Fourier model is a well-known scheme of dissipative fluids where viscosity, heat conduction, and compressibility occur in a linear way. The model is however too simple in many circumstances and is claimed to result in a parabolic system of differential equations. In the literature the Oldroyd-B model is frequently employed for incompressible fluids; the Cauchy stress tensor \(\mathbf{T}\) is assumed to satisfy the constitutive equation

$$\mathbf{T}+ \lambda _1 \mathop {\mathbf{T}}\limits^{\triangledown} = 2 \mu (\mathbf{D}+ \lambda _2 \mathop {\mathbf{D}}\limits^{\triangledown}),$$
(1)

the superposed triangle \({^{\triangledown }}\) denoting the upper convected time derivative; as with any tensor \(\buildrel {\scriptscriptstyle {\triangledown }}\over {\mathbf{T}} = \dot{\mathbf{T}} - \mathbf{L}\mathbf{T}- \mathbf{T}\mathbf{L}^T .\) The assumption (1) shows some disadvantages and, in particular, it leads to a parabolic equation for the velocity \(\mathbf{v}\). Further there are nontrivial questions about the thermodynamic consistency. This is so because the natural description should involve the second Piola-Kirchhoff stress \(\mathbf{T}_{{\scriptscriptstyle R}{\scriptscriptstyle R}}\) and the Green-St.Venant strain \(\mathbf{E}\) so that \(\mathbf{T}_{{\scriptscriptstyle R}{\scriptscriptstyle R}} \cdot \dot{\mathbf{E}}\) is the mechanical power.

In this connection we mention Ref.  [19] where Oldroyd-type fluids are modelled by means of a strain tensor and a tensorial internal variable \({{\varvec{\alpha }}}\) along with the Maxwell–Cattaneo equation for the heat flux \(\mathbf{q}\) so that the whole model is hyperbolic. We remark that the occurrence of the time derivatives \(\dot{\mathbf{q}}, \dot{{{\varvec{\alpha }}}}\) in the constitutive equations looks in contrast with the objectivity principle. This remark is consistent with Ref.  [14] where objective Rivlin–Eriksen tensors are involved and Ref.   [1] where the time derivative of \(\mathbf{q}\) is replaced by a Lie-Oldroyd upper convected derivative.

The upper convected Maxwell model is the particular case of (1) with \(\lambda _2 = 0\). This allows the model to result in a hyperbolic system of equations but still leaves unsolved the thermodynamic derivation of the free energy. To overcome this problem we follow an objective rate equation where the objective derivative is the corotational one. Meanwhile, to emphasize the role of compressibility, we allow a dependence of the time derivative on the divergence of the velocity. In section 4 we let the stress be split into a pressure tensor and an extra stress tensor \({{\varvec{\mathcal {T}}}}\). To account for objectivity and compressibility we let \({{\varvec{\mathcal {T}}}}\) be subject to a rate that combines the corotational dependence and that on the divergence of the velocity; an analogous approach for incompressible fluids is developed in [8].

2 Balance laws

We consider a fluid occupying a time-dependent region \(\Omega \subset \mathscr {E}^3\). Throughout \(\rho\) is the mass density, \(\mathbf{v}\) the velocity, \(\mathbf{T}\) the symmetric stress tensor, \(\varepsilon\) the internal energy, \(\mathbf{q}\) the heat flux, p the pressure. The symbol \(\nabla\) denotes the gradient operator, \(\partial _t\) is the partial time derivative, at a point \(\mathbf{x}\in \Omega\), while a superposed dot stands for the total time derivative, \(\dot{f} = \partial _t f + \mathbf{v}\cdot \nabla f\). Further, \(\mathbf{L}\) is the velocity gradient, \(L_{ij} = \partial _{x_j} v_i\), \(\mathbf{D}= \mathrm{sym}\mathbf{L}\) is the stretching tensor, \(\mathbf{W}= \mathrm{skw}\mathbf{L}\) is the spin tensor. Moreover, \(\mathbf{b}\) is the body force per unit mass, r is the heat supply

The balance of mass, linear momentum, and energy for any region of the fluid results in the local equations

$$\begin{array}{*{20}c} {\dot{\rho } = - \rho \nabla \cdot {\mathbf{v}},} \\ {\rho \mathop {\mathbf{v}}\limits^{.} = \nabla \cdot {\mathbf{T}} + \rho {\mathbf{b}},} \\ {\rho \dot{\varepsilon } = {\mathbf{T}} \cdot {\mathbf{D}} - \nabla \cdot {\mathbf{q}} + \rho r.} \\ \end{array}$$
(2)

The entropy inequality is assumed to be

$$\rho \dot{\eta } + \nabla \cdot \frac{\mathbf{q}}{\theta } - \frac{\rho r}{\theta } \ge 0 ;$$
(3)

the subsequent developments show that an extra-entropy flux would be redundant. We define the entropy production \(\sigma\),

$$\sigma = \rho \dot{\eta } + \nabla \cdot \frac{\mathbf{q}}{\theta } - \frac{\rho r}{\theta }$$
(4)

and accordingly \(\sigma \ge 0\). As is common in Rational Thermodynamics, physically admissible constitutive equations are required to satisfy \(\sigma \ge 0\) for any appropriate set of fields compatible with the balance equations.

Substituting \(\rho r - \nabla \cdot \mathbf{q}\) from (2) into (3) we have

$$\theta \sigma = - \rho (\dot{\varepsilon } - \theta \dot{\eta }) + \mathbf{T}\cdot \mathbf{D}- \frac{1}{\theta }\mathbf{q}\cdot \nabla \theta ,$$

whence, in terms of the Helmholtz free energy \(\psi = \varepsilon - \theta \eta\), we can write the Clausius–Duhem inequality in the form

$$- \rho (\dot{\psi } + \eta \dot{\theta }) + \mathbf{T}\cdot \mathbf{D}- \frac{1}{\theta }\mathbf{q}\cdot \nabla \theta = \theta \sigma \ge 0.$$
(5)

As we show in the next section, we let \(\sigma\) be given by a constitutive function. In this regard two cases may happen. First, \(\sigma\) coincides with what follows from the definition. As a remarkable example, in connection with the Navier–Stokes–Fourier model, where

$$\mathbf{T}= - p \mathbf{1}+ 2\mu \mathbf{D}+ \lambda (\mathrm{tr}\mathbf{D})\mathbf{1}, \qquad \mathbf{q}= - \kappa \nabla \theta ,$$

we find that

$$\theta \sigma = 2\mu \mathbf{D}\cdot \mathbf{D}+ \lambda (\mathrm{tr}\mathbf{D})^2 + \frac{\kappa }{\theta } \vert \nabla \theta \vert ^2.$$

This shows the constitutive property of the entropy production for the Navier–Stokes–Fourier model [10]. More interestingly, we may have cases where \(\sigma\) is assumed to have a constitutive dependence per se. This happens for the hysteretic models developed in [5, 6, 18] where the function \(\sigma\) has to be non-negative and meanwhile to satisfy (4) as an equality, not an identity. This case occurs also in the constitutive theory developed in the next section.

3 Pressure-rate effects in inviscid fluids

We consider compressible fluids where the constitutive equations may depend on the rate of the pertinent variables, including the pressure. For simplicity we neglect viscosity effects and then we let \(\mathbf{T}= - p \mathbf{1}\). Hence we take

$$\Xi = (\rho , \theta , p, \nabla \theta , \dot{\rho }, \dot{p})$$

as the set of independent variables for the functions \(\psi , \eta , \mathbf{q}\) and \(\sigma\). Upon evaluation of \(\dot{\psi }\) and substitution in (5) we obtain

$$\rho (\partial _\theta \psi + \eta )\dot{\theta } + (\rho \partial _\rho \psi - \frac{p}{\rho } )\dot{\rho } + \rho \partial _p \psi \dot{p} + \rho \partial _{\dot{p}}\,\ddot{p} + \rho \partial _{\nabla \theta } \psi \cdot \dot{\overline{\nabla \theta }} + \rho \partial _{\dot{\rho }}\psi \, \ddot{\rho } + \frac{1}{\theta }\mathbf{q}\cdot \nabla \theta = - \theta \sigma \le 0 .$$
(6)

The linearity and arbitrariness of \(\dot{\theta }, \dot{\overline{\nabla \theta }}, \ddot{\rho }, \ddot{p}\) imply that \(\psi\) be independent of \(\nabla \theta , \dot{\rho }, \dot{p}\) and hence

$$\psi = \hat{\psi }(\rho , \theta , p), \qquad \eta = -\partial _\theta \psi .$$

Consequently inequality (6) reduces to

$$(\rho \partial _\rho \psi - \frac{p}{\rho } )\dot{\rho } + \rho \partial _p \psi \dot{p} + \frac{1}{\theta }\mathbf{q}\cdot \nabla \theta \le 0 .$$
(7)

This result indicates that in general \(\dot{\rho }, \dot{p}, \nabla \theta\) might be related so that the inequality holds.

Since \(\psi\) is independent of \(\nabla \theta\), we let \(\nabla \theta = \mathbf{0}\) and write (7) in the form

$$(\partial _\rho \psi - \frac{p}{\rho ^2} )\dot{\rho } + \partial _p \psi \dot{p} = - \theta \sigma _{\rho p},$$
(8)

where \(\sigma _{\rho p}\) is the entropy production when \(\nabla \theta = \mathbf{0}\). Consequently

$$\mathbf{q}\cdot \nabla \theta = - \theta ^2 \sigma _q, \qquad \sigma _q = \sigma - \sigma _{\rho p}.$$

A Fourier-like relation for \(\mathbf{q}\),

$$\mathbf{q}= - \kappa (\rho , \theta , p, \vert \nabla \theta \vert ) \nabla \theta , \qquad \kappa >0,$$

implies that \(\sigma _q = \kappa \vert \nabla \theta \vert ^2/\theta ^2\).

3.1 Duhem-like flows

Equation (8) may be framed within the class of Duhem-like materials [13].

First let \(\sigma _{\rho p} = 0\), that is no entropy dissipation occurs. If \(\partial _p \psi = 0, \psi = \psi (\rho , \theta )\) then

$$(\partial _\rho \psi - p/\rho ^2)\dot{\rho } = 0.$$

The arbitrariness of \(\dot{\rho }\) implies

$$p = \rho ^2 \partial _\rho \psi ,$$
(9)

and hence p and \(\rho\) cannot be independent variables. Otherwise, if \(\psi\) is a \(C^2\) function we obtain a contradiction,

$$\partial _{p \rho }^2 \psi = \frac{1}{\rho ^2}, \qquad \partial _{\rho p}^2 \psi = 0.$$

As it happens for elastic fluids, the dependence of p on \(\rho\) must be expressed by a constitutive relation \(p = \hat{p}(\rho , \theta , p)\), which is named equation of state and is related to \(\hat{\psi }(\rho , \theta )\) by (9).

If, instead, \(\partial _p \psi \ne 0\) then (8) implies

$$\dot{p} = \frac{p/\rho ^2 - \partial _\rho \psi }{\partial _p \psi } \dot{\rho }.$$

We now show that Eq. (8) allows us to describe density-pressure hysteresis. Let \(\partial _p \psi \ne 0\) and assume

$$\theta \sigma _{\rho p} = \gamma (\rho , \theta , p, \dot{\rho })\vert \dot{\rho }\vert , \qquad \gamma > 0.$$

Equation (8) becomes

$$(\partial _\rho \psi - p/\rho ^2)\dot{\rho } + \partial _p \psi \,\dot{p} = - \gamma \vert \dot{\rho }\vert ,$$

whence

$$\dot{p} = \frac{p/\rho ^2 - \partial _\rho \psi }{\partial _p \psi } \dot{\rho } - \frac{\gamma }{\partial _p \psi } \vert \dot{\rho }\vert .$$
(10)

Let

$$\chi _1 = \frac{p/\rho ^2 - \partial _\rho \psi }{\partial _p \psi } , \qquad \chi _2 = -\frac{\gamma }{\partial _p \psi }.$$

Equation (10) becomes

$$\dot{p} = \chi _1 \dot{\rho } + \chi _2 \vert \dot{\rho }\vert .$$

In the classical theory \(\partial _\rho p\) is the square of the speed of sound in isothermal conditions. Here,

$$\partial _\rho p := \frac{\dot{p}}{\dot{\rho }}= \chi _1+\chi _2\,\mathrm{sgn} \,{\dot{\rho }}.$$

We then assume

$$\chi _1 > 0, \qquad \vert \chi _2\vert \le \chi _1.$$
(11)

At constant temperature \(\dot{\psi } = \partial _p \psi \, \dot{p} + \partial _\rho \psi \,\dot{\rho }\) and hence integration of (10), as \(t \in [t_1, t_2]\), along a closed curve in the \(\rho \,\)-\(\,p\) plane results in

$$0\le \int _{t_1}^{t_2} \gamma \vert \dot{\rho }\vert \,dt = \int _{t_1}^{t_2} [-\partial _p \psi \, \dot{p} + (\frac{p}{\rho ^2} - \partial _\rho \psi )\dot{\rho } ]dt = \int _{t_1}^{t_2} \frac{p}{\rho ^2}\,\dot{\rho }\,dt = \oint \frac{p}{\rho ^2}\,d{\rho },$$

\(\oint\) denoting the integral along the closed curve. Accordingly, we have \(\oint p\, d\rho \ge 0\) and this implies that the closed curve is run in the clockwise sense. Moreover, equation (10) is invariant under the time transformation

$$t \rightarrow c\; t, \qquad c > 0,$$

and hence it describes a rate-independent behaviour.

3.1.1 The Helmholtz free energy

To determine the function \(\psi\), thus characterizing the model, we start with the generic assumption

$$\psi (\rho , p)= \mathcal {L}(p - \mathcal {G}(p)) + \mathcal {F}(p) + \mathcal {H}(\rho ),$$

\(\mathcal {L}, \mathcal {G}, \mathcal {F}, \mathcal {H}\) being undetermined differentiable functions, parameterized by \(\theta\); the dependence on the temperature \(\theta\) is understood and not written. Substitution of \(\partial _p \psi\) and \(\partial _\rho \psi\) yields

$$\chi _1= \frac{p/\rho ^2 + \mathcal {L}'(p - \mathcal {G}(\rho ))\mathcal {G}'(\rho ) - \mathcal {H}'(\rho )}{\mathcal {L}'(p - \mathcal {G}(\rho )) + \mathcal {F}'(p)},\qquad \chi _2= - \frac{\gamma }{\mathcal {L}'(p - \mathcal {G}(\rho )) + \mathcal {F}'(p)}.$$

For simplicity we assume

$$\chi _1 = g(\rho ),$$

where g is a positive function parametrized by \(\theta\). Hence it follows the requirement

$$\frac{p}{\rho ^2} - \mathcal {F}'(p) g(\rho ) - \mathcal {L}'(p - \mathcal {G}(\rho )) [g(\rho ) - \mathcal {G}'(\rho )] = \mathcal {H}'(\rho )$$

which is satisfied by letting \(\mathcal {F}'= 0\) and

$$g(\rho ) = \mathcal {G}'(\rho ) + \frac{\alpha }{\rho ^2}, \quad \mathcal {H}'(\rho ) = \frac{\mathcal {G}(\rho )}{\rho ^2}, \mathcal {L}'(p - \mathcal {G}(\rho )) = \frac{1}{\alpha }(p - \mathcal {G}(\rho )),$$

for some \(\alpha \ne 0\). For definiteness we let \(\alpha > 0\). As well as the constant function \(\mathcal {F}(p)\), \(\alpha\) can in fact depend on the temperature \(\theta\).

In summary, we have

$$\psi (\rho , p) = \frac{1}{2 \alpha }[p - \mathcal {G}(\rho )]^2 + \mathcal {H}(\rho ) + \mathcal {F},$$

where \(\mathcal {H}'(\rho ) = \mathcal {G}(\rho )/\rho ^2\). The characteristic functions \(\chi _1, \chi _2\) are given by

$$\chi _1 = \mathcal {G}'(\rho ) + \frac{\alpha }{\rho ^2}, \qquad \chi _2 = - \frac{\alpha \gamma }{p - \mathcal {G}(\rho )},$$

so that \(\alpha\) and \(\mathcal {G}\) completely characterize the model. The term \(\alpha /\rho ^2\) gives the difference between the differential Boyle’s factor \(\chi _1\) and the slope of the function \(\mathcal {G}(\rho )\). The monotonicity condition (11) requires

$$\mathcal {G}'(\rho )>-\frac{\alpha }{\rho ^2}, \ \ \vert p - \mathcal {G}(\rho )\vert \ge \frac{\alpha \gamma \rho ^2}{ \vert \rho ^2\,\mathcal {G}'(\rho ) + {\alpha }\vert }$$
(12)

Moreover, it turns out \(\chi _1 - \mathcal {G}'(\rho ) \rightarrow 0\) as \(\rho \rightarrow \infty\).

3.1.2 A simple example

The function \(\mathcal {G}\) is taken to be positive and quadratic increasing as \(\rho >0\),

$$\mathcal {G}(\rho ) = \kappa _0 \rho ^2+\kappa _1,$$

\(\kappa _0,\kappa _1 >0\) being possibly dependent on the temperature \(\theta\). Hence

$$\psi = \frac{1}{2 \alpha }[p - \kappa _0\rho ^2-\kappa _1]^2 + \kappa _0\rho -\frac{\kappa _1}{\rho }+ \mathcal {C},$$

\(\mathcal {C}\) being a constant possibly dependent on \(\theta\). Moreover, \(\chi _1 = 2\kappa _0\rho + \alpha /\rho ^2\). The hysteretic function \(\gamma\) is taken in the regular form

$$\gamma (\rho , p) = \frac{1}{\beta } (p- \kappa _0 \rho ^2-\kappa _1)^2, \qquad \beta > 0.$$

Hence, the slope of the hysteretic path is given by

$$\frac{dp}{d\rho } =2\kappa _0\rho +\frac{\alpha }{\rho ^2}+ \frac{\alpha }{\beta } (p- \kappa _0 \rho ^2-\kappa _1)\mathrm{sgn}\,\dot{\rho }.$$

Figure 1 shows an example of the loops obtained by solving the system of equations

$$\begin{aligned} \left\{ \begin{array}{l} \dot{\rho } = \Big .\frac{1}{2}\omega {\mathfrak {E}}\sin \omega t, \\ \displaystyle \dot{p} = \Big (2\kappa _0\rho + \frac{\alpha }{\rho ^2}\Big ) \dot{\rho } - \frac{\alpha }{\beta } (p-\kappa _0\rho -\kappa _1)\vert \dot{\rho }\vert , \end{array}\right. \end{aligned}$$

with initial values \(\rho _0, p_0\) (the density oscillates between \(\rho _0\) and \({\mathfrak {E}}+\rho _0\)). The rate independence of the model allows the loops to be independent of the angular frequency \(\omega\). The monotonicity requirement (12) is satisfied provided that

$$\vert p- \kappa _0 \rho ^2-\kappa _1\vert \le \frac{\beta }{\alpha } \vert 2\kappa _0\rho +\frac{\alpha }{\rho ^2} \vert .$$

This inequality identifies a wide region in the \(\rho \,\)-\(\,p\) plane around the characteristic curve \(p=\mathcal {G}(\rho )\). Similar hysteretic cycles are described in some special fields of fluid mechanics (see, for instance, [11, 25]).

Fig. 1
figure 1

Isothermal pressure-density hysteresis loops (solid) and the graph of the function \(\mathcal {G}\) (dashed) corresponding to \(\alpha = 0.2\), \(\kappa _0 = 0.5\), \(\kappa _1 = 4\), \(\beta = 3/2\), with \({\mathfrak {E}}=3.4\) and \(\rho _0=0.2\), \(p_0=0,1\)

Full size image

3.2 A Bingham-like model

By mimicking the loading criteria for elastic-plastic materials (see, for instance, [22]) we develop here a different model of dissipative fluid that does not fall into the Duhem-like class but that nevertheless highlights a hysteretic behavior. For formal convenience we now consider the specific volume \(\upsilon = 1/\rho\) as an independent variable instead of density. Letting \(\tilde{\psi }(\upsilon , \theta ,p) = \psi (\rho , \theta ,p)\) we observe that

$$\partial _\upsilon \tilde{\psi } = - \rho ^2 \partial _\rho \psi , \quad \dot{\upsilon } = - \frac{\dot{\rho }}{\rho ^2}, \quad \dot{\rho } = - \frac{\dot{\upsilon }}{\upsilon ^2}.$$

Equation (8) can then be written as

$$(\partial _\upsilon \tilde{\psi } + p)\dot{\upsilon } + \partial _p \tilde{\psi } \dot{p} = - \theta \sigma _{\rho p}$$
(13)

At constant temperature \(\dot{\psi } = \partial _p \psi \, \dot{p} + \partial _\upsilon \psi \,\dot{\upsilon }\) and hence integration of (13), as \(t \in [t_1, t_2]\), along a closed curve in the \(\upsilon \,\)-\(\,p\) plane results in

$$0\le \int _{t_1}^{t_2} \theta \sigma _{\rho p} \,dt = \int _{t_1}^{t_2} [-\partial _p \psi \, \dot{p} - (p + \partial _\upsilon \psi )\dot{\upsilon } ]dt {}=- \int _{t_1}^{t_2} p\,\dot{\upsilon }\,dt = - \oint p\,d{\upsilon },$$

\(\oint\) denoting the integral along the closed curve. Hence we have \(\oint p\, d\upsilon \le 0\) and this implies that the closed curve is run in the counterclockwise sense.

We then take the pressure p and the specific volume \(\upsilon = 1/\rho\) as the independent variables and we assume

$$\theta \sigma _{\rho p} =- \gamma H(\vert p-p^*\vert -\beta ) H(-p\dot{\upsilon }) p \dot{\upsilon } ,$$

where \(p^*>\beta>0,\,\gamma > 0\) and H denotes the Heaviside step function. Equation (13) then becomes

$$(\partial _\upsilon {\tilde{\psi }} + p - \gamma H(\vert p-p^*\vert -\beta )H(-p \dot{\upsilon }) p]\dot{\upsilon } + \partial _p {\tilde{\psi }} \dot{p} = 0.$$

Hence we obtain \(dp/d\upsilon = \dot{p}/\dot{\upsilon }\) in the form

$$\frac{dp}{d\upsilon } = - \frac{\partial _\upsilon {\tilde{\psi }} + p[1 - \gamma H(\vert p-p^*\vert -\beta ) H(-p\dot{\upsilon })]}{\partial _p {\tilde{\psi }}}.$$

The simple case where \(\psi =\frac{1}{2\alpha }p^2 + \psi _0\), \(\psi _0\) and \(\alpha\) being possibly dependent on \(\theta\), leads to

$$\frac{dp}{d\upsilon } = - \alpha [1 - \gamma H(\vert p-p^*\vert -\beta ) H(-p\dot{\upsilon })],$$

where \(-\alpha\) denotes the linear compliance inside the elastic region \(\mathcal {E}=\{(\upsilon ,p): \vert p-p^*\vert <\beta \}\). If \(p^*=0\) then \(\beta\) plays the role of a yield pressure and the model looks like a Bingham fluid.

4 Compressible fluids with objective rates

To describe dissipative effects in fluids the symmetric stress tensor \(\mathbf{T}\) is often decomposed in a pressure tensor \(-p \mathbf{1}\) and an extra stress tensor,

$${{\varvec{\mathcal {T}}}}= \mathbf{T}+ p \mathbf{1}\in \mathrm{Sym}.$$

The extra stress \({{\varvec{\mathcal {T}}}}\) is then assumed to be governed by a rate-type constitutive equation as it happens for the Oldroyd fluid model [21]. Borrowing from this decomposition here we describe dissipative fluids by assuming that the free energy \(\psi\), the pressure p, and appropriate objective derivatives \(\buildrel \diamond \over {{{\varvec{\mathcal {T}}}}}\) and \(\buildrel \diamond \over {\mathbf{q}}\), of \({{\varvec{\mathcal {T}}}}\) and \(\mathbf{q}\), are functions of the set of variables

$$\Xi = (\rho , \theta , {{\varvec{\mathcal {T}}}}, \mathbf{q}, \mathbf{D}, \nabla \theta ).$$

Some remarks are in order about the choice of the objective derivative. The Oldroyd derivative is specially suited for viscoelastic, incompressible fluids. Compressibility is involved in the Truesdell derivative \(\buildrel {\scriptscriptstyle {\Box }}\over {{{\varvec{\mathcal {T}}}}}\) [17]. However \(\buildrel {\scriptscriptstyle {\Box }}\over {{{\varvec{\mathcal {T}}}}}\) is strictly associated with the time derivative of the second Piola-Kirchhoff stress \(\mathbf{T}_{{\scriptscriptstyle R}{\scriptscriptstyle R}}\) [12] and hence mainly appropriate for the modelling of solids. The simplest objective derivative is the corotational one namely [17, 18]

$$\buildrel \circ \over {{{\varvec{\mathcal {T}}}}} = \dot{{{\varvec{\mathcal {T}}}}} - \mathbf{W}{{\varvec{\mathcal {T}}}}+ {{\varvec{\mathcal {T}}}}\mathbf{W}, \qquad \buildrel \circ \over {\mathbf{q}} = \dot{\mathbf{q}} - \mathbf{W}\mathbf{q},$$

\(\mathbf{W}\) being the spin tensor of the fluid. To emphasize the effects of compressibility we add an objective term directly related to the divergence of the velocity. We then consider

$$\begin{aligned} \buildrel \diamond \over {{{\varvec{\mathcal {T}}}}}&:= \dot{{{\varvec{\mathcal {T}}}}} - \mathbf{W}{{\varvec{\mathcal {T}}}}+ {{\varvec{\mathcal {T}}}}\mathbf{W}+ \nu (\nabla \cdot \mathbf{v}){{\varvec{\mathcal {T}}}}, \\ \buildrel \diamond \over {\mathbf{q}}\,&:= \dot{\mathbf{q}} - \mathbf{W}\mathbf{q}+ \nu (\nabla \cdot \mathbf{v}) \mathbf{q}.\end{aligned}$$
(14)

If \(\mathbf{W}\) is replaced by \(\mathbf{L}\) and \(\nu =1\) then (14) becomes the Truesdell [17, 18] or Oldroyd-Lie [3] derivative; if \(\nu = 0\) then (14) becomes the corotational derivative.

We now evaluate the thermodynamic restrictions placed on the functions \(\psi , p, \buildrel \diamond \over {{{\varvec{\mathcal {T}}}}}, \buildrel \diamond \over {\mathbf{q}}\) of \(\Xi\). Evaluation of \(\dot{\psi }\) and substitution in the Clausius–Duhem inequality (5) results in

$$-\rho (\partial _\theta \psi + \eta ) \dot{\theta } - \rho \partial _\rho \psi \dot{\rho } - \rho \partial _{{{\varvec{\mathcal {T}}}}} \psi \cdot \dot{{{\varvec{\mathcal {T}}}}}- \rho \partial _\mathbf{q}\psi \cdot \dot{\mathbf{q}} - \rho \partial _\mathbf{D}\psi \cdot \dot{\mathbf{D}} -\rho \partial _{\nabla \theta } \psi \cdot \dot{\overline{\nabla \theta }}- p \nabla \cdot \mathbf{v}+ {{\varvec{\mathcal {T}}}}\cdot \mathbf{D}- \frac{1}{\theta }\mathbf{q}\cdot \nabla \theta \ge 0.$$

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

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

The Clausius–Duhem inequality reduces to

$$\begin{aligned}- \rho \partial _\rho \psi \dot{\rho } - \rho \partial _{{{\varvec{\mathcal {T}}}}} \psi \cdot \dot{{{\varvec{\mathcal {T}}}}} - \rho \partial _\mathbf{q}\psi \cdot \dot{\mathbf{q}} - p \nabla \cdot \mathbf{v}+ {{\varvec{\mathcal {T}}}}\cdot \mathbf{D}- \frac{1}{\theta }\mathbf{q}\cdot \nabla \theta&\ge 0. \end{aligned}$$
(15)

The time derivatives \(\dot{{{\varvec{\mathcal {T}}}}}\) and \(\dot{\mathbf{q}}\) are given by

$$ \dot{\mathbf{T}}=\, \buildrel \diamond \over {{{\varvec{\mathcal {T}}}}}(\Xi ) + \mathbf{W}\mathbf{T}-\mathbf{T}\mathbf{W}- \nu (\nabla \cdot \mathbf{v}){{\varvec{\mathcal {T}}}}, \dot{\mathbf{q}}=\, \buildrel \diamond \over {\mathbf{q}}(\Xi ) + \mathbf{W}\mathbf{q}- \nu (\nabla \cdot \mathbf{v})\mathbf{q}. $$

Hence inequality (15) becomes

$$(\rho ^2 \partial _\rho \psi - p)\nabla \cdot \mathbf{v}+ {{\varvec{\mathcal {T}}}}\cdot \mathbf{D}- \frac{1}{\theta }\mathbf{q}\cdot \nabla \theta - \rho \partial _{{{\varvec{\mathcal {T}}}}}\psi \cdot [ \buildrel \diamond \over {{{\varvec{\mathcal {T}}}}} + \mathbf{W}{{\varvec{\mathcal {T}}}}- {{\varvec{\mathcal {T}}}}\mathbf{W}- \nu (\nabla \cdot \mathbf{v}){{\varvec{\mathcal {T}}}} ] - \rho \partial _\mathbf{q}\psi \cdot [\buildrel \diamond \over {\mathbf{q}} + \mathbf{W}\mathbf{q}-\nu (\nabla \cdot \mathbf{v})\mathbf{q}] \ge 0.$$
(16)

where we have replaced \(\dot{\rho }\) with \(-\rho \nabla \cdot \mathbf{v}\). Inequality (16) can be viewed as

$$\rho [{{\varvec{\mathcal {T}}}}\partial _{{{\varvec{\mathcal {T}}}}} \psi -({{\varvec{\mathcal {T}}}}\partial _{{{\varvec{\mathcal {T}}}}} \psi )^T + \mathbf{q}\otimes \partial _\mathbf{q}\psi ]\cdot \mathbf{W}+ ... \ge 0$$

the dots denoting terms independent of \(\mathbf{W}\). The linearity and arbitrariness of \(\mathbf{W}\) imply

$${{\varvec{\mathcal {T}}}}\partial _{{{\varvec{\mathcal {T}}}}} \psi -({{\varvec{\mathcal {T}}}}\partial _{{{\varvec{\mathcal {T}}}}} \psi )^T + \mathbf{q}\otimes \partial _\mathbf{q}\psi \in \mathrm{Sym}$$

whence

$${{\varvec{\mathcal {T}}}}\partial _{{{\varvec{\mathcal {T}}}}} \psi -({{\varvec{\mathcal {T}}}}\partial _{{{\varvec{\mathcal {T}}}}} \psi )^T = \mathbf{0}, \quad \mathbf{q}\otimes \partial _\mathbf{q}\psi \in \mathrm{Sym}.$$
(17)

The requirements (17) hold in general while further restrictions hold depending on the functions \(\buildrel \diamond \over {{{\varvec{\mathcal {T}}}}}, \buildrel \diamond \over {\mathbf{q}}\). For definiteness let

$$ \tau _{\scriptscriptstyle T}\buildrel \diamond \over {{{\varvec{\mathcal {T}}}}} \,= -\mathbf{F}({{\varvec{\mathcal {T}}}}) + 2\mu \mathbf{D}+ \lambda (\nabla \cdot \mathbf{v})\mathbf{1}, \tau _q \buildrel \diamond \over {\mathbf{q}} \,= -\mathbf{f}(\mathbf{q}) - \kappa \nabla \theta , $$
(18)

where \(\tau _{\scriptscriptstyle T},\tau _q, \mathbf{F}, \mathbf{f}, \mu , \lambda , \kappa\) are functions of \(\rho\) and \(\theta\) with \(\tau _{\scriptscriptstyle T}, \tau _q > 0\).

Consider the terms dependent on \(\nabla \theta\) in (16). We have

$$(\frac{\kappa \rho }{\tau _q} \partial _\mathbf{q}\psi - \frac{1}{\theta }\mathbf{q} ) \cdot \nabla \theta + ... \ge 0.$$

It follows

$$\frac{\kappa \rho }{\tau _q} \partial _\mathbf{q}\psi - \frac{1}{\theta }\mathbf{q}= \mathbf{0}$$

whence

$$\psi = \hat{\psi }(\rho , \theta , {{\varvec{\mathcal {T}}}}) + \frac{\tau _q}{2\kappa \rho \theta } \mathbf{q}^2,$$

which is consistent with the symmetry of \(\mathbf{q}\otimes \partial _\mathbf{q}\psi\) required by (17).

Inequality (16) reduces to

$$(\rho ^2 \partial _\rho \psi - p)\nabla \cdot \mathbf{v}+ {{\varvec{\mathcal {T}}}}\cdot \mathbf{D} - \rho \partial _{{{\varvec{\mathcal {T}}}}}\psi \cdot [ \frac{2\mu }{\tau _{\scriptscriptstyle T}} \mathbf{D}- \frac{1}{\tau _{\scriptscriptstyle T}} \mathbf{F}+ \frac{\lambda }{\tau _{\scriptscriptstyle T}} (\nabla \cdot \mathbf{v})\mathbf{1}- \nu (\nabla \cdot \mathbf{v}){{\varvec{\mathcal {T}}}} ] - \rho \partial _\mathbf{q}\psi \cdot [ - \frac{1}{\tau _q}\mathbf{f}- \nu (\nabla \cdot \mathbf{v})\mathbf{q}] \ge 0.$$
(19)

If \(\mathbf{D}= \mathbf{0}\) then we find

$$g(\rho , \theta ,{{\varvec{\mathcal {T}}}}, \mathbf{q}) := \frac{1}{\tau _{\scriptscriptstyle T}} \partial _{{{\varvec{\mathcal {T}}}}}\psi \cdot \mathbf{F}+ \frac{1}{\tau _q} \partial _\mathbf{q}\psi \cdot \mathbf{f}\ge 0.$$
(20)

Now look at (19) when \(\nabla \cdot \mathbf{v}= 0\) and hence \(\mathbf{D}= \mathbf{D}_0\), the subscript 0 denoting the deviator. We have

$$(-\frac{2\mu \rho }{\tau _{\scriptscriptstyle T}} \partial _{{{\varvec{\mathcal {T}}}}} \psi + {{\varvec{\mathcal {T}}}} ) \cdot \mathbf{D}_0 + \rho g \ge 0.$$

The linearity and arbitrariness of \(\mathbf{D}_0\) imply

$$[- \frac{2\mu \rho }{\tau _{\scriptscriptstyle T}} \partial _{{{\varvec{\mathcal {T}}}}} \psi + {{\varvec{\mathcal {T}}}} ]_0 = \mathbf{0},\quad g\ge 0.$$

If instead we let \(\mathbf{D}_0 = \mathbf{0}\), \(\mathbf{D}= {\tfrac{1}{3}} (\nabla \cdot \mathbf{v}) \mathbf{1}\) then we obtain

$$[\rho ^2 \partial _\rho \psi - p - \frac{\rho (\lambda + 2\mu /3)}{\tau _{\scriptscriptstyle T}} \,\mathrm{tr}\,\partial _{{{\varvec{\mathcal {T}}}}} \psi + {\tfrac{1}{3}}\mathrm{tr}{{\varvec{\mathcal {T}}}} +\nu \rho (\partial _{{{\varvec{\mathcal {T}}}}} \psi \cdot {{\varvec{\mathcal {T}}}}+ \partial _\mathbf{q}\psi \cdot \mathbf{q}) ]\nabla \cdot \mathbf{v}+ \rho g\ge 0.$$

The linearity and arbitrariness of \(\nabla \cdot \mathbf{v}\) imply the vanishing of the quantities in brackets together with \(g\ge 0\). This shows that the distinction between p and \(\mathrm{tr}{{\varvec{\mathcal {T}}}}\) is subjective and this is consistent with the fact that p and \(\mathrm{tr}{{\varvec{\mathcal {T}}}}\) belong to the spherical part of \(\mathbf{T}\). We look at \({{\varvec{\mathcal {T}}}}\) as the dissipative stress and hence we let

$$p = \rho ^2 \partial _\rho \psi + \nu \rho (\partial _{{{\varvec{\mathcal {T}}}}} \psi \cdot {{\varvec{\mathcal {T}}}}+ \partial _\mathbf{q}\psi \cdot \mathbf{q}),$$

and

$$\mathrm{tr}\partial _{{{\varvec{\mathcal {T}}}}} \psi = \frac{\tau _{\scriptscriptstyle T}}{\rho (2\mu + 3 \lambda )} \mathrm{tr}{{\varvec{\mathcal {T}}}}.$$

Hence we find the free energy \(\psi\) in the form

$$\psi =\Psi (\rho , \theta ) + \frac{\tau _{\scriptscriptstyle T}}{2\rho (2\mu + 3\lambda )} (\mathrm{tr}{{\varvec{\mathcal {T}}}})^2+ \frac{\tau _{\scriptscriptstyle T}}{4\mu \rho } {{\varvec{\mathcal {T}}}}_0\cdot {{\varvec{\mathcal {T}}}}_0+ \frac{\tau _q}{2 \kappa \rho \theta } \mathbf{q}^2.$$
(21)

The functions \(\mathbf{F}, \mathbf{f}\) are required to satisfy inequality (20). The simplest choice of \(\mathbf{F}, \mathbf{f}\) might be

$$\mathbf{F}= {{\varvec{\mathcal {T}}}}, \qquad \mathbf{f}= \mathbf{q},$$

in which case eqs. (18) become an objective version of the Maxwell model. Now from (20) and (21) we obtain

$$\rho g = \frac{1}{2\mu } {{\varvec{\mathcal {T}}}}_0 \cdot {{\varvec{\mathcal {T}}}}_0 + \frac{1}{2\mu + 3 \lambda } (\mathrm{tr}{{\varvec{\mathcal {T}}}})^2 + \frac{1}{2\kappa \theta } \mathbf{q}^2.$$

Positive values of the entropy production \(\rho g\) occur if and only if

$$\mu> 0, \qquad 2\mu + 3 \lambda> 0, \qquad \kappa > 0,$$
(22)

namely the classical conditions of the Navier-Stokes-Fourier model. Incidentally, inequalities (22) make the free energy minimum at equilibrium (\({{\varvec{\mathcal {T}}}}= \mathbf{0}, \mathbf{q}= \mathbf{0}\)).

It is worth remarking that the choice of the objective derivative influences the constitutive functions. In the present case \(\nu \ne 0\) implies an additive pressure-term

$$\nu \rho (\partial _{{{\varvec{\mathcal {T}}}}} \psi \cdot {{\varvec{\mathcal {T}}}}+ \partial _\mathbf{q}\psi \cdot \mathbf{q}).$$

In this sense the customary use of the Oldroyd fluid model, where \(\nabla \cdot \mathbf{v}= 0\), does not affect significantly the constitutive properties. Indeed, \({\hat{g}}=\partial _{{{\varvec{\mathcal {T}}}}} \psi \cdot {{\varvec{\mathcal {T}}}}+ \partial _\mathbf{q}\psi \cdot \mathbf{q}\) is just the entropy production density g provided that \(\tau _{\scriptscriptstyle T}=\tau _q=1\). Now, since

$$\rho ^2 \partial _\rho \psi = \rho ^2 \partial _\rho \Psi - {\tfrac{1}{2}}\rho {\hat{g}}$$

from (21) it follows that

$$p = \rho ^2 \partial _\rho \Psi + (\nu -{\tfrac{1}{2}})\rho {\hat{g}}.$$

So the pressure is the classical term \(\rho ^2 \partial _\rho \Psi\) plus \(\nu -1/2\) times the entropy production corresponding to unit relaxation times, \(\tau _{\scriptscriptstyle T}=\tau _q=1\).

5 Relation to other approaches

There are similarities and differences with other approaches in the literature which deserve a brief review. First we ask about the connection between the entropy production as considered in Sect. 3 and dissipation potentials.

In Lagrangian mechanics with generalized coordinates \(q_j\), \(j=1,...,n\), frictional forces are possibly modelled by a quadratic function \(\mathcal {R}\) of the generalized velocities. It is then required that the frictional forces Y are given by

$$Y_j = \partial _{\dot{q}_j} \mathcal {R};$$

the function \(\mathcal {R}\) is called the Rayleigh dissipation potential. If the corresponding power is \(Y \cdot \dot{q}\) and \(\mathcal {R}\) is a homogeneous function of degree two then

$$Y \cdot \dot{q} = (\partial _{\dot{q}_j}\mathcal {R}) \dot{q}_j = 2 \mathcal {R}.$$

Variational approaches are then developed for dissipative systems provided the power is the product of generalized forces (Y) by generalized velocities (\(\dot{q}\)) [23]. Based on this analogy, dissipative effects are modelled by letting \(\mathbf{I}\) a set of internal variables such that \(- \partial _\mathbf{I}\psi\) is the generalized force in that \(- \partial _\mathbf{I}\psi \cdot \dot{\mathbf{I}}\) is an additional entropy production (see, e.g., [15]). Eventually the internal variables have to be related to pertinent physical variables. In our approach Sect. 4, instead, the entropy production \(\sigma\) is a function of the common set of physical variables involved in the model, and \(\sigma\) is determined so that the Clausius–Duhem inequality holds.

Dissipative processes are also described in a Hamiltonian setting [16]. Despite the conceptual differences, some consequences are of interest. Observe that the balance of entropy can be written in the form (4). Assume that the entropy production is given by

$$\theta \sigma = {{\varvec{\mathcal {T}}}}\cdot \mathbf{D}- \frac{1}{\theta }\mathbf{q}\cdot \nabla \theta$$

while \(\mathbf{q}/\theta\) is the entropy flux. Hence the evolution of \(\rho , \mathbf{v}, \eta\) is governed by the differential equations

$$ \partial _t \rho= - \nabla \cdot (\rho \mathbf{v}), \rho \partial _t \mathbf{v}= -\rho (\mathbf{v}\cdot \nabla )\mathbf{v}- \nabla p + \nabla \cdot {{\varvec{\mathcal {T}}}}, \rho \theta \partial _t \eta= - \rho \theta (\mathbf{v}\cdot \nabla )\eta + {{\varvec{\mathcal {T}}}}\cdot \mathbf{D}- \nabla \cdot \mathbf{q}.$$
(23)

Moreover observe that

$$(\mathbf{v}\cdot \nabla ) = 2 \mathbf{W}\mathbf{v}+ \nabla {\tfrac{1}{2}} \mathbf{v}^2.$$

The heat flux \(\mathbf{q}\) is assumed to be given by Fourier law \(\mathbf{q}= - \mathbf{K}\nabla \theta\), where \(\mathbf{K}\) is a positive definite tensor. The free energy \(\psi\) is assumed to be independent of \({{\varvec{\mathcal {T}}}}, \mathbf{q}\). We observe that

$$\eta = - \partial _\theta \psi , \qquad p = \rho ^2 \partial _\rho \psi .$$

By

$$\varepsilon (\rho , \eta ) = \psi (\rho , \theta (\rho , \eta )) + \theta (\rho , \eta ) \eta$$

we have

$$ \partial _\rho \varepsilon (\rho , \eta )= \partial _\rho \psi (\rho , \theta ) = p/\rho ^2, \partial _\eta \varepsilon (\rho , \eta )= \partial _\theta \psi (\rho , \theta ) + \partial _\theta \psi (\rho , \theta ) \partial _\eta \theta + \partial _\eta \theta \,\eta = \theta .$$

Let

$$\mathcal {H}= \int _\Omega H(\rho , \mathbf{v}, \eta )dv, \qquad H = {\tfrac{1}{2}} \rho \mathbf{v}^2 + \rho \varepsilon (\rho , \eta ) .$$

It follows

$$\partial _\rho H = {\tfrac{1}{2}}\mathbf{v}^2 + \varepsilon + \partial _\rho \varepsilon = {\tfrac{1}{2}}\mathbf{v}^2 + \varepsilon + p/\rho = {\tfrac{1}{2}}\mathbf{v}^2 + h,$$

\(h = \varepsilon + p/\rho\) being the enthalpy. Moreover,

$$\partial _\mathbf{v}H = \rho \mathbf{v}, \qquad \partial _\eta H = \rho \theta .$$

Hence we can write (23) in the form

$$ \partial _t \rho =- \nabla \cdot \partial _\mathbf{v}H, \partial _t \mathbf{v}=- \nabla \partial _\rho H - \frac{2\mathbf{W}}{\rho } \partial _\mathbf{v}H + \nabla (\frac{\eta }{\rho }\partial _\eta H ) - \frac{1}{\rho }\nabla \cdot (\frac{{{\varvec{\mathcal {T}}}}}{\rho \theta } \partial _\eta H ), \partial _t \eta =- \frac{1}{\rho }\nabla \eta \cdot \partial _\mathbf{v}H - \frac{1}{\rho \theta } {{\varvec{\mathcal {T}}}}\cdot \frac{\partial _\mathbf{v}H}{\rho } + \nabla \frac{\partial _\eta H}{\rho } \cdot \frac{\mathbf{K}}{\rho \theta ^2} \nabla \frac{\partial _\eta H}{\rho } + \frac{1}{\rho }\nabla \cdot \frac{\mathbf{K}}{\theta } \nabla \frac{\partial _\eta H}{\rho }. $$

This set of equations is then used to characterize the evolution of the fluid, not to investigate the constitutive properties.

6 Conclusion

This paper develops two unusual approaches to the modelling of dissipative compressible fluids. Both approaches are based on the view that the entropy production too is given by a constitutive function.

First a hysteretic effect is considered in the relation between mass density an pressure. This is realized by Eq. (10) where the entropy production is assumed to be proportional to \(\vert \dot{\rho }\vert\). Next the dissipative character is modelled by letting the extra stress \({{\varvec{\mathcal {T}}}}= \mathbf{T}+ p \mathbf{1}\) and the heat flux be expressed by rate-type constitutive equations. For definiteness and simplicity the corotational rate is considered along with the isotropic terms \((\nabla \cdot \mathbf{v}){{\varvec{\mathcal {T}}}}\) and \((\nabla \cdot \mathbf{v})\mathbf{q}\). The classical restrictions on the viscosity coefficients and the heat conductivity are obtained along with the dependence of pressure on the entropy production.

This paper shows the conceptual role of the entropy production \(\sigma\) as a constitutive function. As with the Navier–Stokes–Fourier model, in many cases Eq. (4) holds for \(\sigma\) as an identity. Instead, especially in the models of hysteresis, the selection of a constitutive function \(\sigma \ge 0\) characterizes the dissipative properties of the continuum.