Rate-type models of dissipative compressible fluids

Two models of dissipative compressible fluids are established within the general setting of Rational Thermodynamics. It is a common feature underlying the analysis of the thermodynamic consistency that the entropy production and the free energy potential are viewed as constitutive functions of the same set of physical variables and no internal variable is involved. Also in light of late experimental observations, hysteretic models are established for the dependence of pressure on the mass density. Next relaxation properties are modelled via rate equations with an objective derivative which combines the corotational rate with the divergence of the velocity.


Introduction
Compressible fluids are usually described by a function p( , ) specifying the pressure p in terms of the mass density and the (absolute) temperature . In practice we should account also for relaxation phenomena where the connection between p and , or , 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 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 is independent of the constitutive prescription of 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 is assumed to satisfy the constitutive equation the superposed triangle ▿ denoting the upper convected time derivative; as with any tensor The assumption (1) shows some disadvantages and, in particular, it leads to a parabolic equation for the velocity . Further there are nontrivial questions about the thermodynamic consistency. This is so because the natural description should involve the second Piola-Kirchhoff stress RR and the Green-St.Venant strain so that RR ⋅̇ 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 along with the Maxwell-Cattaneo equation for the heat flux so that the whole model is hyperbolic. We remark that the occurrence of the time derivatives ̇ ,̇ 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 is replaced by a Lie-Oldroyd upper convected derivative.
The upper convected Maxwell model is the particular case of (1) with 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 T . To account for objectivity and compressibility we let 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].

Balance laws
We consider a fluid occupying a time-dependent region Ω ⊂ E 3 . Throughout is the mass density, the velocity, the symmetric stress tensor, the internal energy, the heat flux, p the pressure. The symbol ∇ denotes the gradient operator, t is the partial time derivative, at a point ∈ Ω , while a superposed dot stands for the total time derivative, ̇f = t f + ⋅ ∇f . Further, is the velocity gradient, L ij = x j v i , = sym is the stretching tensor, = skw is the spin tensor. Moreover, 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 The entropy inequality is assumed to be the subsequent developments show that an extraentropy flux would be redundant. We define the entropy production , and accordingly ≥ 0 . As is common in Rational Thermodynamics, physically admissible constitutive equations are required to satisfy ≥ 0 for any appropriate set of fields compatible with the balance equations.
Substituting r − ∇ ⋅ from (2) into (3) we have whence, in terms of the Helmholtz free energy = − , we can write the Clausius-Duhem inequality in the form As we show in the next section, we let be given by a constitutive function. In this regard two cases may happen. First, coincides with what follows from the definition. As a remarkable example, in connection with the Navier-Stokes-Fourier model, where we find that This shows the constitutive property of the entropy production for the Navier-Stokes-Fourier model [10]. More interestingly, we may have cases where is assumed to have a constitutive dependence per se. This happens for the hysteretic models developed in [5,6,18] where the function 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.

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 = −p . Hence we take as the set of independent variables for the functions , , and . Upon evaluation of ̇ and substitution in (5) we obtain The linearity and arbitrariness of ̇,∇ ,̈,p imply that be independent of ∇ ,̇,ṗ and hence

Consequently inequality (6) reduces to
This result indicates that in general ̇,ṗ, ∇ might be related so that the inequality holds.
Since is independent of ∇ , we let ∇ = and write (7) in the form where p is the entropy production when ∇ = . Consequently A Fourier-like relation for ,

Duhem-like flows
Equation (8) may be framed within the class of Duhem-like materials [13].
First let p = 0 , that is no entropy dissipation occurs. If p = 0, = ( , ) then The arbitrariness of ̇ implies and hence p and cannot be independent variables. Otherwise, if is a C 2 function we obtain a contradiction, As it happens for elastic fluids, the dependence of p on must be expressed by a constitutive relation p =p( , , p) , which is named equation of state and is related to ̂( , ) by (9). If, instead, p ≠ 0 then (8) implies We now show that Eq. (8) allows us to describe density-pressure hysteresis. Let p ≠ 0 and assume Equation (8) becomes whence Let Equation (10) becomes In the classical theory p is the square of the speed of sound in isothermal conditions. Here, We then assume At constant temperature ̇= pṗ +̇ and hence integration of (10), as t ∈ [t 1 , t 2 ] , along a closed curve in the -p plane results iṅ ∮ denoting the integral along the closed curve. Accordingly, we have ∮ p d ≥ 0 and this implies that the closed curve is run in the clockwise sense. Moreover, equation (10) is invariant under the time transformation and hence it describes a rate-independent behaviour.

The Helmholtz free energy
To determine the function , thus characterizing the model, we start with the generic assumption L, G, F, H being undetermined differentiable functions, parameterized by ; the dependence on the temperature is understood and not written. Substitution of p and yields For simplicity we assume where g is a positive function parametrized by .
Hence it follows the requirement which is satisfied by letting F � = 0 and for some ≠ 0 . For definiteness we let > 0 . As well as the constant function F(p) , can in fact depend on the temperature . In summary, we have where H � ( ) = G( )∕ 2 . The characteristic functions 1 , 2 are given by so that and G completely characterize the model. The term ∕ 2 gives the difference between the differential Boyle's factor 1 and the slope of the function G( ) . The monotonicity condition (11) requires Moreover, it turns out 1 − G � ( ) → 0 as → ∞.

A simple example
The function G is taken to be positive and quadratic increasing as > 0, 0 , 1 > 0 being possibly dependent on the temperature . Hence C being a constant possibly dependent on . Moreover, 1 = 2 0 + ∕ 2 . The hysteretic function is taken in the regular form Hence, the slope of the hysteretic path is given by Figure 1 shows an example of the loops obtained by solving the system of equations with initial values 0 , p 0 (the density oscillates between 0 and + 0 ). The rate independence of the model allows the loops to be independent of the angular frequency . The monotonicity requirement (12) is satisfied provided that This inequality identifies a wide region in the -p plane around the characteristic curve p = G( ) . Similar hysteretic cycles are described in some special fields of fluid mechanics (see, for instance, [11,25]).

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 = 1∕ as an independent variable instead of density. Letting ̃( , , p) = ( , , p) we observe that Equation (8) can then be written as (13) (̃+ p)̇+ p̃ṗ = − p Fig. 1 Isothermal pressure-density hysteresis loops (solid) and the graph of the function G (dashed) corresponding to = 0.2 , At constant temperature ̇= pṗ +̇ and hence integration of (13), as t ∈ [t 1 , t 2 ] , along a closed curve in the -p plane results in ∮ denoting the integral along the closed curve. Hence we have ∮ p d ≤ 0 and this implies that the closed curve is run in the counterclockwise sense.
We then take the pressure p and the specific volume = 1∕ as the independent variables and we assume where p * > > 0, > 0 and H denotes the Heaviside step function. Equation (13)

Compressible fluids with objective rates
To describe dissipative effects in fluids the symmetric stress tensor is often decomposed in a pressure tensor −p and an extra stress tensor, The extra stress 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 , the pressure p, and appropriate objective derivatives  [12] and hence mainly appropriate for the modelling of solids. The simplest objective derivative is the corotational one namely [17,18] 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 If is replaced by and = 1 then (14) becomes the Truesdell [17,18] or Oldroyd-Lie [3] derivative; if = 0 then (14) becomes the corotational derivative.
Vol.: (0123456789) The Clausius-Duhem inequality reduces to The time derivatives Ṫ and ̇ are given by Hence inequality (15)  Consider the terms dependent on ∇ in (16). We have = , It follows whence which is consistent with the symmetry of ⊗ required by (17). Inequality (16) reduces to If = then we find Now look at (19) when ∇ ⋅ = 0 and hence = 0 , the subscript 0 denoting the deviator. We have The linearity and arbitrariness of 0 imply If instead we let 0 = , [− 2 The linearity and arbitrariness of ∇ ⋅ imply the vanishing of the quantities in brackets together with g ≥ 0 . This shows that the distinction between p and trT is subjective and this is consistent with the fact that p and trT belong to the spherical part of . We look at T as the dissipative stress and hence we let and Hence we find the free energy in the form The functions , are required to satisfy inequality (20). The simplest choice of , might be in which case eqs. (18) become an objective version of the Maxwell model. Now from (20) and (21)  It is worth remarking that the choice of the objective derivative influences the constitutive functions. In the present case ≠ 0 implies an additive pressure-term In this sense the customary use of the Oldroyd fluid model, where ∇ ⋅ = 0 , does not affect significantly the constitutive properties. Indeed, (22) > 0, 2 + 3 > 0, > 0, ( T ⋅ T + ⋅ ).ĝ = T ⋅ T + ⋅ is just the entropy production density g provided that T = q = 1 . Now, since from (21) it follows that So the pressure is the classical term 2 Ψ plus − 1∕2 times the entropy production corresponding to unit relaxation times, T = q = 1.

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 R of the generalized velocities. It is then required that the frictional forces Y are given by the function R is called the Rayleigh dissipation potential. If the corresponding power is Y ⋅q and R is a homogeneous function of degree two then Variational approaches are then developed for dissipative systems provided the power is the product of generalized forces (Y) by generalized velocities ( ̇q ) [23]. Based on this analogy, dissipative effects are modelled by letting a set of internal variables such that − is the generalized force in that − ⋅̇ 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 is a function of the common set of physical variables involved