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Derivation of Equations for ContinuumMechanics and Thermodynamics of Fluids

Josef Málek and Vít Pruša

Abstract

The chapter starts with overview of the derivation of the balance equations formass, momentum, angular momentum, and total energy, which is followed by adetailed discussion of the concept of entropy and entropy production. While thebalance laws are universal for any continuous medium, the particular behaviorof the material of interest must be described by an extra set of material-specificequations. These equations relating, for example, the Cauchy stress tensor andthe kinematical quantities are called the constitutive relations.

The core part of the chapter is devoted to the presentation of a modernthermodynamically based phenomenological theory of constitutive relations. Thekey feature of the theory is that the constitutive relations stem from the choiceof two scalar quantities, the internal energy and the entropy production. This istantamount to the proposition that the material behavior is fully characterized bythe way it stores the energy and produces the entropy.

The general theory is documented by several examples of increasing com-plexity. It is shown how to derive the constitutive relations for compressible andincompressible viscous heat-conducting fluids (Navier–Stokes–Fourier fluid),Korteweg fluids, and compressible and incompressible heat-conducting vis-coelastic fluids (Oldroyd-B and Maxwell fluid).

KeywordsContinuum mechanics • Constitutive relations • Thermodynamics

Dedicated to professor K. R. Rajagopal on the occasion of his 65th birthday.

J. Málek (�) • V. PrušaFaculty of Mathematics and Physics, Charles University in Prague, Praha 8 – Karlín, CzechRepublice-mail: [email protected]; [email protected]

© Springer International Publishing AG 2016Y. Giga, A. Novotny (eds.), Handbook of Mathematical Analysis in Mechanicsof Viscous Fluids, DOI 10.1007/978-3-319-10151-4_1-1

1

mailto:[email protected]

mailto:; [email protected]

2 J. Málek and V. Pruša

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Balance of Mass, Momentum, and Angular Momentum. . . . . . . . . . . . . . . . . . . . . . . 52.2 Balance of Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Stress Power and Its Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Stability of the Rest State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Compressible and Incompressible Viscous Heat-Conducting Fluids . . . . . . . . . . . . . 294.3 Compressible Korteweg Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Compressible and Incompressible Viscoelastic Heat-Conducting Fluids . . . . . . . . . . 444.5 Beyond Linear Constitutive Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.6 Boundary Conditions for Internal Flows of Incompressible Fluids . . . . . . . . . . . . . . . 62

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2000 Mathematics Subject Classification. 76A02, 76A05, 74A15, 74A20

1 Introduction

Continuum mechanics and thermodynamics are based on the idea of continuouslydistributed matter and other physical quantities. Originally, continuum mechanicswas equated with hydrodynamics, aerodynamics, and elasticity. The scope of thestudy was the motion of water and air and the deformation of some special solidsubstances. However, the concept of continuous medium has been shown to beuseful and extremely viable even in the modelling of the behavior of much morecomplex systems such as polymeric solutions, granular materials, rock and landmasses, special alloys, and many others. Moreover, the range of physical processesmodelled in the continuum framework nowadays goes beyond purely mechanicalprocesses. Processes such as phase transitions or growth and remodeling of biologi-cal tissues are routinely approached in the setting of continuum thermodynamics.Finally, the systems studied in continuum thermodynamics range from the verysmall ones studied in microfluidics (see, e.g., Squires and Quake [89]) up to thegigantic ones studied in planetary science; see, for example, Karato and Wu [48].

It seems that the laws governing the motion of a continuous medium must beextremely complicated in order to capture such a wide range of physical systemsand phenomena of interest. This is only partially true. In principle, the lawsgoverning the motion of a continuous medium can be seen as reformulations and

The authors were supported by the project LL1202 in the program ERC-CZ funded by the Ministryof Education, Youth and Sports of the Czech Republic.

Equations for Continuum Mechanics and Thermodynamics of Fluids 3

generalizations/counterparts of the classical physical laws (the laws of Newtonianphysics of point particles and the laws of classical thermodynamics). Surprisingly,the continuum counterparts of the classical laws are relatively easy to derive. Theselaws are – in the case of a single continuum medium – the balance equations forthe mass, momentum, angular momentum, and energy and the evolution equationfor the entropy. The balance equations are supposed to be universally valid for anycontinuous medium, and their derivation is briefly discussed in Sect. 2.

The critical and the most difficult part in formulating the system of governingequations for a given material is the specification of the response of the materialto the given stimuli. The sought stimulus–response relation can be, for example,a relation between the deformation and the stress or a relation between the heatflux and the temperature gradient. The task of finding a description of the materialresponse is the task of finding the so-called constitutive relations.

Apparently, the need to specify the constitutive relation for a given material callsfor an investigation of the microscopic structure of the material. The reader who isinterested in examples of the derivation of constitutive relations from microscopictheories is referred to Bird et al. [4] or Larson [52] to name a few. However, theinvestigation of the microscopic structure of the material is not the only option.

The constitutive relations can be specified staying entirely at the phenomenologi-cal level. Here the phenomenological level means that it is possible to deal only withphenomena directly accessible to the experience and measurement without trying tointerpret the phenomena in terms of ostensibly more fundamental (microscopic)physical theories. Indeed, the possible class of constitutive relations is in factseverely restricted by physical requirements stemming, for example, from therequirement on Galilean invariance of the governing equations, perceived symmetryof the material, or the second law of thermodynamics. As it is apparent from thediscussion in Sect. 4, such restrictions allow one to successfully specify constitutiverelations.

2 Balance Equations

Before discussing the theory of constitutive relations, it will be convenient to brieflyrecall the fundamental balance equations for a continuous medium. The reader whois not yet familiar with the field of continuum mechanics and thermodynamics isreferred to Truesdell and Toupin [93], Müller [62], Truesdell and Rajagopal [92],or Gurtin et al. [35] for a detailed treatment of balance equations and kinematics ofcontinuous medium. Note that the same formalism can be applied even for severalinteracting continuous media which constitute the continuum approach to the theoryof mixtures; see, for example, Samohýl [84] and Rajagopal and Tao [81].

The continuous body B is assumed to be a part of Euclidean space R3. The motionof the body is described by the function � that maps the positions X of points attime t0 to their respective positions x at a later time instant t , such that x D �.X ; t /.Concerning a suitable framework for the description of processes in a continuousmedium, one can, in principle, choose from two alternatives.


Either one expresses the quantities of interest as functions of time and the initialpositionX or as functions of time and the chosen position x in space R3. The formeralternative is called the Lagrangian description, while the latter is referred to as theEulerian description.

The Lagrangian description is especially suitable for the description of changeof form or shape, since in order to describe the change, a reference point is needed.(Change with respect to some initial state.) As such, the Lagrangian description is apopular choice for studying the motion of solids.

On the other hand, the Eulerian description is not primarily focused on changewith respect to some initial state, but on the rate of change. Naturally, the rate ofchange (the time derivative) can be captured by referring to the state of the materialat the current time and in its infinitesimally small-time neighborhood. In otherwords, if one is interested in the instantaneous velocity field, then there is no need toknow complete trajectories of the individual points from the possibly distant initialstate. As such, the Eulerian approach is useful for the description of the motion offluids, where one is primarily interested in the velocity field, and the knowledgeof the motion (individual trajectories) is of secondary importance. (Note, however,that the Eulerian description can be advantageous even in the description of solids,in particular in problems that involve phenomena like fluid–structure interaction orgrowth; see, e.g., Frei et al. [29,30].) In what follows a theory for fluids is of primaryinterest; hence the Eulerian description is used.

The balance laws are derived by applying the classical laws of Newtonian physicsand classical thermodynamics to a volume of the moving material V .t/ Ddef

�.V .t0/; t/, where V .t0/ is an arbitrarily chosen part of the continuous body B.The main mathematical tool is the Reynolds transport theorem; see, for example,Truesdell and Toupin [93].

Theorem 1 (Reynolds transport theorem). Let �.x; t / be a sufficiently smoothfunction describing the motion of the body B, and let V .t0/ be an arbitrary part ofB at the inital time t0. Let �.x; t / be a sufficiently smooth scalar Eulerian field, andlet V .t/ D �.V .t0/; t/ be the volume transported by the motion �. Then

d

dt

ˆV .t/

�.x; t / dv DˆV .t/

�d�.x; t /

dtC �.x; t / div v.x; t /

�dv;

where

d�.x; t /

dtDdef

@�.x; t /

@tC v.x; t / � r�.x; t / (1)

denotes the material time derivative, and v.x; t / is the Eulerian velocity field,

v.x; t / Ddef@�.X ;t/

@t

ˇXD��1.x;t/

.


2.1 Balance of Mass, Momentum, and Angular Momentum

2.1.1 Balance of MassClassical Newtonian physics assumes conservation of massm, which can be writtenas dm

dt D 0. The balance of mass is a generalization of this equation to the continuummechanics setting. The mass mV .t/ of volume V .t/ is expressed in terms of theEulerian density field �.x; t / as mV .t/ Ddef

´V .t/

�.x; t / dv. Using the notion of thedensity field, the Reynolds transport theorem, and the requirement

dmV .t/

dtD 0 (2)

then immediately yield the integral form of the balance of mass

ˆV .t/

�d�.x; t /

dtC �.x; t / div v.x; t /

�dv D 0; (3)

where V .t/ is an arbitrary volume in the sense that it is the volume obtainedby tracking an arbitrarily chosen initial volume V .t0/. Since the volume in (3)is arbitrary, and the considered physical quantities are assumed to be sufficientlysmooth, the integral form of the balance of mass leads to the pointwise evolutionequation

d�.x; t /

dtC �.x; t / div v.x; t / D 0: (4)

Having (4), it follows that Reynolds transport theorem for the quantity �.x; t / Ddef

�.x; t /w.x; t /, where w is an arbitrary Eulerian field, in fact reads

d

dt

ˆV .t/

�.x; t /w.x; t / dv DˆV .t/

�.x; t /dw.x; t /

dtdv: (5)

This simple identity will be useful in the derivation of the remaining balanceequations.

Note that the integral form of the balance of mass and the other balance equationsas well can be reformulated in a weak form without the need to use the pointwiseequation (4) as an intermediate step. See Feireisl [27] or Bulícek et al. [8] for details.

2.1.2 Balance of MomentumBalance of momentum is the counterpart of Newton second law dp

dt D F forpoint particles, where p D mv is the momentum of particle with mass m movingwith velocity v. In continuum setting the momentum pV .t/ of volume V .t/ isexpressed in terms of the Eulerian density and the velocity field as pV .t/ Ddef´V .t/

�.x; t /v.x; t / dv. The balance of momentum for V .t/ then reads


dpV .t/dt

D F : (6)

The force F on the right-hand side of the continuum counterpart of the Newtonsecond law consists of two contributions,

F D F volume C F contact: (7)

The first physical mechanism that contributes to the force acting on the volumeV .t/ is the specific body force b. This is the force that acts on every part of V .t/,hence the specific body force contributes to the total force F via the volume integral

F volume Ddef

ˆV .t/

�.x; t /b.x; t / dv: (8)

A particular example of specific body force is the electrostatic force or thegravitational force.

The second contribution to the force F acting on the volume V .t/ is the forceF contact due to the resistance of the material surrounding the volume V .t/. Since thisforce contribution arises due to wading of the volume V .t/ through the surroundingmaterial, it acts on the surface of the volume V .t/. Consequently, it is referred toas the contact or surface force. The contact force contribution is a new physicalmechanism that goes beyond the concept of forces between point particles, and it isthe key concept in mechanics of continuous media.

The total contact force F contact acting on V .t/ is assumed to take the form

F contact Ddef

ˆ@V .t/

t.x; t;n.x; t // ds; (9)

where t is the contact force density and n denotes the unit outward normal to thesurface of volume V .t/. It is worth emphasizing that the formula for the contactforce is a fundamental and nontrivial assumption concerning the nature of the forcesacting in a continuous medium.

Assuming that the contact force is given in terms of the contact force densityt.x; t;n.x; t //, one can proceed further and prove that the contact force density isin fact given by the formula

t.x; t;n.x; t // D T.x; t /n.x; t /; (10)

where T.x; t / is a tensorial quantity that is referred to as the Cauchy stresstensor. The corresponding theorem is referred to as the Cauchy stress theorem,and the interested reader will find the proof, for example, in the classical treatiseby Truesdell and Toupin [93]. Note that the proof of the Cauchy stress theorem


could be rather subtle if one wants to work with functions that lack smoothness;see, for example, Šilhavý [87] and references therein for the discussion of this issue.

Using (10) and the formulae for the body force and the contact force, the equalitydpV .t/

dt D F volume C F contact that holds for arbitrary volume V .t/ can be rewritten inthe form

d

dt

ˆV .t/

�.x; t /v.x; t / dv DˆV .t/

�.x; t /b.x; t / dvCˆ@V .t/

T.x; t /n.x; t / ds;

(11)

which upon using the Reynolds transport theorem, the Stokes theorem, and theidentity (4) reduces to

ˆV .t/

�.x; t /dv.x; t /

dtdv D

ˆV .t/

Œdiv T.x; t /C �.x; t /b.x; t /� dv; (12)

where dv.x;t /dt denotes the material time derivative of the Eulerian velocity field. (The

divergence of the tensor field is defined as the operator that satisfies .div A/ � w Ddiv.A>w/ for an arbitrary constant vector field w.) Since the considered physicalquantities are assumed to be sufficiently smooth and the volume V .t/ can bechosen arbitrarily, the integral form of the balance of momentum (12) reduces tothe pointwise equality

�.x; t /dv.x; t /

dtD div T.x; t /C �.x; t /b.x; t /: (13)

2.1.3 Balance of Angular MomentumBalance of angular momentum is for a single point particle a simple consequenceof Newton laws of motion, and it is in fact redundant. In the context of a continuousmedium, the balance of angular momentum however provides a nontrivial piece ofinformation concerning the structure of the Cauchy stress tensor. In the simplestsetting it reduces to the requirement on the symmetry of the Cauchy stress tensor

T.x; t / D T>.x; t /; (14)

See, for example, Truesdell and Toupin [93] for details.

2.2 Balance of Total Energy

The need to work with thermal effects inevitably calls for reformulating the lawsof classical thermodynamics in continuum setting. Classical thermodynamics isessentially a theory applicable to a volume of a material wherein the physicalfields are homogeneous and undergo only infinitesimal (slow in time) changes. This


g

d

cb b

a

c

h

f

cb b

a

d d dd

e

ke

k

d d dc

d

ba

c

ba

a b

Fig. 1 Joule experiment (Original figures from Joule [47] with edits). (a) Overall sketch of Jouleexperimental apparatus. (b) Horizontal and vertical cross section of the vessel with a paddle wheel

is clearly insufficient for the description of the behavior of a moving continuousmedium. In particular, the concepts of energy and entropy as introduced in classicalthermodynamics (see, e.g., Callen [15]) need to be revisited.

A good demonstration of the (in)applicability of classical thermodynamics andits continuum counterpart is the analysis of the famous Joule experiment; see Joule[46]. In the experiment concerning the mechanical equivalence of heat, Joule studiedthe rise of temperature due to the motion of a paddle wheel rotating in a vessel filledwith a fluid. The motion of the paddle wheel was driven by the descent of weightsconnected via a system of pulleys to the paddle wheel axis; see Fig. 1. The potentialenergy of the weights is in this experiment transformed to the kinetic energy of thepaddle wheel and due to the resistance of the fluid in the vessel also to the kineticand thermal energy of the fluid.

Classical thermodynamics is restricted to the description of the initial and finalstate where the weights are at rest. At the beginning, the temperature is constant allover the vessel. This is the initial state. As the weights descend, the fluid moves, andits temperature rises. Once the weights are stopped, the temperature reaches, aftersome time, homogeneous distribution in the vessel. This is the final state. In theclassical setting, it is impossible to say anything about the intermediate states sincethe descent of the weight induces substantial motion in the fluid and inhomogeneousdistribution of the temperature in the vessel. (Recall that the energy, entropy, andtemperature are in the classical setting defined only for the whole vessel.)

On the other hand, the ambition of continuum thermodynamics is to describethe whole process and the time evolution of the spatial distribution of the physicalquantities of interest. In order to describe the spatial distribution of the physicalquantities of interest, one obviously needs to talk about specific internal energye.x; t / and the specific entropy �.x; t /. The internal energy or the entropy ofthe volume V .t/ of the material is then obtained by volume integration of thecorresponding densities.


Besides the introduction of the specific internal energy e.x; t / and the specificentropy �.x; t /, it is necessary to identify the energy exchange mechanisms in thecontinuous medium. Naturally, one part of the energy exchange is due to processesof mechanical origin.

In order to identify the energy exchange due to mechanical processes, it sufficesto recall the same concept in classical Newtonian physics. The energy balance fora single particle of constant mass m is in the classical setting obtained via themultiplication of the Newton’s second law d

dt .mv/ D F by the particle velocity v,

which yields ddt

�12mjvj2

�D F � v. Repeating the same steps in the setting of

continuum mechanics – see the pointwise equality (13) – yields

ˆV .t/

�.x; t /dv.x; t /

dt� v.x; t / dv D

ˆV .t/

Œdiv T.x; t /C �.x; t /b.x; t /� � v.x; t / dv;

(15)

where v.x; t / is the Eulerian velocity field.Further, one needs to introduce a quantity describing the nonmechanical part of

the energy exchange between the given volume of the material and its surroundings.It is assumed that this part of the energy exchange can be described using the energyflux je and that the energy exchange through the volume boundary is then given bythe means of a surface integral

�

ˆ@V .t/

je.x; t / � n.x; t / ds; (16)

where n denotes the unit outward normal to the surface of the volume. (The minussign is due to the standard sign convention. If the energy flux vector je points intothe volume V .t/ – that is, in the opposite direction to the unit outward normal –then the energy is transferred from the surrounding into the volume, and the surfaceintegral is positive.) The energy flux je is, in the simplest setting, tantamount tothe heat flux jq.x; t /. Consequently, for the sake of clarity and specificity of thepresentation, it is henceforward assumed that je � jq . Using the heat flux jq , thenonmechanical energy exchange – the transferred heat – between the volume V .t/and its surrounding is given by the surface integral

�

ˆ@V .t/

jq.x; t / � n.x; t / ds: (17)

Alternatively, one can also introduce volumetric heat source contribution to theenergy exchange,

ˆV .t/

�.x; t /q.x; t / dv; (18)


but the volumetric contribution shall not be considered here for the sake ofsimplicity of presentation. (Similarly, volumetric contribution is not considered inthe discussion on the concept of entropy and entropy production.)

Now one is ready to formulate the total energy balance for the volume V .t/. Thetotal energy of the continuous medium in the volume V .t/ is assumed to be givenby the volume integral

´V .t/

�.x; t /etot.x; t / dV of the specific total energy etot.Further, the specific total energy is assumed to be given as the sum of the specifickinetic energy of the macroscopic motion 1

2jvj2 and the specific internal energy e,

�.x; t /etot.x; t / Ddef �.x; t /e.x; t /C1

2�.x; t / jv.x; t /j2 : (19)

(In the simplest setting, the internal energy represents the thermal energy, i.e., theenergy of the microscopic motion; see Clausius [17].) The integral form of thebalance law then reads

d

dt

ˆV .t/

�.x; t /etot.x; t / dv DˆV .t/

�.x; t /b.x; t / � v.x; t / dv

C

ˆ@V .t/

ŒT.x; t /n.x; t /� � v.x; t / ds

�

ˆ@V .t/

jq.x; t / � n.x; t / ds; (20)

where the right-hand side contains all possible contributions to the energy exchange,namely, the mechanical ones (see the right-hand side of (15)) and the nonmechanicalones (see (17)).

If the physical quantities of interest are sufficiently smooth, then (20) can bein virtue of the Reynolds theorem, the Stokes theorem, and the symmetry of theCauchy stress tensor (14) reduced to the pointwise equation

�.x; t /detot.x; t /

dtD �.x; t /b.x; t / � v.x; t /C div ŒT.x; t /v.x; t /� � div jq.x; t /:

(21)

Equation (21) can be further manipulated in order to get an evolution equationfor the specific internal energy e only. Multiplying the balance of momentum (13)by v and subtracting the arising equation from the balance of total energy (21) yield,in virtue of the identity div.A>a/ D .div A/ � aC A W ra, the equation

�.x; t /de.x; t /

dtD T.x; t / W D.x; t / � div jq.x; t /; (22)

where the symmetry of the Cauchy stress tensor has been used. The symbol Dstands for the symmetric part of the velocity gradient, D Ddef

12

�rvCrv>

�, and


A W B Ddef Tr�AB>

�. The term T.x; t / W D.x; t / in (22) is called the stress power

and plays a fundamental role in thermodynamics of continuous medium. A few hintson its relevance are given in Sect. 3. Later, in the discussion concerning constitutiverelations for viscous fluids, it is shown that the stress power is one of the terms inthe entropy production.

Note that although equations (22) and (21) are, in virtue of the balance ofmomentum (13), equivalent for sufficiently smooth functions, they are differentfrom the perspective of a weak solution; see Feireisl and Málek [28], Bulícek et al.[8], and Bulícek et al. [9] for details.

2.3 Entropy

Concerning the concept of entropy in the setting of continuum mechanics, twoassumptions must be made.

First, it is assumed, following the classical setting, that an energetic equation ofstate holds even for the specific internal energy and the specific entropy. Note thatthe particular form of the energetic equation of state depends on the given material;hence, the energetic equation of state is in fact a constitutive relation.

In the simplest setting, the energetic equation of state reads

e.x; t / D e.�.x; t/; �.x; t//; (23)

which is clearly a straightforward generalization of the classical relation E DE.S; V /; see, for example, Callen [15]. A general energetic equation of state canhowever take a more complex form

e.x; t / D e.�.x; t /; y1.x; t /; : : : ; ym.x; t //; (24)

where y1; : : : ; ym, m 2 N, m � 1 are other state variables.If the energetic equation of state is obtained as a direct analogue of a classical

equilibrium energetic equation of state, then it is said that the system underconsideration satisfies the assumption of local equilibrium and that the system isstudied in the framework of classical irreversible thermodynamics. However, if onewants to describe the phenomena that go beyond the classical setting (see, e.g.,Sect. 4), then the energetic equation of state must be more complex. In particular,the energetic equation of state can contain the spatial gradients as variables. In sucha case, it is said that the system is studied in the framework of extended irreversiblethermodynamics.

Second, the energetic equation of state is assumed to hold at every time instant inthe given class of processes of interest. This is again a departure from the classicalsetting, where the relations of type E D E.S; V / hold only in equilibrium or inquasistatic (infinitesimally slow) processes. In particular, in the current setting, it is


assumed that one can take the time derivative of the equation of state. For example,if the equation of state is (23), then

de.x; t /

dtD@e.�; �/

@�

ˇˇ�D�.x;t/; �D�.x;t/

d�.x; t /

dtC@e.�; �/

@�

ˇˇ�D�.x;t/; �D�.x;t/

d�.x; t /

dt(25)

is assumed to be valid in the given class of processes of interest, no matter how rapidthe time changes are or how large the spatial inhomogenities are.

2.3.1 Thermodynamic TemperatureBesides the energetic equation of the state, one can, following classical equilibriumthermodynamics, specify the entropy as a function of the internal energy e and theother state variables y1; : : : ; ym,m 2 N,m � 1, which leads to the entropic equationof state

�.x; t / D �.e.x; t /; y1.x; t /; : : : ; ym.x; t //: (26)

The specific entropy � is assumed to be a differentiable function, and further it isassumed that

@�

@e.e; y1; : : : ; ym/

ˇˇeDe.x;t/;y1Dy1.x;t/;:::;ymDym.x;t /

> 0 (27)

holds for any fixed m-tuple y1; : : : ; ym and point .x; t /. This allows one toinvert (26) and get the energetic equation of state

e.x; t / D e.�.x; t /; y1.x; t /; : : : ; ym.x; t //: (28)

Further, following classical equilibrium thermodynamics, the thermodynamic tem-perature � is defined as

�.x; t / Ddef@e

@�.�; y1; : : : ; ym/

ˇˇ�D�.x;t /;y1Dy1.x;t/;:::;ymDym.x;t/

: (29)

2.3.2 Clausius–Duhem InequalityThe last ingredient one needs in the development of thermodynamics of continuousmedium is a counterpart of the classical Clausius inequality

dS �dQ

�; (30)

where � is the thermodynamic temperature; see Clausius [18]. Recall that in theideal situation of reversible processes, the inequality reduces to the equality


dS DdQ

�: (31)

The generalization of the Clausius inequality is the Clausius–Duhem inequality

d

dt

ˆV .t/

�.x; t /�.x; t / dv � �ˆ@V .t/

jq.x; t /

�.x; t /� n dsC

ˆV .t/

�.x; t /q.x; t /

�.x; t /dv;

(32)

which can be obtained from (30) by appealing to the same arguments as inthe discussion of the concept of the energy. (Recall that the transferred heat isexpressed in terms of the surface integral (17).) The last term in (32) accounts forentropy changes due to volumetric heat sources. As before (see the discussion inSect. 2.2), the volumetric term shall not be henceforward considered for the sakeof simplicity of presentation. In what follows, the Clausius–Duhem inequality istherefore considered in the form

d

dt

ˆV .t/

�.x; t /�.x; t / dv � �ˆ@V .t/

jq.x; t /

�.x; t /� n ds; (33)

where V .t/ is again an arbitrary volume in the sense that it is the volume obtainedby tracking an arbitrarily chosen initial volume V .t0/.

Since the classical Clausius (in)equality (30) is the mathematical formalizationof the second law of thermodynamics, the Clausius–Duhem inequality (33) canbe understood as the reinterpretation of the second law of thermodynamics in thesetting of continuum thermodynamics. As such the Clausius–Duhem inequality canbe expected to play a crucial role in the theory which is indeed the case.

It is convenient, for the sake of later reference, to rewrite the inequality in aslightly different form. Using the Stokes theorem, the Reynolds transport theorem,and the identity (5), the inequality (33) can be rewritten in the form

ˆV .t/

��.x; t /

d�

dt.x; t /C div

�jq.x; t /

�.x; t /

�dv � 0: (34)

The first term in (34) is the change of the net entropy,

dSV .t/dt

Ddef

�d

dt

ˆV .t/

�.x; t /�.x; t / dv

; (35)

while the other term is the entropy exchange – the entropy flux – with thesurrounding of volume V .t/,

J@V .t/ Ddef

ˆ@V .t/

jq.x; t /

�.x; t /� n ds: (36)


The general statement of a balance law in continuum mechanics and thermody-namics is that the time change of the given quantity in the volume V .t/ plus the fluxJ@V .t/ of the quantity through the boundary of V .t/ is equal to the production of thegiven quantity in the volume V .t/,

dSV .t/dtC J@V .t/ D „V .t/: (37)

Using this nomenclature, it follows that the left-hand side of (34) deserves to bedenoted as the net entropy production in the volume V .t/.

Further, it is assumed that the net entropy production can be obtained via avolume integral of the corresponding spatially distributed quantity �.x; t /,

„V .t/ Ddef

ˆV .t/

�.x; t / dv: (38)

In such a case, the Clausius–Duhem inequality can be rewritten as

„V .t/ DdSV .t/

dtC J@V .t/ � 0: (39)

Now it is clear that the Clausius–Duhem inequality in fact states that the net entropyproduction in the volume V .t/ is nonnegative,

„V .t/ � 0: (40)

Note that if the heat flux jq vanishes on the boundary of V .t/, then the flux termin (39) vanishes, and it follows that the net entropy of the volume V .t/ increases intime. This is the classical statement concerning the behavior of isolated systems.

Finally, using the concept of entropy production density (38), it follows that thelocalized version of (34) reads

�.x; t / D �.x; t /d�

dt.x; t /C div

�jq.x; t /

�.x; t /

�� 0: (41)

The fact that (33) is indeed a generalization of (30) is best seen in the case ofa vessel filled with a fluid with “almost” constant uniform temperature distribution�.x; t / Ddef � that is subject to heat exchange with its surrounding. In such a case,time integration of (33) from t1 to t2 yields

ˆV .t2/

�.x; t2/�.x; t2/ dv�ˆV .t1/

�.x; t1/�.x; t1/ dv��1

�

ˆ t2

t1

�@V .t/

jq.x; t / � nds

�dt:

(42)


The left-hand side is the difference between the net entropy of the fluid in the vesselat times t2 and t1, and the right-hand side is the heat exchanged with the surroundingin the time interval Œt1; t2�; hence, one gets

S.t2/ � S.t1/ ��Q

�; (43)

which is tantamount to (30).

2.3.3 Entropy ProductionIn the setting of continuum mechanics, it is assumed that the energetic equationof state is known, which means that a relation of the type (25) holds for the timederivative of the entropy. The knowledge of the energetic equation of state for thegiven material allows one to identify the entropy production mechanisms in thematerial, that is, the quantity �.x; t / in (41). In other words, one can rewrite the left-hand side of (41) in terms of more convenient quantities than the entropy. Indeed,inspecting carefully (25) it is easy to see that (25) is in fact a formula for the materialtime derivative of the specific entropy d�.x;t/

dt ,

d�.x; t /

dtD

1

�.x; t /

�de.x; t /

dt�pth.x; t /

Œ�.x; t /�2d�.x; t /

dt

; (44)

where the thermodynamic temperature �.x; t / and the thermodynamic pressurepth.x; t / have been defined by formulae

�.x; t / Ddef@e.�; �/

@�

ˇˇ�D�.x;t/;�D�.x;t/

; (45a)

pth.x; t / Ddef Œ�.x; t /�2 @e.�; �/

@�

ˇˇ�D�.x;t/;�D�.x;t /

: (45b)

(These relations are counterparts of the well-known classical relations; see,e.g., Callen [15].)

The time derivatives de.x;t/dt and d�.x;t/

dt are known from the balance laws (22)and (4), and using the balance laws in (44) yields

�.x; t /d�.x; t /

dtD

1

�.x; t /

T.x; t / W D.x; t / � div jq.x; t /C pth.x; t / div v.x; t /

�:

(46)Finally, using (46) in (41) leads one, after some manipulation, to

1

�.x; t /ŒT.x; t / W D.x; t /C pth.x; t / div v.x; t /� �

1

Œ�.x; t /�2jq.x; t / � r�.x; t /

D �.x; t /d�

dt.x; t /C div

�jq.x; t /

�.x; t /

�� 0: (47)


The expression on the left-hand side of (47), that is,

�.x; t / Ddef1

�.x; t /ŒT.x; t / W D.x; t /C pth.x; t / div v.x; t /�

�1

Œ�.x; t /�2jq.x; t / � r�.x; t / (48)

determines the entropy production in the given material; see the local version ofthe Clausius–Duhem inequality (41). In other words, in the continuum setting,one is able to explicitly evaluate, for the given material, the “uncompensatedtransformation”

N Ddef dS �dQ

�(49)

introduced by Clausius [18].The Clausius–Duhem inequality (41) requires �.x; t / to be nonnegative;

hence (47) can be expected to impose some restrictions on the form of theconstitutive relations. This is indeed the case, and this observation is the key conceptexploited in the theory of constitutive relations discussed in Sect. 4.

Furthermore, relations of the type (47) provide one an explicit criterion for thevalidity of the Clausius equality (31). In particular, a process in a material withenergetic equation of state of the type (23) is reversible, that is, (31) holds, if theentropy production (48) vanishes. From (48) it follows that the entropy productionvanishes if there are no temperature and velocity gradients in the material. Thismeans that the process in the material is very close to an ideal reversible processif the gradients r� and D are during the whole process kept extremely small. Theother possibility is that the entropy production identically vanishes because of theparticular choice of the constitutive relation for T and jq ; see Sect. 4.2.2 for details.

3 Stress Power and Its Importance

The second law of thermodynamics requires the entropy production to be anonnegative quantity. However, such statement concerning the entropy productiondoes not seem to be very intuitive. In what follows it is shown that the nonnegativityof the entropy production is in fact closely related to some fundamental directobservations concerning the qualitative behavior of real materials.

Let us, for example, consider the entropy production for an incompressiblehomogeneous viscous heat nonconducting fluid. The incompressibility means thatdiv v D 0 is required to hold in the class of considered processes, while the fact thata heat nonconducting fluid is considered translates into the requirement jq D 0. Insuch a case, the entropy production (48) reduces to


�.x; t / D1

�.x; t /T.x; t / W D.x; t /: (50)

(Details concerning the derivation of the entropy production in an incompressiblefluid can be found in Sect. 4. Note that for an incompressible fluid, one getsT W D D Tı W Dı; see the discussion in Sect. 4.2.3, in particular formula (119).)Since the thermodynamic temperature in (50) is a positive quantity, it follows thatthe requirement on the nonnegativity of the entropy production simplifies to

T W D � 0: (51)

This means that the second law is satisfied provided that the stress power is apointwise nonnegative quantity.

The importance of the requirement concerning the nonnegativity of the stresspower term T W D – and consequently the nonnegativity of the entropy production –is documented below by means of two simple mechanical examples. These twoexamples show that the seemingly esoteric requirement on the nonnegativity ofentropy production leads to very natural consequences. First, it is shown that thepointwise positivity of the stress power is sufficient for establishing the fact thatthe drag force acting on a rigid body moving through an incompressible fluiddecelerates the moving body; see Sect. 3.1. Second, it is shown that the pointwisepositivity of the stress power is sufficient to establish the stability of the rest state ofan incompressible fluid; see Sect. 3.2.

3.1 Drag

Let us first consider the setting shown in Fig. 2. A rigid body B moves with constantvelocity U in an infinite domain, and it is assumed that no body force is acting onthe surrounding fluid. Further, it is assumed that the velocity field in the fluid that isgenerated by the motion of the body decays sufficiently fast at infinity and that thefluid adheres to the surface of the body, that is, vj@B D U .

The drag force is the projection of the force acting on the body to the directionparallel to the velocity U ,

Fig. 2 Body moving in afluid

Fdrag

Flift

BU


F drag DdefU ˝ U

jU j2

�ˆ@B

Tn ds

�: (52)

The key observation is that if the fluid is assumed to be a homogeneous incom-pressible fluid with symmetric Cauchy stress tensor, then the formula for the dragforce (52) can be rewritten in the form

F drag D �

�ˆexterior of B

T W D dv

�U

jU j2: (53)

The derivation of (53) proceeds as follows. The balance of momentum (13) andthe balance of mass (5) for the motion of the fluid outside the body B are

�dv

dtD div T; (54a)

div v D 0; (54b)

and the boundary conditions read

vj@B D U ; (54c)

v.x; t /! 0 as jxj ! C1: (54d)

Note that the velocity field is considered to be a steady velocity field; hence, thematerial time derivative is dv

dt D Œrv� v.Let D

˚x 2 R3; jxj < R

�n B denote the ball centered at origin with

radius R (large number) with excluded body B and its boundary @B. Governingequations (54) hold in this domain. Now one can multiply (54a) by v and integrateover ,

ˆ

�.Œrv� v/ � v dv Dˆ

.div T/ � v dv: (55)

Application of standard identities yields

ˆ

1

2�v �

�rjvj2

�dv D

ˆ

div�T>v

�dv �

ˆ

T W D dv: (56)

The first term on the right-hand side can be in virtue of the Stokes theorem rewrittenas a sum of integrals over the boundary of the ball and the boundary of the body


ˆ

div�T>v

�dv D

ˆfx2R3; jxjDRg

�T>v

�� n ds �

ˆ@B

�T>v

�� n ds

D

ˆfx2R3; jxjDRg

�T>v

�� n ds � U �

ˆ@B

Tn ds; (57)

where the boundary condition (54c) has been used. (Note that the last integral istaken over the surface of the body; hence, one needs to add the minus sign in orderto compensate the opposite orientation of the surface of the body and the surfaceof .)

Concerning the left-hand side of (56), integration by parts implies

ˆ

1

2�v �

�rjvj2

�dv D

ˆ@

1

2�jvj2v � n ds �

ˆ

1

2�jvj2 div v dv

D

ˆfx2R3; jxjDRg

1

2�jvj2v � n ds; (58)

where the boundary condition (54c) and the incompressibility, div v D 0, have beenused.

Summing up all partial results, it follows that (56) reduces to

ˆfx2R3; jxjDRg

1

2�jvj2v � n ds D

ˆfx2R3; jxjDRg

�T>v

�� n ds

� U �

ˆ@B

Tn ds �ˆ

T W D dv: (59)

Now one can take the limit R ! C1 and use the fact that the velocity vanishessufficiently fast for R ! C1. This means that the surface integrals in (59) vanishfor R!C1 and that (59) reduces to

U �

ˆ@B

Tn ds D �ˆ

T W D dv; (60)

which yields the proposition.The implication of formula (53) is, that if the stress power T W D is pointwise

nonnegative, then the drag direction is opposite to the direction of the motion. (Thedrag force is acting against the motion of the body.) This is definitely a desirableoutcome from the point of everyday experience, and, as it has been shown, thisoutcome is closely related to the second law of thermodynamics.


3.2 Stability of the Rest State

Let denote a vessel occupied at time t0 by a homogeneous incompressible viscousfluid, and let no external body force act on the fluid. The Cauchy stress tensor isagain assumed to be symmetric. Further, it is assumed that the fluid adheres to thesurface of the vessel,

vj@ D 0: (61)

The everyday experience is that whatever has been the initial state of the fluid, thefluid comes, after some time, to the state of rest, that is, v D 0 in. The question iswhether it is possible to recover this fact for a general homogeneous incompressibleviscous fluid.

A good measure of the deviation from the rest state is the net kinetic energy ofthe fluid

Ekin Ddef1

2

ˆ

�.x; t /jv.x; t /j2 dv: (62)

In order to assess the stability of the rest state, one needs to find an evolutionequation for a measure of the deviation from the rest state. This is an easy task ifthe measure of the deviation from the rest state is the net kinetic energy introducedin (62).

The evolution equations for the motion of the fluid are

�dv

dtD div T; (63a)

div v D 0; (63b)

and the multiplication of (63a) with v followed by integration over the domain yields

ˆ

d

dt

��1

2v � v

�dv D

ˆ

.div T/ � v dv: (64)

(See identity (5).) The left-hand side is equal to the time derivative of the net kineticenergy (62). Concerning the right-hand side, one can use the identity div.A>a/ D.div A/ � aCA W ra, the Stokes theorem, and the fact that v vanishes on the boundaryto get

ˆ

.div T/ � v dv D �ˆ

T W D dv: (65)

This implies that the evolution equation for the net kinetic energy reads


dEkin

dtD �

ˆ

T W D dv: (66)

It follows that if the stress power T W D is pointwise positive, then the right-hand side of (66) is negative, and the net kinetic energy decays in time. If one wasinterested in the rate of decay, and in the proof that the net kinetic energy decays tozero, then one would need further information on the relation between the Cauchystress tensor and the symmetric part of the velocity gradient. (The details are notelaborated here; interested reader is referred to Serrin [86] for the detailed treatmentof the Navier–Stokes fluid, T D �pIC 2D. Note that the net kinetic energy doesnot completely vanish in finite time.) The decay of the kinetic energy of the fluidcontained in a closed vessel is definitely a desirable outcome from the point of viewof everyday experience, and, as it has been shown, the decay is closely related to thesecond law of thermodynamics.

4 Constitutive Relations

The motion of a single continuous medium is governed by the following system ofpartial differential equations

d�.x; t /

dtC �.x; t / div v.x; t / D 0; (67a)

�.x; t /dv.x; t /

dtD div T.x; t /C �.x; t /b.x; t /; (67b)

T.x; t / D T>.x; t /; (67c)

�.x; t /de.x; t /

dtD T.x; t / W D.x; t / � div jq.x; t / (67d)

that are mathematical expressions of the balance law for the mass, linear momen-tum, angular momentum, and total energy. (See equations (4), (13), (14), and (22).)The unknown quantities are the density �, the velocity field v, and the specificinternal energy e. (For the first reading, the internal energy can be thought of asa proxy for the temperature field �.x; t /.)

These equations are insufficient for the description of the evolution of thequantities of interest. The reason is that the Cauchy stress tensor T and the heatflux jq act in (67) as additional unknowns a priori unrelated to the density �, thevelocity field v, and the specific internal energy e. Therefore, in order to get aclosed system of governing equations, system (67) must be supplemented by a set ofequations relating the stress and the heat flux to the other quantities. These relationsare called the constitutive relations, and they describe the response of the materialto the considered stimuli.


It is worth emphasizing that the constitutive relations are specific for the givenmaterial. For example, the Cauchy stress tensor is a quantity that describes howthe given volume of the material is affected by its surrounding; see Sect. 2.1.2.Clearly, the interaction between the given volume of the material and its surroundingdepends on the particular type of the material. Similarly, the heat flux describesthe thermal energy transfer capabilities of the given material; hence, it is materialspecific. The fact that the constitutive relations are material specific means thatthey indeed provide an extra piece of information supplementing the balance laws.Consequently, the constitutive relations can be derived only if one appeals tophysical concepts that go beyond the balance laws.

Moreover, the constitutive relations are inevitably simplified and reduceddescriptions of the physical reality, and as such they are designed to describeonly the behavior of the given material in a certain class of processes. For example,the same material, say steel, can be processed by hot forging, and it can deform asa part of a truss bridge or move as a projectile. The mathematical models for theresponse of the material in these processes are however different despite the factthat the material remains the same. The specification of the constitutive relationstherefore depends on two things – the material and the considered processes.

The most convenient form of the constitutive relations would be a set of equationsof the form

T D T.�; v; e/; (68a)

jq D jq.�; v; e/: (68b)

This would allow one to substitute for T and jq into (67) and get a system ofevolution equations for the unknown fields �, v, and e. An example of a systemof constitutive relations of the type (68) are the well-known constitutive relationsfor the compressible viscous heat-conducting Navier–Stokes fluid,

T D �pth.�; �/IC � .div v/ IC 2�D; (69a)

jq D � r�; (69b)

where pth.�; �/ denotes the thermodynamic pressure determined by an equation ofstate. (See Sect. 4.2 for details.) Substituting (69) into (67) then leads to the well-known Navier–Stokes–Fourier system of partial differential equations.

Besides the constitutive relations, the system of balance laws must be supple-mented with initial and boundary conditions. The boundary conditions can be seenas a special case of constitutive relations at the interface between two materials.As such the boundary conditions can be very complex. In fact, the processesat the interface can have, for example, their own dynamics, meaning that theycan be governed by an extra set of partial differential equations at the interface.Unfortunately, discussion of complex boundary conditions goes beyond the scopeof the present contribution, and the issue is only briefly touched in Sect. 4.6.Consequently, overall the discussion of boundary conditions is mainly restricted


to the claim that the considered boundary conditions are the standard ones such asthe no-slip boundary condition for the velocity field. The reader should be howeveraware of the fact that the specification of the boundary conditions is far from beingtrivial and that it again requires additional physical insight into the problem.

The early efforts concerning the theory of constitutive relations for fluids (see,e.g., Rivlin and Ericksen [83], Oldroyd [65], and Noll [64] or the historical essayby Tanner and Walters [90]) were mainly focused on the specification of the Cauchystress tensor in terms of the kinematical variables. These early approaches werebased almost exclusively on mechanical considerations, such as the symmetry ofthe material and the requirement of the invariance of the constitutive relations withrespect to the change of the observer. Thermodynamic considerations were rarelythe leading theme and were mainly reduced to the a posteriori verification of thenonnegativity of the stress power.

Since then thermodynamics started to play an increasingly important role in thetheory of constitutive relations; see, for example, Coleman [19] and Truesdell andNoll [91]. This paradigm shift went hand in hand with the development of thefield of nonequilibrium thermodynamics; see, for example, de Groot and Mazur[34], Glansdorff and Prigogine [32], Ziegler [97], and Müller [62]. In what followsa modern thermodynamics based approach to the phenomenological theory ofconstitutive relations is presented. The presented approach is essentially based onthe work of Rajagopal and Srinivasa [78] (see also Rajagopal and Srinivasa [80]),who partially took inspiration from various earlier achievements in the field, mostnotably from the work of Ziegler [97] and Ziegler and Wehrli [98]. The reader whois interested into some comments concerning the precursors of the current approachis referred to Rajagopal and Srinivasa [78].

The advantage of the presented approach is that the second law of thermody-namics plays a key role in the theory of constitutive relations. Unlike in some otherapproaches, the second law is not used a posteriori in checking whether the derivedconstitutive relations conform to the second law. In fact, the opposite is true; thesecond law is the starting point of the presented approach. The constitutive relationsfor the Cauchy stress tensor, the heat flux, and the other quantities of interest are inthe presented approach derived as consequences of the choice of the specific formfor the internal energy e and the entropy production � .

If the entropy production � is chosen to be nonnegative, then the second lawis satisfied. Since the relations of the type (68) are in the presented approach theconsequences of the choice of e and � , then it is clear that the arising constitutiveequations cannot violate the second law. The second law is automatically built inthe arising constitutive relations of the type (68). Further, the specification of twoscalar quantities e and � is apparently much easier than the direct specification ofthe constitutive relations between the vectorial and tensorial quantities as in (68).This brings simplicity into the theory of constitutive relations. Two scalar quantitiesdetermine everything.

Finally, the advocated approach is robust enough to handle complex materials.Nonlinear constitutive relations and constitutive relations for constrained materialsas well as constitutive relations for materials reacting to the stimuli of various


origins (thermal, mechanical, electromagnetic, chemical) are relatively easy to workwith in the presented approach.

A general outline of the advocated approach is given in Sect. 4.1, and then it isshown how the general approach can be used in various settings. First, the approachis applied in a very simple setting, namely, the well-known constitutive relations forthe standard compressible Navier–Stokes–Fourier fluid are derived in Sect. 4.2. Thisexample should allow the reader to get familiar with the basic concepts. Then thediscussion proceeds to more complex settings, namely, that of Korteweg fluid; seeSect. 4.3. Finally (see Sect. 4.4), the discussion of the approach is concluded by thederivation of constitutive equations for viscoelastic fluids.

4.1 General Framework

The main idea behind the presented approach is that the behavior of the materialin the processes of interest is determined by two factors, namely, its ability tostore energy and produce entropy. The energy storage mechanisms are specifiedby the choice of the energetic equation of state, that is, by expressing the internalenergy e as a function of the state variables. (Other thermodynamic potentials suchas the Helmholtz free energy, Gibbs potential, or the enthalpy can be used as well.The discussion below is however focused exclusively on the internal energy.) Theentropy production mechanisms are specified by the choice of the formula for theentropy production � .

The derivation of the constitutive relations (68) from the knowledge of e and �then proceeds in the following steps:

STEP 1: Specify the energy storage mechanisms by fixing the constitutive relationfor the specific internal energy e in the form of the energetic equation of state

e D e.�; y1; : : : ; ym/ (70)

or in the form of the entropic equation of state (26). At this level it is sufficientto determine the state variables yi that enter (70).

STEP 2: Find an expression for the material time derivative of the specificentropy �. This can be achieved by the application of the material timederivative to (70), and by the multiplication of the result by �.x; t /, whichyields

�.x; t /de.x; t /

dtD

�.x; t /@e.�; y1; : : : ; ym/

@�

ˇˇ�D�.x;t/;y1Dy1.x;t/;:::;ymDym.x;t/

d�.x; t /

dt


C

mXiD1

�.x; t /@e.�; y1; : : : ; ym/

@yi

ˇˇ�D�.x;t/;y1Dy1.x;t/;:::;ymDym.x;t/

dyi .x; t /

dt:

(71)

The definition of the thermodynamic temperature (see (29)) and (71) then yield

�.x; t /�.x; t /d�.x; t /

dtD �.x; t /

de.x; t /

dt

�

mXiD1

�.x; t /@e.�; y1; : : : ; ym/

@yi

ˇˇyjDyj .x;t /; �D�.x;t /

dyi .x; t /

dt; (72)

which is the sought formula for d�dt .

STEP 3: Identify the entropy production. Use the balance equations and kinematicsto write down (derive) the formulae for the material time derivative of the statevariables dyi .x;t /

dt , i D 1; : : : ; m, and substitute these formulae into (72). Rewrite

the equation for �.x; t / d�.x;t/dt in the form

�.x; t /d�.x; t /

dtC div j�.x; t/ D

1

�.x; t /

mX˛D1

j˛.x; t / � a˛.x; t /; (73)

where j˛.x; t / � a˛.x; t / denotes the scalar product of vector or tensor quanti-ties, respectively. The right-hand side of (73) is the entropy production, whereeach summand is supposed to represent an independent entropy-producingmechanism.The quantities j˛.x; t / are called the thermodynamic fluxes, and the quantitiesa˛.x; t / are called the thermodynamic affinities. The affinities are usually thespatial gradients of the involved quantities, for example, r� or D, while thefluxes are, for example, the heat flux jq or the Cauchy stress tensor T. (As a ruleof thumb, the fluxes are the quantities that appear under the divergence operatorin the balance laws (67); see also Rajagopal and Srinivasa [78] for furthercomments.) The quantity j�.x; t/ is called the entropy flux, and in standard

cases, it is tantamount tojq.x;t/

�.x;t /.

STEP 4* (Linear nonequilibrium thermodynamics): The second law of thermo-dynamics states that the entropy production �.x; t / Ddef �.x; t /

d�.x;t/dt C

div j�.x; t/ is nonnegative; see Sect. 2.3.3. Referring to (73), it follows that

�.x; t / D1

�.x; t /

mX˛D1

j˛.x; t / � a˛.x; t / (74)


must be nonnegative. A simple way to fulfill this requirement is to considerlinear relations between each pair of j˛.x; t / and a˛.x; t /, that is,

j˛.x; t / D �˛a˛.x; t /; (75)

where �˛ , ˛ D 1; : : : ; m, are nonnegative constants. Alternatively, one canconsider cross effects by assuming that the relations between j˛.x; t / anda˛.x; t / take the form

j˛.x; t / D

mXˇD1

�˛ˇaˇ.x; t /; (76)

where �˛ˇ is a symmetric positive definite matrix. This is essentially theapproach of linear nonequilibrium thermodynamics; see de Groot and Mazur[34] for details.

STEP 4 (Nonlinear nonequilibrium thermodynamics): Since the linear relation-ships between the fluxes j˛.x; t / and affinities a˛.x; t / can be insufficientfor a proper description of the behavior of complex materials, an alternativeprocedure is needed. In particular, the procedure should allow one to derivenonlinear constitutive relations of the type ji D ji .a1; : : : ; am/ or vice versa.Here it comes to the core of the approach suggested by Rajagopal and Srinivasa[78]. As argued by Rajagopal and Srinivasa [78], one first specifies function �in one of the following forms

� D �a1 ;:::;am.j1; : : : ; jm/; (77a)

or

� D �j1;:::;jm.a1; : : : ; am/: (77b)

Note that the state variables can enter the constitutive relations (77) as well,but they are not, for the sake of compactness of the notation, written explicitlyin (77).Formula (77a) or (77b) is a constitutive relation that determines the entropyproduction � , � Ddef

�

�. Since the constitutive function � is – up to the positive

factor � – tantamount to the entropy production, it must be nonnegative, whichguarantees the fulfillment of the second law of thermodynamics. Further, �should vanish if the fluxes vanish. Other restrictions concerning the formulafor the entropy production can come from classical requirements such as thematerial symmetry and the invariance with respect to the change of the observer.Moreover, the assumed form of the entropy production must be compatiblewith the already derived form of the entropy production (74). Consequently,the following equation must hold


�j1;:::;jm.a1; : : : ; am/ �1

�.x; t /

mX˛D1

j˛.x; t / � a˛.x; t / D 0; (78)

and similarly for �a1;:::;am.j1; : : : ; jm/. In terms of the constitutive function �,this reduces to the requirement

�a1;:::;am.j1; : : : ; jm/ �

mX˛D1

j˛.x; t / � a˛.x; t / D 0; (79a)

or to

�j1;:::;jm.a1; : : : ; am/ �

mX˛D1

j˛.x; t / � a˛.x; t / D 0; (79b)

respectively, depending whether one starts with (77a) or (77b).Relations (79) can be rewritten in the form

�A.J / � J �A D 0; (80a)

�J .A/ � J �A D 0; (80b)

where A Ddefa1 : : : am

�>and J D

j1 : : : jm

�>are vectors in RM

0

,M 0 > m that contain all the affinities and fluxes, respectively. Here the tensorialquantities such as the Cauchy stress tensor T are understood as column vectorsT11 T12 : : : T33

�>.

Having fixed a formula for the constitutive function � via (77a) or (77b), respec-tively, the task is to determine the fluxes j1; : : : ; jm or affinities a1; : : : ; am thatare compatible with the single constraint (80a) or (80b), respectively. If thefluxes/affinities are vectorial or tensorial quantities, then the single constraint isnot sufficient to fully determine the sought relation between the fluxes and theaffinities. (This means that in general there exist many constitutive relations ofthe type ji D ji .a1; : : : ; am/ or vice versa such that (80a) or (80b) holds.)Rajagopal and Srinivasa [78] argued that the choice between the multipleconstitutive relations that fulfill the constraint (80a) or (80b) can be based on theassumption of maximization of the entropy production. The assumption simplyrequires that the sought constitutive relation is the constitutive relation that leadsto the maximal entropy production in the material and that is compatible withother available information concerning the behavior of the material.In more operational terms, the maximization of the entropy production leadsto the following problem. Given the function �A.J / and the values of thestate variables and the values of the affinities A, the corresponding values ofthe fluxes J are those which maximize �A.J / subject to the constraint (80a),


and alternatively to other possible constraints. (Note that since � is a positivequantity, then the task to maximize � with respect to the fluxes is indeedtantamount to the task of maximizing the entropy production � with respectto the fluxes.) A similar reformulation can be made for �J .A/; it suffices toswitch the role of the fluxes and affinities.The constrained maximization problems arising from the assumption on themaximization of the entropy production read

maxJ2J

�A.J / (81)

where J Ddef

nJ 2 RM

0

; �A.J / � J �A D 0o, and

maxA2A

�J .A/ (82)

where A Ddef

nA 2 RM

0

; �j .A/ � J �A D 0o. Assuming that �A or �J are

smooth and strictly convex with respect to their variables, then the correspond-ing values of J and A, respectively, are uniquely determined and can be foundeasily by employing the Lagrange multipliers; see Rajagopal and Srinivasa [78].For example, if the starting point is the constitutive function �A.J /, then theauxiliary function for the constrained maximization reads ˆ.J / Ddef �A.J /C

` Œ�A.J / � J �A�, where ` is a Lagrange multiplier. The condition for the valueof the flux J that corresponds to maximal entropy production is @�.J /

@JD 0

which leads to equation

A D1C `

`

@�A.J /

@J; (83)

where

1C `

`D

�A.J /@�A.J /

@J� J: (84)

Relation (83) is the sought relation between J and A.Consequently, if one accepts the assumption on the maximization of the entropyproduction, then the relations between the fluxes and affinities are due to thestrict convexity of �A.J / indeed uniquely determined by the choice of theconstitutive function �A.J /. Moreover, if the constitutive function �A.J / isquadratic in J , then one recovers the result known from linear nonequilibriumthermodynamics, namely, that the fluxes are linear functions of the affinities;see (76). Apparently, a similar procedure can be followed if the starting point is�J .A/ instead of �A.J /.

STEP 5: Relations (75) or (76) or (83) are the sought constitutive relations.


The rest of the present contribution is focused mainly on linear relations betweenthe fluxes and affinities; hence, a quadratic ansatz for the constitutive function � ispredominantly used in the rest of the text. This means that STEP 4 is rarely referredto in its full complexity and that one basically stays in the framework of STEP 4*.

The reason is that the following text is mainly focused on the most difficult stepof the advocated approach which is the correct specification of the functions e and�. This is a nontrivial task in the development of the mathematical model, since thecorrect specification of these functions requires physical insight into the problem.However, the key physical arguments concerning the choice of the functions e and� can be easily documented in the simplest possible case of a quadratic ansatz;generalization of the quadratic ansatz to a more complex one is then a relativelystraightforward procedure. Also note that once the functions e and � are specified,the derivation of the constitutive relations is only a technical problem not worth oflengthy discussion.

4.2 Compressible and Incompressible Viscous Heat-ConductingFluids

The application of the outlined procedure is first documented in a very simplecase of the derivation of constitutive relations for the compressible Navier–Stokes–Fourier fluid (see Sect. 4.2.1), which is a simple model for a compressible viscousheat-conducting fluid. The next section (see Sect. 4.2.2) is devoted to reducedmodels that are variants of the compressible Navier–Stokes–Fourier fluid model.Finally, the incompressible counterparts of the former models are discussed inSect. 4.2.3.

4.2.1 Compressible Navier–Stokes–Fourier FluidSTEP 1: The specific internal energy e is assumed to be a function of the specific

entropy � and the density �,

e D e.�; �/: (85)

This is a straightforward generalization of the classical energetic equation ofstate E D E.S; V / known for the equilibrium thermodynamics; see, forexample, Callen [15]. (For the sake of clarity of the notation, the spatial andtemporal variable is omitted in (85). If written in full, the energetic equation ofstate should read e.x; t / D e.�.x; t /; �.x; t //. The same approach is applied inthe rest of this section.)


STEP 2: Taking the material derivative of (85) yields

�@e

@�

d�

dtD �

de

dt� �

@e

@�

d�

dt: (86)

Recalling the definition of the temperature (see (29)) and introducing thethermodynamic pressure pth through

� Ddef@e

@�; (87a)

pth Ddef �2 @e

@�; (87b)

which are again the counterparts of the classical relations known from equilib-rium thermodynamics, one can rewrite (86) as

��d�

dtD �

de

dt�pth

�

d�

dt: (88)

STEP 3: Using the evolution equation for the internal energy (67d) and the balanceof mass (67a), one can in (88) substitute for de

dt and d�dt which yields

��d�

dtD T W D � div jq C pth div v: (89)

The right-hand side can be further rewritten as

T W D � div jq C pth div v D Tı W Dı C .mC pth/ div v � div jq; (90)

where

m Ddef1

3Tr T (91)

denotes the mean normal stress, and

Tı Ddef T �1

3.Tr T/ I; (92)

Dı Ddef D �1

3.Tr D/ I (93)

denote the traceless part of T and D, respectively. Since the thermodynamictemperature is positive, one can, after some manipulation, rewrite (89) as

�d�

dtC div

�jq

�

�D1

�

�Tı W Dı C .mC pth/ div v � jq �

r�

�

(94)

which is the evolution equation for � in the desired form (73).


The rationale for using the traceless parts of T and D in (94) is the following.The quantities D and div v that appear in the first and the last term on the right-hand side of (89) are not independent quantities. (Recall that Tr D D div v.)However, the fluxes one would like to identify when writing (89) in theform (73) should be independent quantities. Therefore, it is necessary to split Dto mutually independent quantities Dı and div v, which requires one to split theCauchy stress tensor in a similar manner.A closer inspection of (94) indicates that there exist three “independent”entropy-producing mechanisms in the material. The first one, �jq �

r��

, isentropy production due to heat transfer; the second one, .mC pth/ div v, isentropy production due to volume changes; and the last entropy productionmechanism, Tı W Dı , is due to isochoric processes such as shearing.

STEP 4: The entropy production � – that is, the right-hand side of (94) – takes theform

� D1

�

3X˛D1

j˛ � a˛ D1

�

�Tı W Dı C .mC pth/ div v � jq �

r�

�

: (95)

Using the flux/affinity nomenclature, the fluxes are the traceless part of theCauchy stress tensor Tı , the mean normal stress plus the thermodynamicpressure m C pth, and the heat flux jq . The affinities are Dı , div v, and �r� .

(Strictly speaking the last affinity is not �r� but �r��

. However, the positivefactor 1

�is of no importance.) The entropy production is a positive quantity if

the constitutive relations are chosen as follows,

Tı D 2�Dı; (96a)

mC pth D Q� div v; (96b)

jq D � r�; (96c)

where � > 0, Q� > 0, and > 0 are given positive functions of the statevariables � and �. This choice of constitutive relations follows the templatespecified in (75). (Note that since Dı is a symmetric tensor, then Tı is alsosymmetric; hence, the balance of angular momentum (67c) is satisfied.) Thecoefficients �, Q�, and are called the shear viscosity, the bulk viscosity, and theheat conductivity. Traditionally, the bulk viscosity is written in the form

Q� D3�C 2�

3; (97)

where � is a function of the state variables such that Q� remains positive.


STEP 5: It follows from (96) that the constitutive relation for the full Cauchy stresstensor T D Tı CmI is

T D 2�Dı C Q� .div v/ I � pthI (98)

which can be in virtue of (97) rewritten as

T D �pthIC 2�DC �.div v/I: (99a)

This is the standard formula for the Cauchy stress tensor in the so-calledcompressible Navier–Stokes fluid. The constitutive relation for the heat fluxreads

jq D � r�; (99b)

which is the standard Fourier law of thermal conduction.

The coefficients in the constitutive relations can be functions of the state variables� and �. This choice of variables is however inconvenient in practice. Solving theequation (87a) for the entropy, one can see that the entropy could be written as afunction of the temperature and the density

� D f .�; �/: (100)

If this is possible, then the coefficients can be rewritten as functions of thetemperature and the density,

pth?.�; �/ D pth .f .�; �/; �/ ; (101a)

�?.�; �/ D � .f .�; �/; �/ ; (101b)

�?.�; �/ D � .f .�; �/; �/ ; (101c)

?.�; �/ D .f .�; �/; �/ ; (101d)

e?.�; �/ D e .f .�; �/; �/ : (101e)

Consequently, the final set of the governing differential equations arising fromthe balance laws (67) and the constitutive relations (99) reads

d�

dtD �� div v; (102a)

�dv

dtD div TC �b; (102b)

�de?.�; �/

dtD T W D � div jq; (102c)


T D �pth?.�; �/IC 2�?.�; �/DC �?.�; �/.div v/I; (102d)

jq D � ?.�; �/r�: (102e)

These are the standard governing equations for the so-called compressible Navier–Stokes–Fourier fluids. If necessary, the evolution equation for the internal totalenergy (102c) can be reformulated as a balance of total energy, that is,

�d

dt

�e?.�; �/C

1

2jvj2

�D �b � vC div .Tv/ � div jq: (103)

Note that if the internal energy as a function of the temperature and the densitytakes the simple form e? D cV � , where cV is a constant referred to as the specificheat capacity, then the evolution equation for the internal energy (102c) takes thewell-known form

�cVd�

dtD T W D � div jq: (104)

4.2.2 Other Linear Models for Compressible FluidsReferring to (95) one can notice that the entropy production � can be written in theform

�� DTı; mC pth;�jq

��

�Dı; div v;

r�

�

D J �A: (105)

If the constitutive relation for any of the quantities Tı , m C pth, jq is trivial, thatis, if Tı D 0, m C pth D 0 or jq D 0, then the corresponding term in theentropy production vanishes. Such models are referred to as the reduced models.For example, Tı D 0 corresponds to an inviscid compressible heat-conducting fluid(Euler–Fourier fluid).

On the other hand, if any of the quantities Dı , div v, r� vanishes, thenthe corresponding entropy production also vanishes regardless of the particularconstitutive equation for Tı ,mCpth, or jq , respectively. The processes in which Dı ,div v, or r� vanishes are the motion with a velocity field in the form v.x; t / Ddef

a.t/ � x C b.t/, where a and b are arbitrary time-dependent vectors, the isochoricmotion, that is, volume-preserving motion, and isothermal process, that is, a processwith no temperature variations, respectively.

Going back to the constitutive relations, the entropy production completelyvanishes if the constitutive relations take the form

Tı D 0; m D �pth; and jq D 0: (106)

This means that

T D �pth?.�; �/I; and jq D 0; (107)


which are the constitutive equations for a compressible Euler fluid. The correspond-ing system of partial differential equations reads

d�


�dv

dtD �rpth

?.�; �/C �b; (108b)

�d

dt

�e?.�; �/C

1

2jvj2

�D � div .pth

?.�; �/v/C �b � v: (108c)

There is a hierarchy of models between the compressible Euler fluid (108) withno entropy production and the compressible Navier–Stokes–Fourier fluid (102) inwhich all entropy production mechanisms are active. These models are, for example,the Euler–Fourier fluid where the constitutive relations read

Tı D 0; m D �pth; and jq D � ?.�; �/r�; (109)

the fluid where thermodynamic pressure coincides (up to the sign) with the meannormal stress, that is, the fluid with constitutive relations

m D �pth; jq D � ?.�; �/r�.x; t/; Tı D 2�

?.�; �/Dı; (110)

which leads to T D �pth?.�; �/I C 2�?.�; �/Dı , or the fluid dissipating due to

volume changes but not due to shearing, that is, the fluid with constitutive relations

Tı D 0; m D �pth?.�; �/C �?.�; �/ div v; jq D �

?.�; �/r�: (111)

Finally, one can observe that if one deals with the ideal gas, that is, with the fluidgiven by constitutive relations (102) and e?.�; �/ D cV � , where cV is a constant,then one can in some cases consider isothermal processes. (The temperature is aconstant �.x; t / D �?.) Indeed, if the initial temperature distribution is uniform,and if the fluid undergoes only small volume changes, and if the entropy productionmechanisms in the fluid are weak, then the terms

T W D D �pth div vC Q� .div v/2 C 2�Dı W Dı; (112)

and div jq do not significantly contribute to the right-hand side of (104). In sucha case, and under favorable boundary conditions for the temperature, the evolutionequation (104) for the internal energy/temperature is of no interest, and one canfocus only on the mechanical part of the system, which leads to the compressibleNavier–Stokes equations


d�


�dv

dtD �rfpth.�/C div .2 Q�.�/D/Cr

�Q�.�/ div v

�C �b; (113b)

with Q�.�/ > 0 and 2 Q�.�/ C 3 Q�.�/ > 0. Here fpth.�/ Ddef pth?.�?; �/, Q�.�/ Ddef

�?.�?; �/, and Q�.�/ Ddef �?.�?; �/. Note that genuine isothermal processes are in

fact impossible in a fluid with nonvanishing entropy production due to mechanicaleffects. Once there is entropy production due to the term T W D, kinetic energy is lostin favor of thermal energy, and the temperature must change; see (102c) or (104).The isothermal process is in this case only an approximation.

Other simplified systems that arise from (113) are the compressible Eulerequations

d�


�dv

dtD �rfpth.�/C �b; (114b)

the compressible “bulk viscosity” equations

d�


�dv

dtD �r

fpth.�/C

2 Q�.�/C 3 Q�.�/

3div v

!C �b: (115b)

and the compressible “shear viscosity” equations

d�


�dv

dtD �rfpth.�/ � div .2 Q�.�/D/C �b: (116b)

4.2.3 Incompressible Navier–Stokes–Fourier FluidIncompressibility means that the density of any given material point X is constantin time. In terms of the Eulerian density field �.x; t /, this requirement translatesinto the requirement of vanishing material time derivative of �.x; t /,

d�

dtD 0: (117)


(Note that the density can still vary in space, i.e., the density assigned to differentmaterial points can be different. In such a case, the material is referred to asinhomogeneous material.) Using the balance of mass (67a), it follows from (117)that

div v D 0; (118)

which means that the material can undergo only isochoric – that is, volumepreserving – motions. In this sense, the incompressible material can be understoodas a constrained material, meaning that the class of motions possible in the givenmaterial is constrained by the requirement div v D 0.

If one takes into account the constraint div v D 0, then the term

T W D D Tı W Dı Cm div v (119)

in the evolution equation for the internal energy (67d) reduces to Tı W Dı . (Recallthat m denotes the mean normal stress, m Ddef

13

Tr T, and that Tr D D div v.)This implies that the mean normal stress cannot be directly present in the entropyproduction in the form of a flux/affinity pair, because the corresponding fluxdiv v identically vanishes. This observation is easy to document by inspecting theentropy production formula (95) for compressible heat-conducting fluids, where theconstraint div v D 0 would imply that .mC pth/ div v D 0.

However, the specification of the constitutive relations is based on the presenceof the quantities of interest in the formula for the entropy production in the formof a flux/affinity product. In this context one could say that the mean normal stresscannot be specified constitutively. (This means that the mean normal stress m isnot – in contrast to the traceless part Tı of the Cauchy stress tensor – given by aconstitutive relation in terms of other variables such as the kinematical quantities.)

If thought of carefully, this is a natural consequence of the concept of incom-pressibility. The mean normal stress in an incompressible fluid must incorporate theforce mechanism that prevents the fluid from changing its volume. (In mathematicalterms this means that it must depend on the Lagrange multiplier enforcing theincompressibility constraint (118).) As such it cannot be specified without theknowledge of boundary conditions. Think, for example, of a vessel completely filledwith an incompressible fluid at rest. The value of the mean normal stress in the vesselcannot be determined without the knowledge of the forces acting on the vessel walls.

This makes the normal stress in an incompressible fluid a totally differentquantity than the mean normal stress in a compressible fluid. In a compressiblefluid (see, e.g., (96b)), the mean normal stress is a function of the thermodynamicpressure pth and the velocity field, and the thermodynamic pressure is a function ofthe density and the temperature. Therefore, the mean normal stress in a compressiblefluid is fully specified in terms of the local values of the state variables and thevelocity field. On the other hand, the mean normal stress in an incompressible fluidis not a function of the local values of the state variables and the velocity field; itconstitutes another unknown in the governing equations.


Unfortunately, the simplest model for an incompressible viscous fluid is theNavier–Stokes fluid where the Cauchy stress tensor is given by the formula

T D �pIC 2�D; (120)

where p is called the “pressure.” This could be misleading since the quantity called“pressure” also appears in the compressible Navier–Stokes fluid, albeit the pressurein the compressible Navier–Stokes fluid is the thermodynamic pressure, which isa totally different quantity than the pressure p in (120). Further, the mean normalstress m in the incompressible Navier–Stokes fluid model (120) is m D �p; henceit differs from the “pressure” p by a mere sign. Consequently, the notions of themean normal stress and the “pressure” are often used interchangeably, which is abad practice since these notions do not coincide if one deals with incompressiblefluid models that go beyond (120).

Concerning the compressible Navier–Stokes fluid model (102d), the meannormal stress is given by the formula m D �pth C

3�C2�3

div v; hence the meannormal stress differs from the thermodynamic pressure by more than a sign. Onthe other hand, the mean normal stress coincides with the thermodynamic pressureif one adopts so-called Stokes assumption 3� C 2� D 0. In such a case, one isallowed to use interchangeably terms thermodynamic pressure and mean normalstress, which is otherwise not correct.

The subtleties concerning the notion of the pressure, Lagrange multiplier enforc-ing the incompressibility constraint, thermodynamic pressure, and the mean normalstress can be of importance if one, for example, wants to talk about viscoelasticmaterials with “pressure”-dependent coefficients. The different contexts in whichthe term pressure is used are discussed in detail by Rajagopal [76] (see alsocomments by Huilgol [43] and Dealy [22]), and this issue is not further commentedin the present treatise. The reader should be however aware of the fact that the notionof “pressure” becomes complicated in particular if one deals with non-Newtonianfluids.

Concerning the possible approaches to the derivation of constitutive relations forincompressible materials, one can either opt for the strategy outlined in Sect. 4.1and enforce the incompressibility constraint via an additional Lagrange multiplier.Examples of such approaches can be found in Málek and Rajagopal [54] (incom-pressible Navier–Stokes fluid) and Málek et al. [58] (incompressible viscoelasticfluids). This approach allows one to find a relation between the Lagrange multiplierenforcing the incompressibility constraint and the mean normal stress. However,this relation is usually of no interest in the final constitutive relations. Therefore, adifferent and more pragmatic approach is adopted in the rest of the text.

The constitutive relations are derived in a standard manner with the con-straint (118) in mind, which leads to a constitutive relation for the traceless partof the Cauchy stress tensor Tı . As shown above, the incompressibility constraintmakes the search for a constitutive relation for the mean normal stress pointless;hence one can leave the final formula for the full Cauchy stress tensor in the from

T D mIC Tı; (121)


where m is understood as a new unknown quantity to be solved for. (In (121) onecan further exploit the fact that the incompressibility constraint (118) allows one towrite D D Dı .)

In particular, if the outlined approach is applied in the case of the constitutiverelations derived in Sect. 4.2.1, one gets T D mIC Tı D mIC 2�?.�; �/Dı; hence

T D mIC 2�?.�; �/D; (122a)

where the relation D D Dı has been exploited. The constitutive relation for the heatflux remains formally the same,

jq D � ?.�; �/r�: (122b)

These are the constitutive relations for the incompressible Navier–Stokes–Fourierfluid. Note that unlike in the majority of the literature, the constitutive relation forthe Cauchy stress tensor is written as (122a) and not as

T D �pIC 2�?.�; �/D: (123)

This notation emphasizes the different nature the thermodynamic pressure pth incompressible fluids, which is frequently denoted as p, and the “pressure” p Ddef

�m in incompressible fluids.The governing equations for the incompressible Navier–Stokes–Fourier fluid

then take the form

d�

dtD 0; (124a)

div v D 0; (124b)

�dv

dtD rmC div .2�?.�; �/D/C �b; (124c)

�d

dt

�e?.�; �/C

1

2jvj2

�D div .mvC 2�?.�; �/DvC ?.�; �/r�/C �b � v:

(124d)

The first equation in (124), @�@tC v � r� D 0, implies that � is transported along

the characteristics,

�.x; t / D �0 .X/ D �0��1.x; t /

�; (125)

where �0 is the density in the initial (reference) configuration. Consequently thedensity �.x; t / is constant in space and time if the initial (reference) density �0 isuniform, that is, if

�0.X 1/ D �0.X 2/ (126)


holds for all X 1, X 2 2 B. If (126) does not hold, then the fluid at its initial(reference) state is inhomogeneous. This is why the system (124) is sometimescalled inhomogeneous incompressible Navier–Stokes–Fourier equations.

Obviously the incompressible Navier–Stokes–Fourier fluid model can be furthersimplified. Setting Tı D 0, one gets a model for a fluid without shear viscosity.Such a model can be referred to as inhomogeneous incompressible Euler–Fourierfluid model, and the governing equations reduce to

d�

dtD 0; (127a)

div v D 0; (127b)

�dv

dtD rmC �b; (127c)

�d

dt

�e?.�; �/C

1

2jvj2

�D div .mvC ?.�; �/r�/C �b � v: (127d)

Note that equations (127a)–(127c) are not coupled with the temperature and canprovide a solution for � and v independently of the energy equation. The fluiddescribed by (127a)–(127c) is called the inhomogeneous incompressible Euler fluid.

If the viscosity in (124) does not depend on the temperature, �?.�; �/ D Q�.�/,then energy equation again decouples from the system, and one can solve onlythe mechanical part of (124). This leads to inhomogeneous incompressible Navier–Stokes equations

d�

dtD 0; (128a)

div v D 0; (128b)

�dv

dtD rmC div .2 Q�.�/D/C �b: (128c)

If the density at initial (reference) configuration is uniform, then as a consequenceof (126) and (125), one finds that

�.x; t / Ddef �? 2 .0;C1/ (129)

for all t > 0, x 2 B. System (124) then simplifies to the so-called homogeneousincompressible Navier–Stokes–Fourier equations

div v D 0; (130a)

�dv

dtD rmC div .2 O�.�/D/C �b; (130b)

�d

dt

�Oe.�/C

1

2jvj2

�D div .mvC 2 O�.�/DvC O .�/r�/C �b � v; (130c)


where O�.�/ Ddef �?.�; �?/, Oe.�/ Ddef e

?.�; �?/, and O .�/ Ddef ?.�; �?/. Fol-

lowing the same template as above, one can introduce homogeneous incompressibleEuler–Fourier fluids.

If the viscosity does not depend on the temperature, �? Ddef O�.�?/ D �?.�?; �?/,

then (130c) decouples from the rest of the system (130), and one can solve only themechanical part of (130) for the mechanical variables m and v. This leads to theclassical incompressible Navier–Stokes equations

div v D 0; (131a)

�dv

dtD rmC div .2�?D/C �b; (131b)

where the right-hand side of (131b) can be rewritten as rmC div .2�?D/C �b Drm C �?�v C �b. Finally, if Tı D 0, then one ends up with the incompressibleEuler equations

div v D 0; (132a)

�dv

dtD rmC �b: (132b)

4.3 Compressible Korteweg Fluids

Another popular model describing the behavior of fluids is the model introducedby Korteweg [51]. The model has been developed with the aim to describe phasetransition phenomena, but it is also used for the modelling of the behavior of somegranular materials. The key feature of the model is that the Cauchy stress tensorincludes terms of the type r�˝r�, which is natural given the fact that the modelis designed to take into account steep density changes that can occur, for example,at a fluid/vapour interface. In what follows it is shown how to derive the modelusing the thermodynamical procedure outlined in Sect. 4.1. The reader interested invarious applications of the model and the details concerning the derivation of themodel is referred to Málek and Rajagopal [55] and Heida and Málek [38].

Since the density gradient is the main object of interest, for later use, it isconvenient to observe that the balance of mass (67a) implies

d

dt.r�/ D �r .� div v/ � Œrv�> r�: (133)

STEP 1: Consider the constitutive equation for the specific internal energy – or thespecific entropy � – in the form

� D Q�.e; �;r�/ or e D Qe.�; �;r�/: (134)


The presence of the density gradient r� in the formula for the internalenergy/entropy means that in this specific example, one goes, for the firsttime in this text, beyond classical equilibrium thermodynamics. Moreover, theenergetic equation of the state is considered in a special form

e Ddef Qe.�; �;r�/ D e.�; �/C�

2�jr�j2; (135)

where � is a positive constant. This choice is motivated by the fact that oneexpects the density gradient r� to contribute to the energy storage mechanismsand the fact that the norm of the gradient r� is the simplest possible choice ifone wants the internal energy to be a scalar isotropic function of r�.

STEP 2 and 3: Applying the material time derivative to (135), multiplying the resultby � and using the evolution equation for the internal energy (22), the balanceof mass (4) and identity (133) yield

�@e

@�

d�

dtDTWD�div jqC�

2 @Qe

@�div vC�.r�/ �r .� div v/C�.rv/W.r�˝r�/:

(136)Introducing the thermodynamic pressure and temperature via

� Ddef@Qe

@�; (137a)

pKth Ddef �

2 @Qe

@�; (137b)

one ends up with a formula for the material time derivative of the entropy

��d�

dtD .Tı C �.r�˝r�/ı/ W Dı C

�mC pK

th C�

3jr�j2

�div v � div jq

C div .�� .div v/r�/ � �� div v; (138)

where the relevant tensors have been split to their traceless part and the rest; seeSect. 4.2 for the rationale of this step. Notice however that this thermodynamicpressure pK

th is different from the thermodynamic pressure pNSEth Ddef �

2 @e@�

forthe Navier–Stokes fluid as they are related through

pKth D p

NSEth �

�

2jr�j2: (139)

Equation (138) can be further rewritten as

��d�

dtD .Tı C �.r�˝r�/ı/ W Dı C

�mC pK

th C�

3jr�j2 � ��

�div v

� div�jq � ��.r�/ div v

�; (140)


and using the notation

jQq Ddef jq � ��r�; (141)

and applying the standard manipulation finally yields

�d�

dtC div

�jQq

�

�D1

�

�.Tı C �.r�˝r�/ı/ W Dı

C�mC pK

th C�

3jr�j2 � ��

�div v � jq �

r�

�C �� .div v/ .r�/ �

r�

�

:

(142)

The affinities are in the present case Dı , div v, and r� . Apparently, the last termon the right-hand side of (142) can be interpreted either as

��.r�/ �

r�

�

div v (143)

and grouped with the second term�mC pK

th C�3jr�j2 � ��

�div v, or it can

be read as

Œ�� .div v/ .r�/� �r�

�(144)

and grouped with the third term jq �r��

. The question is left unresolved, and aparameter ı 2 Œ0; 1� that splits the last term as

�� .div v/ .r�/ �r�

�D.1�ı/

��.r�/ �

r�

�

div vCıŒ�� .div v/ .r�/� �

r�

�;

(145)

is introduced. The arising terms are then grouped with the corresponding termsin (142). This yields

�d�

dtC div

�j�

�

�D1

�

�.Tı C �.r�˝r�/ı/ W Dı

C

�mC pth

K C�

3jr�j2 � ��C .1 � ı/��.r�/ �

r�

�

�div v

��jq � ı�� .div v/r�

��r�

�

: (146)


STEP 4: The entropy production � – that is, the right-hand side of (146) – takes theform

� D1

�

3X˛D1

j˛ � a˛; (147)

where the affinities are Dı , div v, and r� . Requiring the linear relationsbetween the fluxes and affinities in each term contributing to the rate of entropyproduction leads to

Tı C �.r�˝r�/ı D 2�Dı; (148a)

mC pKth C

�

3jr�j2 � ��C .1 � ı/��.r�/ �

r�

�D2� C 3�

3div v;

(148b)

jq � ı�� .div v/r� D � r� (148c)

where � > 0, 2�C3�3

> 0, and > 0 are positive constants. This choice ofconstitutive relations follows the template specified in (75).

STEP 5: It follows from (148) that the constitutive relation for the full Cauchystress tensor T D Tı CmI is

TD�pKthIC2�DC� .div v/ I�� .r�˝r�/C�� .��/ I�.1 � ı/��.r�/ �

r�

�I

(149)that can be upon introducing the notation

� Ddef pNSEth �

�

2jr�j2 � ��C .1 � ı/��.r�/ �

r�

�(150a)

rewritten as

T D ��IC 2�DC � .div v/ I � � .r�˝r�/ : (150b)

Concerning the heat/energy flux, one gets

jq D � r� C ı�� .div v/r�: (150c)

Note that if only isothermal processes are considered, then (149) reduces to

T D �pNSEth IC 2�DC � .div v/ IC � .r�˝r�/C �

jr�j2

2C � .��/

!I;

(151)which is the constitutive relation obtained by Korteweg [51].


4.4 Compressible and Incompressible ViscoelasticHeat-Conducting Fluids

Viscoelastic materials are materials that exhibit simultaneously two fundamentalmodes of behavior. They can store the energy in the form of strain energy; hencethey deserve to be called elastic, and they dissipate the energy; hence they deserveto be called viscous. Since the two modes are coupled, the nomenclature viscoelasticis obvious.

A rough visual representation of materials that exhibit viscoelastic behavioris provided by systems composed of springs and dashpots. The springs representthe elastic behavior, while the dashpots represent the dissipation. Such a visualrepresentation is frequently used in the discussion of viscoelastic properties ofmaterials; see, for example, Burgers [14] or more recently Wineman and Rajagopal[95]. The use of the visual representation as a motivation for the subsequent studyof the constitutive relations for viscoelastic fluids is also followed in the subsequentdiscussion.

A simple example of a spring–dashpot system is the Maxwell element consistingof a viscous dashpot and an elastic spring connected in series; see Fig. 3a. Theresponse of the element to step loading is the following. (The step loading is aloading that is constant and is applied only over time interval Œt0; t1�; otherwise theloading is zero.) Initially, the element is unloaded, and it is in an initial state; seeFig. 3a. Once the loading is applied at time t0, the spring extends, and the dashpotstarts to move as well; see Fig. 3b. When the loading is suddenly removed at sometime t1 (see Fig. 3c), the spring, that is, the elastic element, instantaneously shrinksto its initial equilibrium length. On the other hand, the sudden unloading does not

ls

l0

ls

l0

ls

l0

a b

c

Fig. 3 Maxwell element. (a) Initial state. (b) Loaded state. (c) Unloaded state


change the length of the dashpot at all. (The term “length of the dashpot” denotesthe distance between the piston and the dashpot left wall.)

This means that the instantaneous response to the sudden load removal is purelyelastic, while the configuration (length) to which the material relaxes after thesudden load removal is determined by the length of the dashpot. Therefore, theunloaded configuration is determined by the behavior of the purely dissipativeelement.

4.4.1 Concept of Natural Configuration and Associated KinematicsNow the question is whether one can use this observation in the development of theconstitutive relations for viscoelastic materials. It turns out that this is possible. Onecan associate these three states (initial/loaded/unloaded) of the Maxwell elementwith the three states of a continuous body:

1. the initial (reference) configuration 0.B/,2. the current configuration t .B/ that the body takes at time t > 0 during the

deformation process due to the external load,3. the natural configuration p.t/.B/ that would be taken by the considered body at

time t > 0 upon sudden load removal.

The evolution from the initial configuration to the current configuration is thenvirtually decomposed to the evolution of the natural configuration (dissipative pro-cess) and instantaneous elastic response (non-dissipative process) from the naturalconfiguration to the current configuration. Figure 4 depicts the situation. Referringback to the spring–dashpot analogue, the evolution of the natural configuration plays

current configuration

reference configurationdissipativeresponse

elasticresponse

time

natural configuration

κ0(B)

κt(B)

κp(t)(B)

0 t

Fig. 4 Initial (reference), current and natural configurations associated with a material body


the role of the dashpot, while the instantaneous elastic response from the currentconfiguration to the natural configuration plays the role of the spring.

Derivation of constitutive relations for viscoelastic materials by appealing to theprocedure outlined in Sect. 4.1 can then proceed as follows. Since the energy storagemechanisms in a viscoelastic fluid are determined by the elastic part – the spring –the plan is to enrich the constitutive relation for the internal energy (see (70)) by astate variable that measures the deformation of the elastic part (deformation fromthe natural to the current configuration).

The next step of the thermodynamic procedure requires one to take material timederivative of the internal energy; hence a formula for the material time derivativeof the chosen measure of the deformation of the elastic part is needed. This taskrequires a careful analysis of the kinematics of continuous media with multipleconfigurations.

Recall that the key quantities in the kinematics of continuous media are thedeformation gradient

F.X ; t / Ddef@� 0.X ; t /

@X; (152)

and the spatial velocity field

v.x; t / Ddef@� 0.X ; t /

@t

ˇˇXD��1 0 .x;t /

: (153)

(The subscript 0 recalls that the deformation gradient is taken with respect to theinitial configuration.) Other quantities of interest are the left Cauchy–Green tensorand the right Cauchy–Green tensor B Ddef FF>, C Ddef F>F, the (spatial) velocitygradient L Ddef

@v@x

, and its symmetric part D. The material time derivative of F isthen given by the formula

dF

dtD LF: (154)

Within the framework consisting of three configurations 0.B/, t .B/, and p.t/.B/, the standard kinematical setup is extended by the deformation gradientF p.t/ describing the deformation between the natural and the current configurationand the deformation gradient G describing the deformation between the initial(reference) configuration and the natural configuration.

Obviously (see Fig. 4), the following relation holds

F D F p.t/G: (155)


For later use, one also introduces notation

B p.t/ Ddef F p.t/F> p.t/

; (156a)

C p.t/ Ddef F> p.t/F p.t/ ; (156b)

for the left and right Cauchy–Green tensors for the deformation from the natural tothe current configuration. The left Cauchy–Green tensor B p.t/ is the sought measureof the deformation associated with the instantaneous elastic part of the deformation.

The reason is the following. In the standard finite elasticity theory, one uses theleft Cauchy–Green tensor B Ddef FF> in order to characterize the finite deformationof a body. In the current setting, the elastic response is the response from the naturalto the current configuration; hence the left Cauchy–Green B p.t/ tensor built usingthe relative deformation gradient F p.t/ instead of the full deformation gradient F isused.

Now one needs to find an expression for the material time derivative of B p.t/ .(The knowledge of an expression for the time derivative is needed in the STEP 2 ofthe procedure discussed in Sect. 4.1.) The aim is to express B p.t/ as a function ofF p.t/ and a rate quantity associated with the dissipative part of the response and arate quantity associated with the total response.

Noticing that (154) implies that L D dFdt F�1, a new tensorial quantity L p.t/ is

introduced through

L p.t/ DdefdG

dtG�1; (157)

and its symmetric part is denoted as D p.t/ ,

D p.t/ Ddef1

2

�L p.t/ C L> p.t/

�: (158)

Relations (155), (154), and (157) together with the formula

d

dt

�G�1

�D �G�1

dG

dtG�1 (159)

then yield

dF p.t/dt

DdF

dtG�1 C F

dG�1

dtD LFG�1 � FG�1

dG

dtG�1 D LF p.t/ � F p.t/L p.t/ ;

(160)which implies

dB p.t/dt

D LB p.t/ C B p.t/L> � 2F p.t/D p.t/F

> p.t/

: (161)


This is the sought expression for the time derivative of the chosen measure of thedeformation. As required, the time derivative is a function of F p.t/ and the ratequantities D p.t/ and L that are associated to the dissipative part of the response andthe total response, respectively.

Introducing the so-called upper convected time derivative through the formula

OA Ddef

dA

dt� LA � AL>; (162)

where A is a second order tensor, it follows from (161) that

O

B p.t/ D �2F p.t/D p.t/F> p.t/

: (163)

For later reference it is worth noticing that (162) implies

O

I D �2D; (164)

and that (161) implies

d

dtTr B p.t/ D 2B p.t/ W D � 2C p.t/ W D p.t/ : (165)

Since ddt det A D .det A/Tr

�dAdt A�1�, one can also observe that

d

dt

�lndet B p.t/

��D Tr

�dB p.t/

dtB p.t/

�1

�D 2I W D � 2I W D p.t/ : (166)

Finally, the upper convected derivative of the traceless part of B p.t/ reads

O�B p.t/

�ıD

d�B p.t/

�ı

dt� L

�B p.t/

�ı��B p.t/

�ıL>

DdB p.t/

dt�1

3

dTr B p.t/dt

I � LB p.t/ � B p.t/L> C

2

3

�Tr B p.t/

�D

DO

B p.t/ �2

3

�B p.t/ W D

�IC

2

3

�C p.t/ W D p.t/

�IC

2

3

�Tr B p.t/

�D

D �2F p.t/D p.t/F p.t/> �

2

3

�B p.t/ W D

�IC

2

3

�C p.t/ W D p.t/

�IC

2

3

�Tr B p.t/

�D:

(167)

4.4.2 Application of the Thermodynamic Procedure in the Context ofNatural Configuration

The application of the thermodynamic procedure discussed in Sect. 4.1 in the settingof a material with an evolving natural configuration goes as follows:


STEP 1: The fact that the energy storage mechanisms are required to be related tothe elastic part of the deformation (see Fig. 4) suggests that the internal energye should be enriched by a term measuring the elastic part of the deformation.As it has been already noted, a suitable measure of the deformation is the leftCauchy–Green tensor B p.t/ ; hence the entropic/energetic equation of state isassumed to take the form

� D Q��e; �;B p.t/

�or e D Qe

��; �;B p.t/

�: (168)

For the sake of simplicity of the presentation, the energetic equation of state isfurther assumed to take the form

eDOe��; �;Tr B p.t/ ; det B p.t/

�Ddef e .�; �/C

2�

�Tr B p.t/ � 3 � ln det B p.t/

�;

(169)

where is a constant. The motivation for this particular choice comes from thetheory of constitutive relations for isotropic compressible elastic materials; seefor example Horgan and Saccomandi [41] and Horgan and Murphy [40] for alist of some frequently used constitutive relations in the theory of compressibleelastic materials.

STEP 2: Applying the material time derivative to (169), multiplying the result by �and using the balance of mass (67a) and the evolution equation for the internalenergy (67d) together with the identities (165) and (166), one gets

��d�

dtD T W D � div jq C p

Mth div v � B p.t/ W DC C p.t/ W D p.t/

C D W I � D p.t/ W I (170)

where

pMth Ddef �

2 @Oe

@�D �2

@e

@��

2


�

D pNSEth �

2


�(171)

denotes the “pressure”, and pNSEth Ddef �

2 @e@�

.STEP 3: Splitting the tensors on the right-hand side of (170) into the traceless part

and the rest yields

��d�

dtD�Tı �

�B p.t/

�ı

�W Dı

C�mC pM

th �

3Tr B p.t/ C

�div vC

�C p.t/

�ıW�D p.t/

�ı

C

�Tr B p.t/3

� 1

�Tr D p.t/ � div jq; (172)


where the notation m Ddef13

Tr T for the mean normal stress has been used.(See Sect. 4.2.1 for the rationale of the splitting.) The standard manipulationfinally leads to

�d�

dtC div

�jq

�

�

D1

�

��Tı �

�B p.t/

�ı

�W Dı C

�mC pM

th �

3Tr B p.t/ C

�div v

C�C p.t/

�ıW�D p.t/

�ıC

�Tr B p.t/3

� 1

�Tr D p.t/ � jq �

r�

�

(173)

which is the sought formula that allows one to identify the entropy production.The flux/affinity pairs are Tı �

�B p.t/

�ı

versus Dı and so on.STEP 4: The entropy production – the right-hand side of (173) – is positive

provided that the linear relations between the fluxes and affinities are

Tı � �B p.t/

�ıD 2�Dı; (174a)

mC pMth �

3Tr B p.t/ C D

2� C 3�

3div v; (174b)

�C p.t/

�ıD 2�1

�D p.t/

�ı; (174c)

�Tr B p.t/3

� 1

�D2�1 C 3�1

3Tr D p.t/ ; (174d)

jq D � r�; (174e)

where � > 0, 2�C3�3

> 0, �1 > 0, 2�1C3�13

> 0, and > 0 are constants. Thischoice of constitutive relations follows the template specified in (75).

STEP 5: The first two equations in (174) allow one to identify the constitutiverelation for the full Cauchy stress tensor T D mIC Tı ,

T D 2�Dı C �B p.t/

�ıC2� C 3�

3.div v/ I � pM

th IC

3

�Tr B p.t/

�I � I;

(175)which can be further rewritten as

T D �pMth IC 2�DC � .div v/ IC

�B p.t/ � I

�; (176)

or, in virtue of (171), as

T D �pNSEth IC

2


�IC 2�DC � .div v/ I

C �B p.t/ � I

�: (177)


The last step leading to the complete description of the material behavior is theelimination of D p.t/ from the constitutive relations. Note that the Cauchy stressis given in terms of D, that is, the velocity field v, and the left Cauchy–Greentensor B p.t/ . The evolution equation for B p.t/ is (163), that is,

O

B p.t/ D �2F p.t/D p.t/F> p.t/

: (178)

The equation expresses the time derivative of B p.t/ as a function of L, F p.t/ ,and D p.t/ . If the right-hand side of (163) can be rewritten as a function ofB p.t/ and L, then the time evolution of B p.t/ would be given in terms of theother quantities directly present in the governing equations, and the process ofspecification of the constitutive relation would be finished. This can be achievedas follows.Summing (174c) and (174d) multiplied by the identity tensor yields

�C p.t/ � I

�D 2�1D p.t/ C �1

�Tr D p.t/

�I (179)

and

Tr D p.t/ D3

2�1 C 3�1

�Tr C p.t/3

� 1

�; (180)

which leads to

�C p.t/ � I

�D 2�1D p.t/ C

3�1

2�1 C 3�1

�Tr C p.t/3

� 1

�I: (181)

This is an explicit relation between D p.t/ and C p.t/ . Recalling the definition ofC p.t/ and B p.t/ (see (156)) and noticing that Tr C p.t/ D Tr B p.t/ , it follows thatthe multiplication of (181) from the left by F p.t/ and from the right by F> p.t/yields

�B2 p.t/ � B p.t/

�D 2�1F p.t/D p.t/F

> p.t/C

3�1

2�1 C 3�1

�Tr C p.t/3

� 1

�B p.t/ :

(182)

Substituting the formula just derived for F p.t/D p.t/F> p.t/

into (178) then gives

�1

O

B p.t/ C �B2 p.t/ � B p.t/

�D

3�1

2�1 C 3�1

�Tr B p.t/3

� 1

�B p.t/ ; (183)

which is the sought evolution equation for B p.t/ that contains only B p.t/ and L.

The set of equations (67) supplemented with the constitutive equations for theheat flux (see (174e)), the Cauchy stress tensor (see (177)), and the evolution


equation for the left Cauchy–Green tensor B p.t/ (see (183)) forms a closed systemof equations for the density �, the velocity field v, the specific internal energy e (orthe temperature � ), and the left Cauchy–Green tensor B p.t/ .

The constitutive relation for the Cauchy stress tensor (see (177)) can be furthermanipulated as follows. Introducing the notation

� Ddef �pNSEth C

2


�; (184a)

and

S Ddef �B p.t/ � I

�; (184b)

the equations (177) and (183) take the form

T D �IC 2�DC � .div v/ IC S; (185)

and

�1O

SC S2 C S D 2�1DC�1

2�1 C 3�1.Tr S/ .SC I/ : (186)

Note that if �1 D 0, then (186) simplifies to

�1O

SC S2 C S D 2�1D; (187)

hence the derived model could be seen as a compressible variant of the classicalmodel for a viscoelastic incompressible fluid developed by Giesekus [31].

If the material is constrained in such a way that

Tr D D Tr D p.t/ D 0; (188)

which means that the material is assumed to be an incompressible material and thatthe response from the initial to the natural configuration is assumed to be isochoric,it is clear that neither Tr B p.t/ norm can be specified constitutively. (These quantitiescorrespond to the forces that make the material resistant to the volumetric changes;see Sect. 4.2 for a discussion.) However, formula (174a) still holds, and the onlydifference is that D D Dı; hence

T D mIC Tı D mIC 2�DC �B p.t/

�ı: (189)

Similarly, (174c) is also valid; hence

�C p.t/ �

1

3

�Tr C p.t/

�I

�D 2�1

�D p.t/

�ı; (190)


where, in virtue of (188), D p.t/ D�D p.t/

�ı. Multiplication of (190) by F> p.t/ from

the right and F p.t/ from the left and the evolution equation for the left Cauchy–Green tensor (163) then yield

�1

O

B p.t/ C

�B2 p.t/ �

1

3

�Tr B p.t/

�B p.t/

�D 0 (191)

as the evolution equation for B p.t/ .

4.4.3 Application of the Maximization of the Entropy Production:Constitutive Relations for Oldroyd-B and Maxwell ViscoelasticFluids

The last example is devoted to discussion of the derivation of incompressibleOldroyd-B and Maxwell models for viscoelastic fluids and their compressiblecounterparts. Unlike in the previous example, the full thermodynamic procedurebased on the assumption of the maximization of the entropy production is nowapplied.

The starting point is the formula for the entropy production derived in theprevious section (see (173)) that can be rewritten in the form

� D1

�

��Tı �

�B p.t/

�ı

�W Dı C

�mC pM

th �

3Tr B p.t/ C

�div v

C �C p.t/ � I

�W D p.t/ � jq �

r�

�

: (192)

Using this formula one can continue with STEP 4 of the full thermodynamicprocedure discussed in Sect. 4.1.

STEP 4: At this point the constitutive function �, � D �� , is chosen as

� DdefQ��Dı; div v;D p.t/ ;r�

�D 2�jDıj

2 C2� C 3�

3.div v/2

C 2�1D p.t/ W C p.t/D p.t/ C jr� j2

�; (193)

where the right Cauchy–Green tensor C p.t/ is understood as a state variableand �, 2�C3�

3, �1, and are positive constants. Using the definition of the right

Cauchy–Green tensor (see (156b)) and the properties of the trace, it is easy tosee that D p.t/ W C p.t/D p.t/ D

ˇF p.t/D p.t/

ˇ2� 0; hence the chosen constitutive

function is nonnegative as required by the second law of thermodynamics.Further it is a strictly convex smooth function.Introducing the auxiliary function ˆ for constrained maximization problem

maxDı ;div v;D p.t/ ;r�

Q��Dı; div v;D p.t/ ;r�

�(194)


subject to the constraint (192) as

ˆ DdefQ��Dı; div v;D p.t/ ;r�

�C `

hQ��Dı; div v;D p.t/ ;r�

�

��Tı �

�B p.t/

�ı

�W Dı �

�mC pM

th �

3Tr B p.t/ C

�div v

��C p.t/ � I

�W D p.t/ C jq �

r�

�

; (195)

where ` is the Lagrange multiplier, the conditions for the extrema are

1C `

`

@ Q�

@DıD Tı �

�B p.t/

�ı; (196a)

1C `

`

@ Q�

@div vD mC pM

th �

3Tr B p.t/ C ; (196b)

1C `

`

@ Q�

@D p.t/D

�C p.t/ � I

�; (196c)

1C `

`

@ Q�

@r�D �

jq

�: (196d)

On the other hand, direct differentiation of (193) yields

@ Q�

@DıD 4�Dı; (197a)

@ Q�

@div vD2

3.2� C 3�/ div v; (197b)

@ Q�

@D p.t/D 2�1

�C p.t/D p.t/ C D p.t/C p.t/

�; (197c)

@ Q�

@r�D 2

r�

�: (197d)

Now it is necessary to find a formula for the Lagrange multiplier `. This can bedone as follows. First, each equation in (196) is multiplied by the correspondingaffinity, that is, by Dı , div v, D p.t/ , and r� , respectively, and then the sum ofall equations is taken. This manipulation yields

1C `

`

"@ Q�

@DıW Dı C

@ Q�

@div vdiv vC

@ Q�

@D p.t/W D p.t/ C

@ Q�

@r�� r�

#


D�Tı �

�B p.t/

�ı

�W Dı C

�mC pM

th �

3Tr B p.t/ C

�div v

C �C p.t/ � I

�W D p.t/ � jq �

r�

�: (198)

The right-hand side is identical to Q� D �� (see (192)), while the term on the left-

hand side reduces, in virtue of (193) and (197), to 2 Q�. Consequently 1C``DQ�

2Q�,

which fixes the value of the Lagrange multiplier

1C `

`D1

2: (199)

Inserting (197) and (199) into (196), one finally concludes that

Tı � �B p.t/

�ıD 2�Dı; (200a)

mC pMth �

3Tr B p.t/ C D

2� C 3�

3div v; (200b)

�C p.t/ � I

�D �1

�C p.t/D p.t/ C D p.t/C p.t/

�; (200c)

jq D � r�: (200d)

Further, if (200c) holds, then it can be shown that the symmetric positive definitematrix C p.t/ and the symmetric matrix D p.t/ commute. (In order to prove thatC p.t/ and D p.t/ commute, it suffices to show that the eigenvectors of C p.t/ andD p.t/ coincide, which is in virtue of (200c) an easy task. See the Appendix inRajagopal and Srinivasa [77] for details.) If the matrices commute, then (200c)in fact reads

�C p.t/ � I

�D 2�1C p.t/D p.t/ : (201)

STEP 5: The first two equations in (200) allow one to identify the constitutiverelation for the full Cauchy stress tensor T D mIC Tı ,

T D �pMth IC 2�DC

�B p.t/ � I

�C � .div v/ I: (202)

Further, (201) can be rewritten as

�C p.t/ � I

�D 2�1F

> p.t/

F p.t/D p.t/ : (203)

which upon multiplication by F> p.t/ from the right and by F�> p.t/ from the left

yields a formula for F p.t/D p.t/F> p.t/

. This formula can be substituted into (163)that yields the evolution equation for the left Cauchy–Green tensor,

�1

O

B p.t/ C �B p.t/ � I

�D 0: (204)


The set of equations (67) supplemented with the constitutive equations for theheat flux (see (200d)), the Cauchy stress tensor (see (202)), and the evolutionequation for the left Cauchy–Green tensor B p.t/ (see (204)) forms a closed systemof equations for the density �, the velocity field v, the specific internal energy e (orthe temperature � ), and the left Cauchy–Green tensor B p.t/ .

Introducing the extra stress tensor S by S Ddef �B p.t/ � I

�, equations (202)

and (204) can be rewritten as

T D �pMth IC 2�DC SC � .div v/ I; (205a)

�1

O

SC S D 2�1D: (205b)

Further manipulation based on the redefinition of the extra stress tensor QS Ddef

SC 2�D allows one to rewrite system (205) as

T D �pMth IC QSC � .div v/ I; (206a)

�1

O

QSC QS D 2 .�1 C �/DC2�1�

O

D: (206b)

The expressions (206) suggest that the derived model can be seen as a compressiblevariant of the classical Oldroyd-B model for viscoelastic incompressible fluidsdeveloped by Oldroyd [65].

On the other hand, if the constitutive relation for the traceless part of the Cauchystress tensor reads

Tı � �B p.t/

�ıD 0; (207)

which corresponds to � D 0 in (200a), and the constitutive relation for the meannormal stress is

mC pMth �

3Tr B p.t/ C D 0; (208)

which corresponds to the choice 2�C3�3D 0 in (200b), then the formula for the full

Cauchy stress tensor reads

T D �pMth IC

�B p.t/ � I

�: (209)

Introducing the extra stress tensor S Ddef �B p.t/ � I

�, it follows that the

system (205) reduces to

T D �pMth IC S; (210a)


�1

O

SC S D 2�1D: (210b)

This model can be denoted as a compressible variant of the classical Maxwell modelfor viscoelastic incompressible fluids; see, for example, Tanner and Walters [90].

Finally, if the fluid is assumed to be incompressible, then it is subject to constraint

Tr D D 0: (211)

However, then the same procedure as above can be still applied. One startsfrom (193) to (194) with the only modification that the terms with div v disappear,and one ends up with the same system of equations as those given in (200) (withthe only exception that the second equation in (200b) is missing). The derivedconstitutive relations read

Tı D 2�Dı C �B p.t/

�ı; (212a)

�C p.t/ � I

�D 2�1C p.t/D p.t/ ; (212b)

jq D � r�; (212c)

which translates to

T D mIC 2�Dı C �B p.t/

�ı

(213)

where the mean normal stress m is a quantity that cannot be specified via a consti-tutive relation. (See Sect. 4.2 for discussion.) Concerning the alternative approachbased on the enforcement of the incompressibility constraint by an addition of anextra Lagrange multiplier to the maximization procedure, the interested reader isreferred to Málek et al. [58].

Introducing the notation

� Ddef m �

3Tr B p.t/ C ; (214)

it follows that (213) can be rewritten as

T D �IC 2�DC �B p.t/ � I

�: (215a)

(Recall that in virtue of (211), one has D D Dı .) Further, the evolution equation forthe left Cauchy–Green tensor B p.t/ reads

�1

O

B p.t/ C �B p.t/ � I

�D 0: (215b)


Defining the extra stress tensor as S Ddef �B p.t/ � I

�or as QS Ddef 2�DC S, it

follows that (215) can be converted into the equivalent form

T D �IC 2�DC S; (216a)

�1

O

SC S D 2�1D; (216b)

or into the following equivalent form

T D �IC QS; (217a)

�1

O

QSC QS D 2 .�1 C �/DC2�1�

O

D; (217b)

which are the frequently used forms of constitutive relations for the Oldroyd-B fluid,which is a popular model for viscoelastic fluids derived by Oldroyd [65]. (Oldroyd[65] has used formulae (217).) Further, if one formally sets � D 0, then one gets thestandard incompressible Maxwell fluid.

Note however that the classical derivation by Oldroyd [65] is based only onmechanical considerations and that the compatibility of the model with the secondlaw of thermodynamics is not discussed at all. Second, the present approachnaturally gives one an evolution equation for the internal energy that automaticallytakes into account the storage mechanisms related to the “elastic” part of thedeformation. If the internal energy is expressed as a function of the temperature andother variables, then the evolution equation for the internal energy leads, upon theapplication of the chain rule, directly to the evolution equation for the temperature.(See Sect. 4.2.1 for the same in the context of a compressible Navier–Stokes–Fourierfluid.) Moreover, the inclusion of the storage mechanisms in the internal energy iscompatible with the specification of the constitutive relation for the Cauchy stresstensor. Again, such issues have not been discussed in the seminal contributionby Oldroyd [65] or for that matter by many following works on viscoelasticity.

Obviously, more complex viscoelastic models can be designed by appealing tothe analogy with more involved spring–dashpot systems; see, for example, Karraand Rajagopal [49, 50], Hron et al. [42], Pruša and Rajagopal [72], or Málek et al.[59].

4.5 Beyond Linear Constitutive Theory

In the previous parts, the development of the constitutive equations has been basedeither on linear relationships between the affinities and fluxes or on postulatingquadratic constitutive equations for the entropy production. In general, this limita-tion to linear constitutive relations or to a quadratic ansatz for the entropy productionis not necessary. In fact a plethora of nonlinear models can be developed in astraightforward manner following the method based on the maximization of entropy


production. Their relevance with respect to real material behavior should be howevercarefully justified.

A list of some popular simple nonlinear models for incompressible fluids isgiven below. Clearly, the list is not complete, and other models can be found inthe literature as well. In particular, the list does not include nonlinear viscoelasticmodels at all.

The first class of nonlinear models is the class of models where the Cauchy stresstensor is decomposed as

T D mIC S; (218)

where m is the mean normal stress, and the relation between the traceless extrastress tensor S and the symmetric part of the velocity gradient D takes the form ofan algebraic relation

f.S;D/ D 0; (219)

where f is a tensorial function. This implicit relation is a generalization of thestandard constitutive relation for the incompressible Navier–Stokes fluid (seeSect. 4.2.3) where

S � 2�?D D 0: (220)

Note that the standard way of writing algebraic constitutive relations for a non-Newtonian fluid is S D f.D/. However, it has been argued by Rajagopal [73,74] thatthe standard setting is too restrictive and that (219) should be preferred to S D f.D/.(See, e.g., Málek et al. [57], Pruša and Rajagopal [71], Le Roux and Rajagopal [53],Perlácová and Pruša [69], and Janecka and Pruša [45] for further developments ofthe idea. Mathematical issues concerning some of the models that belong to theclass (219) have been discussed, e.g., by Bulícek et al. [12] and Bulícek et al. [13].)

The implicit relation (219) opens the possibility of describing – in terms of thesame quantities that appear in (220) – various non-Newtonian phenomena suchas stress thickening or stress thinning, shear thickening or shear thinning, yieldstress, and even normal stress differences. None of these important phenomena canbe captured by the standard Navier–Stokes model (220). (The reader is referredto Málek and Rajagopal [54] or any textbook on non-Newtonian fluid mechanicsfor the discussion of these phenomena and their importance.) The mechanics ofcomplex fluids is indeed an unfailing source of qualitative phenomena that gobeyond the reach of the Navier–Stokes model. For interesting recent observationsconcerning the behavior of complex fluids, see, for example, the references in thereviews by Olmsted [66] and Divoux et al. [23].

Particular models that fall into class (219) are the models in the form

S D 2�.D/D; (221)


where the generalized viscosity �.D/ is given by one of the formulae listed below.Model

�.D/ D �0jDjn�1; (222)

where �0 is a positive constant and n is a real constant, is called Ostwald–de Waelepower law model; see Ostwald [67] and de Waele [94]. Models

�.D/ D �1 C�0 � �1

.1C ˛jDj2/n2

; (223)

�.D/ D �1 C .�0 � �1/�1C ˛jDja

� n�1a ; (224)

are called Carreau model (see Carreau [16]) and Carreau–Yasuda model (see Yasuda[96]), respectively. Here �0 and �1 are positive real constants, and n and a arereal constants. Other models with nonconstant viscosity are the Eyring models (seeEyring [26] and Ree et al. [82]), where the generalized viscosity takes the form

�.D/ D �1 C .�0 � �1/arcsinh .˛jDj/

˛jDj; (225)

�.D/ D �0 C �1arcsinh .˛1jDj/

˛1jDjC �2

arcsinh .˛2jDj/

˛2jDj; (226)

and �0, �1, �2, �1, ˛1, and ˛2 are positive real constants. Finally, the model namedafter Cross [21] takes the viscosity in the form

�.D/ D �1 C�0 � �1

1C ˛jDjn(227)

and the model named after Sisko [88] reads

�.D/ D �1 C ˛jDjn�1; (228)

where �0, �1, and ˛ are positive real constants and n is a real constant.Another subset of general models of the type (219) are models where the

generalized viscosity depends on the traceless part of the Cauchy stress tensorTı D S, that is, models, where

S D �.S/D: (229)

Examples are the Ellis model (see, e.g., Matsuhisa and Bird [61])

�.S/ D�0

1C ˛jSjn�1; (230)


where �0 and ˛ are positive real constants and n is a real constant or the modelproposed by Glen [33],

�.S/ D ˛jSjn�1; (231)

where ˛ is a positive real constant and n is a real constant, the model by Seely [85]

�.S/ D �1 C .�0 � �1/ e�jSj�0 ; (232)

or the models used for the description of the flow of the ice,

�.S/ DA�

jSj2 C �20

� n�12

; (233)

See, for example, Pettit and Waddington [70] and Blatter [5]. Here �0, �1, �0, andA are positive real constants, and n is a real constant. Particular parameter valuesfor models (222), (223), (224), (225), (226), (227), (228) and (230), (231), (232),(233) can be found in the referred works and/or in the follow-up works.

Other popular models are the models with activation criterion introducedby Bingham [3] and Herschel and Bulkley [39]. These models are usually (see,e.g., Duvaut and Lions [25]) written down in the form of a dichotomy relation

jSj � ��, D D 0 and jSj > ��, S D��D

jDjC 2�.jDj/D; (234)

where � is a positive function and �� is a positive constant. (Constant �� is calledthe yield stress.) It is worth emphasizing that these models can be rewritten as

D D1

2�.jDj/

.jSj � ��/C

jSjS; (235)

where xC D maxfx; 0g, which shows that the yield stress models also fall intothe class (219). This observation can be exploited in the discussion of the physicaland mathematical properties of the models; see Rajagopal and Srinivasa [79] andBulícek et al. [13].

Note that if (234) holds, then S cannot be considered as a function of D, while(235) gives a functional (continuous) dependence of D on S. Since the function atthe right-hand side of (235) is tacitly supposed to be zero for S D 0, the alternativeviewpoint given by (219) with a continuous tensorial function f defined on theCartesian product of S and D allows one to avoid description of the material viaa multivalued or discontinuous function. See Bulícek et al. [13] and Bulícek et al.[10] for exploiting this possibility as well as the symmetric roles of S and D in(219) in the analysis of the corresponding initial and boundary value problems.


Another advantage of the formulation (235) is that it allows one to replace, in astraightforward way, the constant yield stress �� by a yield function that can dependon the invariants of both S and D, �� D ��.S;D/.

Finally, the class of implicit relations (219) can be viewed as a subclass of modelswhere the full Cauchy stress tensor T and the symmetric part of the velocity gradientD are related implicitly,

f.T;D/ D 0: (236)

This apparently subtle difference has significant consequences. Constitutive rela-tions in the form (236) provide, contrary to (219), a solid theoretical backgroundto incompressible fluid models where the viscosity depends on the pressure (meannormal stress); see Rajagopal [74] and Málek and Rajagopal [56] for an in-depthdiscussion. Such models have been proposed a long time ago by Barus [2], andthe practical relevance of such models has been demonstrated by Bridgman [6] andBridgman [7]. In the simplest settings, the model for a fluid with pressure dependentviscosity can take the form

T D �pIC 2�refeˇ.p�pref/D; (237)

where pref – the reference pressure – and �ref – the reference viscosity – arepositive constants. If necessary, this model can be combined with power law typemodels. This is a popular choice in lubrication theory; see, for example, Málek andRajagopal [56].

4.6 Boundary Conditions for Internal Flows of IncompressibleFluids

As it has been already noted, the specification of the boundary conditions isa nontrivial task. Since the boundary conditions can be seen as a special caseof constitutive relations at the interface between two materials, thermodynamicalconsiderations could be of use even in the discussion of the boundary conditions. Asimple example concerning the role of thermodynamics in the specification of theboundary conditions for incompressible viscous heat nonconducting fluids is givenbelow.

The problem of internal flow in a fixed vessel represented by the domain hasbeen discussed in Sect. 3.2, and it is revisited here in a slightly different setting. InSect. 3.2 the boundary condition on the vessel wall has been the no-slip boundarycondition

vj@ D 0: (238)


In the present case, one enforces only the no-penetration boundary condition

v � nj@ D 0; (239)

where n denotes the unit outward normal to , and the question concerning thevalue of the velocity vector in the tangential direction to the vessel wall is for themoment left open.

Introducing the notation

u� Ddef u � .u � n/n (240)

for the projection of any vector u to the tangent plane to the boundary, it followsthat the product .Tv/ � n can be rewritten as

.Tv/ � n D mv � nCS W .v˝ n/ D S W .n˝ v/ D .Sn/ � v D .Sn/� � v� : (241)

This observation is a consequence of the symmetry of the Cauchy stress tensor andthe decomposition of the Cauchy stress tensor to the mean normal stress and thetraceless part; see (218).

The multiplication of the balance of momentum (63a) by v followed byintegration over the domain yields

ˆ

d

dt

��1

2v � v

�dv D

ˆ

.div T/ � v dv: (242)

The right-hand side can be, following Sect. 3.2, manipulated as

ˆ

.div T/ � v dv Dˆ

.div Tv/ dv�ˆ

T W D dv D �ˆ

T W D dvCˆ@

Tv � n ds:

(243)Consequently, the evolution equation for the net kinetic energy Ekin reads

dEkin

dtD �

ˆ

T W D dvCˆ@

Tv � n ds; (244)

which can be in virtue of (241) and the incompressibility condition div v D 0 furtherrewritten as

dEkin

dtD �

ˆ

S W D dv �ˆ@

s � v� ds; (245)

where the notation

s Ddef � .Sn/� (246)

has been used.


Unlike in Sect. 3.2, one now gets a boundary term contributing to the evolutionequation for the net kinetic energy. The boundary term is the key to the thermo-dynamically based discussion of the appropriate boundary conditions. The desireddecay of the net kinetic energy is in the considered case guaranteed if one enforcespointwise positivity of the product T W D in and the pointwise positivity of theproduct s � v� at @. This restriction can be used to narrow down the class ofpossible relations between the value of the velocity in the tangential direction v�and the projection of the stress tensor .Sn/� , where the relation between v� and.Sn/� is the sought boundary condition.

Note that in the full thermodynamic setting, both the volumetric term´

T W D dvand the boundary term

´@s � v� ds would appear in the entropy production for

the whole system. Both the boundary term and the volumetric term have theflux/affinity structure. In linear nonequilibrium thermodynamics, the fluxes andaffinities in the volumetric term are connected linearly via a positive (nonnegative)coefficient of proportionality in order to guarantee the validity of the second law ofthermodynamics. Application of the same approach to the boundary term leads to alinear relation between s and v�

s D ��v� ; (247)

where �� is a positive constant. This is the Navier slip boundary condition;see Navier [63]. Moreover, one can identify two cases where the boundary termvanishes. If s D 0, then one gets the perfect slip boundary condition, and if v� D 0,then one gets the standard no-slip boundary condition.

Further, a general relation between s and v� can take, following (219), the formof an implicit constitutive relation

f .s; v�/ D 0; (248)

which considerably expands the number of possible boundary conditions. In partic-ular, the threshold-slip (or stick-slip) boundary condition that is usually describedby the dichotomy relation

jsj � ��, v� D 0 and jsj > ��, s D��v�

jv� jC ��v� ; (249)

where �� and �� are positive constants, can be seen as a special case of (248).Indeed, (249) can be rewritten as

v� D1

��

.jsj � ��/C

jsjs: (250)

(Note the similarity between the threshold-slip boundary condition (249) and theBingham/Herschel–Bulkley model for fluids with the yield stress behavior (235).The formulation of the threshold-slip boundary condition in the form (250) can


be again exploited in the mathematical analysis of the corresponding governingequations; see Bulícek and Málek [11].)

An example of more involved thermodynamical treatment of boundary condi-tions can be found in the study by Heida [37]. Concerning a recent review ofthe nonstandard boundary conditions used by practitioners in polymer science, thereader is referred, for example, to Hatzikiriakos [36].

5 Conclusion

Although the classical (in)compressible Navier–Stokes–Fourier fluid models havebeen successfully used in the mathematical modelling of the behavior of varioussubstances, they are worthless from the perspective of modern applications suchas polymer processing. Navier–Stokes–Fourier models are simply incapable ofcapturing many phenomena observed in complex fluids; see, for example, thelist of non-Newtonian phenomena in Málek and Rajagopal [54], the historicalessay by Tanner and Walters [90], or the classical experimentally oriented treatisesby Coleman et al. [20], Barnes et al. [1], or Malkin and Isayev [60] to name a few.

The need to develop mathematical models for the behavior of complex materialsleads to the birth of the theory of constitutive relations. In the early days of thetheory, constitutive relations have been designed by appealing to purely mechanicalprinciples. This turns out to be insufficient as the complexity of the modelsincreases.

Nowadays, the mathematical models aim at the description of an interplaybetween various mechanisms such as heat conduction, mechanical stress, chemicalreactions, electromagnetic field, or the interaction of several continuous mediain mixtures; see, for example, Humphrey and Rajagopal [44], Rajagopal [75],Dorfmann and Ogden [24], and Pekar and Samohýl [68]. In such cases the correctspecification of the energy transfers is clearly crucial. Consequently, one canhardly hope that an ad hoc specification of the complex nonlinear constitutiverelations for the quantities that facilitate the energy transfers is the way to go. Atheory of constitutive relations that focuses on energy transfers and that guaranteesthe compatibility of the arising constitutive relations with the second law ofthermodynamics is needed.

The modern theory of constitutive relations outlined above can handle thechallenge. The theory from the very beginning heavily relies on the conceptsfrom nonequilibrium thermodynamics, and the restrictions arising from the lawsof thermodynamics are automatically built into the derived constitutive relations.Naturally, the models designed to describe the behavior of complex materials in far-from-equilibrium processes are rather complicated. In particular, the arising systemsof partial differential equations are large and nonlinear.

Fortunately, the available numerical methods for solving nonlinear partial differ-ential equations as well as the available computational power are now at the levelthat makes the numerical solution of such systems feasible. This is a substantialdifference from the early days of the mechanics and thermodynamics of continuous


media; see, for example, Truesdell and Noll [91]. Since the quantitative predictionsbased on complex models are nowadays within the reach of the scientific andengineering community, they can actually serve as a basis for answering importantquestions in the applied sciences and technology. This is a favorable situation for amathematical modeller equipped with a convenient theory of constitutive relations.The design of suitable mathematical models for complex materials undergoing far-from-equilibrium processes does matter – from the practical point of view – morethan ever.

Cross-References

�Equations and Various Concepts of Solutions in the Thermodynamics of Com-pressible Fluids

�Equations for Viscoelastic Fluids�Multifluid Models Including Compressible Fluids� Solutions for Models of Chemically Reacting Mixtures

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