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The Thermodynamics of Linear Fluids and Fluid Mixtures pp 67–141Cite as

Continuum Thermodynamics of Single Fluid

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Abstract

In this chapter, we advance our exposition of rational thermodynamics further. The uniformity is abandoned and the description of space effects enters the scene. To keep the explanation simple, we deal with the rational thermodynamics of a single (pure) substance only (i.e., a substance composed of only one constituent as opposed to many constituent substances—mixtures—discussed in the following Chap. 4) and confine our discussion to fluids. We thus study properties changing not only in time but also in the space, but in such a way that the discrete structure of matter may be ignored. That is, we use the methods of continuum (thermo)mechanics by reducing properly the space scale (in comparison with uniform bodies of Chap. 2). On the other hand, the timescale will be similar to that in Chap. 2, i.e., we confine ourselves only to materials with differential memory. Finally, we discuss the linearized case, which is the most important model in applications, in the subsequent chapters of this book.

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Notes

  1. 1.

    For simplicity we do not discuss bodies in which there exist surfaces of discontinuity even though such models are very important in chemical engineering, e.g., they model phase boundaries, surface chemical reactions, or shock waves. But even in such simplified models (ignoring specific surface phenomena like surface tension) [1, 2], we obtain the important results of phase equilibria (like equality of chemical potentials in bulk phases, cf. Sect. 2.5, which may be generalized to Eshelby tensors of chemical potentials, cf. Rem. 38, [2, 3]). Further generalization with surface phenomena uses configurational forces (in fluids these are chemical potentials related to unit volume) see [4, 5].

  2. 2.

    From a molecular point of view such a “macroscopic” particle \(\mathbf{X}\) contains a great number of molecules.

    We fix this reference once and for all, but in the general theory the change in this reference may be used to describe the symmetry inherent to the material of the body; in the special case it may be used for the definition of fluid (cf. Sect. 3.5 and Rem. 30).

    In some continuum theories of more complicated models (e.g., micromorphic or microcontinuous) \(\mathbf{X}\) may have some inner structure (cf. Rem. 26).

  3. 3.

    Defined in the following Rem. 4 generalized here in a (generally) different Cartesian system (here, a space and referential one). Then, e.g., the matrix of \(\mathbf F = \mathbf 1\) need not be the unit one (Kronecker delta), but a so-called shifter; cf. e.g. [9, 15]. Here, for simplicity, both Cartesian systems are mostly chosen as the same.

  4. 4.

    Tensors of 2nd order \(\mathbf{A}\) are linear transformations (matrix \(3\times 3\)) of vector \(\mathbf a\) to vector \(\mathbf b\)

    $$\begin{aligned} \mathbf{b}&= {\mathbf{A}} \mathbf{a}\\ b^i&= A^{ij} a^j \qquad \qquad (a) \end{aligned}$$

    where the second expression is in Cartesian components (with summation rule of course).

    But vectors and tensors are more than matrices \(3\times 1\) and \(3\times 3\): changing (Cartesian) coordinates by orthogonal matrix \(Q^{kl}\) (cf. Rem. 8) the components \(b^i\) of vector \(\mathbf b\) transforms into new (starred) components \(\mathop {b^{j}}\limits ^{*}\) of the same vector \(\mathbf b\) by

    $$\begin{aligned} \mathop {b^{j}}\limits ^{*} = Q^{ji} b^i \qquad \qquad (b) \end{aligned}$$

    That is, \(\mathbf b\) is the same “arrow” looked at from these different (starred and original) coordinates. Transformation (\(b\)) is valid for the usual polar vectors (less usual axial vectors, e.g., those obtainable by vector product [16], are discussed in Rem. 10). Similarly, the components \(A^{ij}\) of tensor \(\mathbf{A}\) transform into starred components \(\mathop {A^{kl}}\limits ^{*}\) of the same \(\mathbf{A}\) by

    $$\begin{aligned} \mathop {A^{kl}}\limits ^{*} = Q^{ki} A^{ij} Q^{lj} \qquad \qquad (c) \end{aligned}$$

    (namely, transformation (\(c\)) guarantees linear transformation (\(a\)) with both vectors transformed by (\(b\))).

    Relations (b), (c) inspire in Sect. 3.2 the more general notion of changes of frame and frame indifference, cf. (3.31), (3.32).

    Generalizations of tensors for nonCartesian coordinates see, e.g.,  [7, 16, 17] and Appendix A.4. Similar to matrices \((3\times 3)\), tensors may be symmetric, skew-symmetric, etc., about vector and outer products. See Rems. 6, 16.

    If \({\mathbf{A}} = {\mathbf{A}}(t)\) is a tensor function of the scalar \(t\) then det\(\mathbf{A}\) is the scalar function of \(t\). Its derivative is

    $$\begin{aligned} \frac{\text {d det}{\mathbf{A}}}{{\text {d}}t} = \frac{\partial {\text {det}}{\mathbf{A}}}{\partial A^{ij}}\frac{{\text {d}}A^{ij}}{{\text {d}}t} = ({\text {det}}{\mathbf{A}}) \frac{{\text {d}}A^{ij}}{{\text {d}}t} \mathop {A^{ji}}\limits ^{-1} \qquad \qquad (d) \end{aligned}$$

    where we use the following derivative of det\(\mathbf{A}\) with respect to its components (using its development according to line)

    $$\begin{aligned} \frac{\partial {\text {det}}{\mathbf{A}}}{\partial A^{ij}} = ({\text {det}}{\mathbf{A}}) \mathop {A^{ji}}\limits ^{-1} \qquad \qquad (e) \end{aligned}$$
  5. 5.

    Rigid motion (3.1) has the general form

    $$\begin{aligned} {\mathbf{x}} = \underline{\Theta }\,\mathbf{X} + \underline{\gamma } \end{aligned}$$

    where \(\underline{\Theta }(t)\) and \(\underline{\gamma } (t)\) are some orthogonal (Rem. 8) and vector functions of time \(t\) respectively. This follows from the preservation of distances of any two particles in reference \(\mathbf{X},\mathbf{X}_0\) and in actual configurations (positions \(\mathbf{x},{\mathbf{x}}_0\)) in rigid motion, i.e.,

    $$\begin{aligned} {\mathbf{x}}-{\mathbf{x}}_0 = \underline{\Theta }(\mathbf{X}-\mathbf{X}_0) \end{aligned}$$

    where orthogonal \(\underline{\Theta }\) and \({\mathbf{x}}_0\) are arbitrary time function, cf. analogous deduction of (3.25).

  6. 6.

    (In fact a cubic matrix) defined by

    $$\begin{aligned} \varepsilon ^{123} = \varepsilon ^{231} = \varepsilon ^{312} = 1 \end{aligned}$$
    $$\begin{aligned} \varepsilon ^{132} = \varepsilon ^{213} = \varepsilon ^{321} = - 1 \end{aligned}$$

    (the remaining elements of this \(3\times 3 \times 3\) matrix are zero)

    From this definition, it follows (by direct calculation) the following properties of the permutation symbol (and its relation to Kronecker delta \(\delta ^{ij}\))

    $$\begin{aligned} \varepsilon ^{ijk} =\varepsilon ^{jki} =\varepsilon ^{kij} = -\varepsilon ^{kji} =-\varepsilon ^{jik} =-\varepsilon ^{ikj} \end{aligned}$$

    (such “cyclic” permutation does not change its value)

    $$\begin{aligned} \varepsilon ^{ijk} \varepsilon ^{ilm} = \delta ^{jl}\delta ^{km} - \delta ^{jm}\delta ^{kl} \;, \quad \varepsilon ^{ijk} \varepsilon ^{jkn} = 2 \delta ^{in} \;,\quad \varepsilon ^{ijk} \varepsilon ^{ijk} = 6 \end{aligned}$$

    With this symbol we can express the vector and triple products as (\(\mathbf{a},\mathbf{b},\mathbf{c}\) are vectors)

    $$\begin{aligned} (\mathbf{a}\times \mathbf{b})^i = \varepsilon ^{ijk} a^j b^k \;, \quad \mathbf{c}.(\mathbf{a}\times \mathbf{b}) = \varepsilon ^{ijk} c^i a^j b^k \end{aligned}$$

    and for the determinant of matrix \(\mathbf{A}\) it is valid that

    $$\begin{aligned} \varepsilon ^{ijk}A^{im}A^{jn}A^{kp} = \varepsilon ^{mnp} {\text {det}}{\mathbf{A}} \end{aligned}$$

    See also Rems. 4, 10, 16.

  7. 7.

    The density \(\psi \) may be deduced assuming \(\Psi \) as primitive and continuous with volume, i.e., when \(\mathcal{V}\rightarrow 0\) also \(\Psi \rightarrow 0\) [7, 10, 18–20].

    The additivity of \(\Psi \) in volume (\(\Psi \) for volume consisting of two separate volumes is the sum of \(\Psi \) of each separate volume) follows from (3.21). Such quantities \(\Psi \) are usual in continuum thermomechanics, cf. mass, energy, entropy, etc.; using mass and mass density we can introduce specific quantities instead of densities (cf. (3.66) and Sects. 3.4, 4.6) and extensivity instead of additivity (cf. Sects. 1.2, 2.4). Similarly, [19, 20] there are quantities continuous in surface with surface densities (cf. (3.58), (3.99) and Rems. 14, 18).

  8. 8.

    Orthogonal tensor \(\mathbf{Q}\) transforms any vector \(\mathbf{a}\) into vector \(\mathbf{Q}\mathbf{a}\) of the same length \(\mathbf{a}.\mathbf{a} = \mathbf{Q}\mathbf{a}.\mathbf{Q}\mathbf{a} \). Then the basic properties of orthogonal tensor \(\mathbf{Q}\) follow:

    $$\begin{aligned} \mathbf{Q}^T\mathbf{Q} = \mathbf{Q}\mathbf{Q}^T = \mathbf{1} \;, \quad {\text {in components}} \quad Q^{ki}Q^{kj} = Q^{ik}Q^{jk} = \delta ^{ij} \end{aligned}$$
    $$\begin{aligned} \mathbf{Q}^{-1} = \mathbf{Q}^{T} \; , \quad ({\text {det}}\mathbf{Q})^2 = 1 \end{aligned}$$

    An example of the orthogonal tensor is (3.29).

    Orthogonal transformations \(\mathbf{Q}\) form a group: generally (cf. [9, 15, 16]) a set of elements with a defined “product” giving another element from this set (here a matrix product of two orthogonal tensors giving again an orthogonal tensor) with inverse and unit elements (here \(\mathbf{Q}^T\) and \(\mathbf{1}\) respectively). This group is called a full orthogonal group with \(\det \,\mathbf{Q}= \pm 1\) which expresses rotation or/and reflection. A proper orthogonal group forms its subgroup with \(\det \,\mathbf{Q}=+1\) (a subgroup is a subset with group properties again).

    The corresponding orthogonal matrix \(Q^{ij}\) may be also used for rotation (and/or inversion) of Cartesian coordinates, cf. (b), (c) in Rem. 4 and Rem. 10.

  9. 9.

    Use of a full or proper orthogonal group puts the additional property of preservation of right- or left handedness on the change of frame; some authors [12, 23–26] (motivated usually by nonmechanical arguments) confine (3.25) only to the rotations. This problem seems not to have been settled. Because it has no influence on the linear models preferred here, we use in the following the full orthogonal group, see Appendix A.2.

  10. 10.

    As distinct from usual polar vectors which by coordinate changes (characterized by orthogonal matrix \(Q^{ji}\), see Rem. 8) transform by (\(b\)) of Rem. 4, the axial vector \(\mathbf{w}\) is defined by transformation

    $$\begin{aligned} {{\mathop {w}\limits ^{*}}^j} = ({\text{ det }}\mathbf {Q}) Q^{ji} w^i \qquad \qquad (a) \end{aligned}$$

    and therefore changes the sign at parity (right-handedness or left-handedness) changes (det\(\mathbf{Q} = -1\), cf. Rem. 8)

    Lemma (equivalency of skew-symmetric tensors with axial vectors): For every skew-symmetric tensor (of second order) \(\mathbf{W}\) it is possible to define an axial vector \(\mathbf{w}\) (both contain three (independent) components) and vice versa by

    $$\begin{aligned} w^i = (1/2)\varepsilon ^{ijk} W^{kj} \qquad (b) \;, \quad W^{jk} = w^i \varepsilon ^{ikj} \qquad (c) \end{aligned}$$

    Indeed, the usual coordinate transformation of tensor \(\mathbf{W}\) (i.e., of the type (\(c\)) in Rem. 4) leads to axiality transformation (\(a\)). Namely, (\(b\)) must be valid also for the new (starred) coordinate system

    $$\begin{aligned} {{\mathop {w}\limits ^{*}}^i} = (1/2)\varepsilon ^{ijk} {{\mathop {W}\limits ^{*}}^{kj}} = (1/2)\varepsilon ^{ijk} Q^{kl}W^{lm}Q^{jm} = (1/2)\varepsilon ^{ijk} Q^{kl}w^p \varepsilon ^{pml}Q^{jm} \end{aligned}$$

    which multiplying by \(Q^{ir}\) gives

    $$\begin{aligned} {{\mathop {w}\limits ^{*}}^i} Q^{ir} = (1/2)w^p\varepsilon ^{pml}\varepsilon ^{ijk} Q^{ir}Q^{jm}Q^{kl} = (1/2)w^p\varepsilon ^{pml}\varepsilon ^{rml} ({\text{ det }}\mathbf {Q}) = ({\text{ det }}\mathbf {Q}) w^r \end{aligned}$$

    where properties of permutation symbol from Rem. 6 were used. Multiplying it by orthogonal \(Q^{jr}\) we obtain (\(a\)) and therefore \(\mathbf{w}\) is an axial vector.

    Axiality of \(\mathbf{w}\) is automatically achieved by the usual transformation ((\(c\)) in Rem. 4) of tensor \(\mathbf{W}\). Therefore the skew-symmetric tensors instead of axial vectors and outer product (see Rem. 16) may be used and we do it this way at the moment of momentum balances in the Sects. 3.3, 4.3, cf. [7, 8, 14, 27]. Generalization of this Lemma to third-order tensors, made by M. Šilhavý, is published in Appendix of [28].

  11. 11.

    Note that functions on both sides of (3.55)\(_1\) are different: \( a^*({\mathbf{x}}^*,t^*) = a^*({\underline{\chi }}^*(\mathbf{X},t^*),t^*) \equiv a^*(\mathbf{X},t^*) \). Remark that the assumption (3.30) is crucial for validity of (3.55); namely, the function \(\alpha ({\mathbf{x}},t)\), defined by (3.25), (3.26) as \( a^*({\mathbf{x}}^*,t^*) = a^*(\mathbf{c} + \mathbf{Q}{\mathbf{x}},t+b) \equiv \alpha ({\mathbf{x}},t) \) is generally different from function \(a({\mathbf{x}},t)\). Similarly (3.31) and (3.32) are crucial for (3.56) and (3.57).

  12. 12.

    Moreover, it should be expected that

    $$\begin{aligned} \left( \int _{\mathcal{{V}}} \psi \,{\text {d}} v \right) ^* = \int _{\mathcal{{V}}} {\psi }^*\,{\text {d}} v \;, \quad \left( \int _{\partial \mathcal{{V}}} \psi \mathbf{n} \,{\text {d}} a \right) ^* = \int _{\partial \mathcal{{V}}} {\psi }^* \mathbf{Q} \mathbf{n} \,{\text {d}} a \end{aligned}$$

    because the objectivity of d\(a\), d\(v\) follows from the objectivity of space intervals (\(\psi \) may even be a component of a vector or a tensor).

    Note also that the following relationships are valid in the starred frame for the time derivative of function \(\varphi (t)\) (see (3.26), (3.28))

    $$\begin{aligned} \mathop {\overline{\dot{\varphi }}}\limits ^{*} \equiv \frac{{\text {d}} \varphi ^* (t^*)}{{\text {d}}t^*} = \frac{{\text {d}} \varphi ^* (t^*)}{{\text {d}}t} \frac{{\text {d}} (t^* - b)}{{\text {d}}t^*} = \frac{{\text {d}} \varphi ^* (t^*)}{{\text {d}}t} = \dot{\overline{\varphi ^*}} \end{aligned}$$

    Such a function \(\varphi \) may be, e.g., \(\psi (\mathbf{X},t)\) or \(\Psi (t)\) in (3.21); for the latter the relation (3.22) and the previous formula (with (3.50)) gives

    $$\begin{aligned} \left( \dot{\overline{\int _{\mathcal{{V}}} \psi \,{\text {d}} v}} \right) ^* = \int _{{\mathcal{{V}}}_0} (\dot{\overline{\psi J}})^*\,{\text {d}}V = \int _{{\mathcal{{V}}}_0} \dot{\overline{\mathop {\psi }\limits ^{*}\mathop {J}\limits ^{*}}}\,{\text {d}}V = \dot{\overline{\int _{\mathcal{{V}}} \mathop {\psi }\limits ^{*} \,{\text {d}} v}} \end{aligned}$$

    because \({{\mathcal{{V}}}_0}\) is a material volume in reference configuration.

  13. 13.

    On the real surface of the whole body the surface forces \(\mathbf t\) (originated from the outside of the whole body) are given by boundary conditions; cf. Rems. 18 and 24 in this chapter, 9 in Chap. 1.

  14. 14.

    Again [7, 10, 18–20] as we noted in Rem. 7, it would be more natural to postulate forces for any part of volume or surface (which bound them) and then to deduce \(\rho \mathbf{b}\) or \(\mathbf t\) as the volume or surface densities.

    In fact, the formulation of balances in Sects. 3.3 and 3.4 for each part of the body is motivated by the solidification principle: we imagine the part of the body isolated from the remainder of the body and interactions with this remainder and surroundings of the body are expressed by appropriate (volume or surface) densities. This principle will be used also in the following, e.g., contact and body forces in formulation of (3.70) are such interactions.

  15. 15.

    E.g., hydrostatic pressure (typical traction in steady fluid) is directed always perpendicularly to any orientation of the surface in a given place. Moreover, assumption (3.71) may be also proved [7, 21, 30, 31]; from this proof it follows that \(\mathbf t\) cannot depend on the other local properties of surface, like curvature, etc.

  16. 16.

    We use the outer product \(\wedge \) defined for two vectors \(\mathbf{a}\), \(\mathbf{b}\) as \(\mathbf{a}\wedge \mathbf{b} \equiv \mathbf{a}\otimes \mathbf{b} - \mathbf{b}\otimes \mathbf{a}\), i.e. \((\mathbf{a}\wedge \mathbf{b})^{ij} = a^i b^j - a^j b^i\). This product is obviously the skew-symmetric tensor which, using the results from Rem. 10, is equivalent to the axial vector created by the vector product of these vectors, see Rem. 6

    $$\begin{aligned} \mathbf{b}\times \mathbf{a} = - \mathbf{a}\times \mathbf{b} \end{aligned}$$

    Then, e.g., the balance of angular moment (3.90) may be written in a more traditional way as

    $$\begin{aligned} \int _{\mathcal{{V}}} ({\mathbf{x}}-\mathbf{y})\times \rho \dot{\mathbf{v}}\,{\text {d}} v = \int _{\partial {\mathcal{{V}}}} ({\mathbf{x}}-\mathbf{y})\times \mathbf{Tn}\, {\text {d}} a + \int _{\mathcal{{V}}}({\mathbf{x}}-\mathbf{y})\times \rho \mathbf{b} \,{\text {d}} v \end{aligned}$$
  17. 17.

    In more general (mechanically) polar materials [13, 34], the local result (3.93) must be changed (cf. also Rems. 32 in this chapter, 9 in Chap. 4). Namely, the balance (3.89) then contains (besides moments of forces) torques expressing the direct exchange of angular moment on a microscopic level (something like heat in energy exchange). These “microscopic” torques may be expressed by the objective field of density of skew-symmetric tensor \(\mathbf M\) adding to the right-hand side of the postulate (3.89) the integral \(\int _{\mathcal{{V}}} \mathbf{M}\, {\text {d}}v \). Then instead of local result (3.93), we obtain

    $$\begin{aligned} \mathbf{T} - \mathbf{T}^T = \mathbf{M}. \end{aligned}$$
  18. 18.

    Exchange of radiation between distant parts of the same body is neglected; \(q\) on the real surface of body is given as a boundary condition. Assuming the validity of such a balance for each part of the body, we use again the principle of solidification and again volume and surface densities (\(\rho u, Q,q \) etc.) could be deduced from more plausible primitives. Cf. Rems. 7, 13 and 14.

  19. 19.

    Surface heating is scalar. Vectorial heat flux in (3.100) will be deduced quite similarly as the stress tensor was obtained from the traction in (3.72). Dependence of \(q\) on \(\mathbf{n}\) may be expected, e.g., in a body under temperature gradient it may be expected in a given place that \(q\) on the surface perpendicular to such a gradient will be greater then on the surface parallel to it.

  20. 20.

    The sign is in accord with convention mentioned in Rem. 7 in Chap. 1: negative heat \(q\) is emitted when \(\mathbf q\) has direction of outer normal \(\mathbf n\). Also Fourier law (3.187) directs heat flux \(\mathbf q\) against temperature gradients, cf. [1, 14, 24, 27, 36, 37].

  21. 21.

    Heat is based on molecular motion; therefore the possible nonobjectivity of heat flux has been discussed [24, 38–40]. Because of the molecular chaos this effect is probably negligible with the exception of very rarefied gases. Cf. also Rem. 33 in Chap. 4.

  22. 22.

    Such is, e.g. the potential \(\Phi = (1/2){\mathbf{x}}^*.\underline{\Omega }^2 {\mathbf{x}}^*\) giving centrifugal force (3.47); \(\underline{\Omega }^2\) (as a product of the identical skew-symmetrical tensors) is symmetrical.

  23. 23.

    Using Reynolds theorem (3.24) in (3.108) we obtain quite naturally the entropy inequality for open systems in (single) continua. Cf. Rems. 14 in Chap. 2, 11 in Chap. 4 and the end of Sect. 3.1.

  24. 24.

    Such are also boundary values \(\mathbf{q}\), \(\mathbf{T}\) on the real surface of the whole body, cf. Rems. 13, 18, see also Rem. 36.

  25. 25.

    Repeating those noted in Sects. 1.1 and 2.1 the name “principles” here is stilted a little: they are rather rules or recommendations which generalize motivation or proposals of such equations in the past [40], cf. exceptions in Rems. 21, 26, and 28.

  26. 26.

    Great numbers of more general models have been studied e.g. with long range memory (as fading memory or with internal variables mentioned in Sects. 2.1, 2.3), where differential memory is not suitable. Its analog for a space coordinate is the nonlocal material [46–50] where the local action is not sufficient. Another type are materials with a microstructure (micromorphic materials) in which the particles have a more complicated structure [11, 45, 48, 51, 52] (cf. Rem. 2). For simplicity we excluded in (3.118) the temperature memory studied in [23, 26, 53] (the influence of which was outlined in Sect. 2.2; cf. Rem. 31 in Chap. 4). The principle of determinism is modified in materials with internal constraints [6, 7, 10, 12, 54–58] manifested usually as some a priori limitation on the motion (but there are also nonmechanical constraints such as perfect heat conductivity). Most important are incompressible materials where the internal constraint is \(J=1\) (by (3.64) density of particles does not change and therefore only isochoric motions are allowed). The limitation is achieved by forces (pressure in incompressible material) which are not determined by the motion and do not work. The remaining part of the stress is given by the usual principle of determinism. Modification of determinism is also given by using pressure as an independent variable (usual in classical thermodynamics); then incompressibility may be also understood as pressure independence here [24, 59], cf. end of Sect. 3.7.

  27. 27.

    Moreover a unique reference configuration was tacitly assumed in the whole body. But there are (nonfluid, usually solid) materials with dislocations which may be just described by nonunique references and dependence on \(\mathbf{X}\) remains even if they are from the “same” material, cf. [6, 8, 41], cf. also Rem. 30.

  28. 28.

    E.g. we tacitly assume such a principle in the assertion that the same force extends by the same amount the loaded spring when it is suspended in gravitational field or it is attached in the centre of rotated disc. Namely, we assume that the constant of the Hook’s law of the spring (i.e., its constitutive equation) is the same in these both frames [6, 7, 61]. But, cf. Rem. 25, even here they are exceptions [62] (from nonclassical physics).

  29. 29.

    Namely, the substitution described below (3.120) gives

    $$\begin{aligned} \{ s,u,\mathbf{Qq},\mathbf{QTQ}^T \} = \breve{\mathcal{F}}(\mathbf{Qx}+\mathbf{c},\mathbf{Qv}+\dot{\mathbf{c}}+ \underline{\Omega }\mathbf{Qx},\mathbf{QF},\mathbf{Q}\text {Grad}\mathbf{F}, \mathbf{QDQ}^T,\mathbf{QWQ}^T+\underline{\Omega },T,\mathbf{Qg},t+b) \end{aligned}$$

    which by choice

    $$\begin{aligned} \mathbf{Q}=\mathbf{1} , \mathbf{c}=-{\mathbf{x}} , \dot{\mathbf{c}}=-\mathbf{v}+\mathbf{Wx} , \underline{\Omega }=-\mathbf{W} , b=-t \end{aligned}$$

    gets

    $$\begin{aligned} \{ s,u,\mathbf{q},\mathbf{T} \} = \breve{\mathcal{F}}(\mathbf{o},\mathbf{o},\mathbf{F},\text {Grad}\mathbf{F},\mathbf{D}, \mathbf{0},T,\mathbf{g},0) \equiv \bar{\mathcal{F}}(\mathbf{F},\text {Grad}\mathbf{F},\mathbf{D},T,\mathbf{g}) \end{aligned}$$

    valid for any independent variables, i.e., giving (3.121).

  30. 30.

    How the principle of symmetry works we outline on simple material (3.123) (see [6, 7, 10, 14, 41, 63, 69] for details); for nonsimple fluid the similar procedure is more complicated, see [14, 70, 71]. Assume for simplicity a unique reference with reference density \(\rho _0\) in the whole body (everywhere is uniform material without dislocations, see Rem. 27) and all responses behave equally (their symmetries are the same). The material symmetry may be expressed by (referential) tensor \(\mathbf H\) (in components \(H^{JK}\)) which, changing deformation \(\mathbf{{F}}\) to \(\mathbf{{FH}}\) in constitutive relation (3.123), gives the same response

    $$\begin{aligned} {\bar{\mathcal{{F}}}}(\mathbf{F}) = {\bar{\mathcal{{F}}}}(\mathbf{{FH}}) \qquad \qquad (a) \end{aligned}$$

    (nonchanging variables are omitted for brevity) and also the same (actual) density \(\rho \) at considered reference density \(\rho _0\), i.e., by (3.65), (3.12), \(\rho _0 =\rho | \text {det}\mathbf{F}| = \rho |\text {det}\mathbf{FH}|\). This latter condition limits tensors \(\mathbf H\) to those which are unimodular

    $$\begin{aligned} \mid \text {det}\mathbf{H} \mid \,= 1 \qquad \qquad (b) \end{aligned}$$

    E.g. indistinguishable rotation may be described by orthogonal \(\mathbf H\) ((\(b\)) is valid, cf. Rem. 8).

    All such \(\mathbf H\) form the symmetry group \(\mathcal G\) (e.g., two such rotations \(\mathbf{{H_1, H_2}}\) give indistinguishable rotation \(\mathbf{{H_1H_2}}\)) which characterize the inherent symmetry of studied material (3.123) in the considered reference configuration. For example, material is isotropic if any rotation (or even inversion) is indistinguishable, i.e., \(\mathcal G\) contains a proper (or even full) orthogonal group.

    Note that a symmetry group depends on a considered reference: its change (which may also alter referential density) generally changes the group. This is described by Noll’s rule; for this and other details see, e.g. [10].

    A symmetry group of simple material divides it in two parts (and each of them in isotropic and anisotropic subparts) [6, 7, 63]:

    • simple solids: isotropic or anisotropic (crystal classes like cubic, hexagonal, triclinic etc.)

    • simple liquid crystals: isotropic (simple fluids, i.e., gases or liquids) or anisotropic (liquid crystals).

    E.g. in simple solids there exists a reference the symmetry group of which is contained in (full) orthogonal group; if they are identical then the material is the simple isotropic solid.

    Simple fluid has a group of symmetry identical to a unimodular group (contains all \(\mathbf H\) with \(\mid {\text {det}}\mathbf{H} \mid \,=\,1\)); this group is therefore the maximal one and fluids are isotropic (because they contain the orthogonal group, cf. Rem. 8; note that unimodular deformations (indistinguishable in fluids) need not be orthogonal, e.g., isochoric shear). Replacement of \(\mathbf F\) by \(\rho \) follows from (\(a\)), (\(b\)) by the choice \(\mathbf{H}= J^{1/3} \mathbf{F}^{-1}\) (unimodular for given \(\mathbf F\): \(\mid \text {det}\mathbf{H}\mid \,=\,\mid \text {det}(J^{1/3} \mathbf{F}^{-1})\mid \,=\,J \mid \text {det}\mathbf{F}\mid ^{-1}\,=\,1\)). Indeed, by (3.65), the response is

    $$\begin{aligned} \bar{\mathcal{F}}(\mathbf{FH}) = \bar{\mathcal{F}}((\rho _0/\rho )^{1/3}) \equiv \hat{\mathcal{F}}(\rho ) \qquad \qquad (c) \end{aligned}$$

    where \(\hat{\mathcal{F}}\) is in fact independent of any reference (and its \(\rho _0\)) because the response (in actual configuration) must remain the same if the reference (and therefore \(\mathbf{F},\rho _0\)) is changed (cf. remark under (3.121) valid also for (3.123)).

  31. 31.

    Besides those based on symmetry in Rem. 30, see e.g., [8], another was used by Haupt [72] according to the size of memory for the stress tensor \(\mathbf T\) in an isothermal body: materials (mostly solids) are

    1. (i)

      elastic: \(\mathbf T\) is (deformation) rate independent without hysteresis, e.g. (3.125).

    2. (ii)

      plastic: \(\mathbf T\) is rate independent with hysteresis (by appropriate internal variables, cf. Sect. 2.3).

    3. (iii)

      viscoelastic: \(\mathbf T\) is rate dependent without hysteresis, e.g. (3.123).

    4. (iv)

      viscoplastic: \(\mathbf T\) is rate dependent with hysteresis (possible even in equilibrium).

  32. 32.

    Note that by analogical calculation for (mechanically) polar materials Rem. 17, the result (3.139) is valid but its skew-symmetric part gives torque \(\mathbf M\).

  33. 33.

    Let \(f\) be a scalar function \(\tilde{f}\) of symmetric tensor \(D\), i.e., a function of 6 independent variables:

    $$\begin{aligned} f = \tilde{f}(\mathbf{D})&= \tilde{f}(D^{11},D^{12},D^{13},D^{22},D^{23},D^{33}) = \tilde{f}\big (D^{11},\frac{1}{2}(D^{12}+D^{21}),\nonumber \\&\quad \frac{1}{2}(D^{13}+D^{31}),D^{22},\frac{1}{2}(D^{23}+D^{32}),D^{33}\big )\\&\equiv \hat{f}(D^{11},D^{12},D^{13},D^{21},D^{22},D^{23}, D^{31},D^{32},D^{33}) = \hat{f}(\mathbf{D}) \end{aligned}$$

    The last definition of function \(\hat{f}\) of 9 variables (allowed by symmetry of \(\mathbf D\)) permits to employ the customary tensor (or matrix) descriptions, e.g. the summation convention in component form. This is the reason for using this definition of \(\hat{f}\) in (3.146), (3.147) and other formulae in this book (similar definitions may be used for skew-symmetric tensor and vector and tensor functions [7, 14, 79]). As may be seen from the definition above, the main property of \(\hat{f}\) is (when \(\mathbf D\) is symmetrical and this is just such a case) that \(\frac{\partial \hat{f}}{\partial \mathbf D}\) is indeed symmetrical, e.g.

    $$\begin{aligned} \frac{\partial \hat{f}}{\partial D^{12}} = \frac{1}{2}\frac{\partial \tilde{f}}{\partial D^{12}} = \frac{\partial \hat{f}}{\partial D^{21}} \end{aligned}$$

    If \(\mathbf B\) is a symmetric tensor then, as may be expected,

    $$\begin{aligned} \text {tr}\frac{\partial \hat{f}}{\partial \mathbf D} \mathbf{B}&= \frac{\partial \hat{f}}{\partial D^{ij}} B^{ji} = \frac{\partial \tilde{f}}{\partial D^{11}} B^{11} + \frac{\partial \tilde{f}}{\partial D^{12}} B^{12} + \frac{\partial \tilde{f}}{\partial D^{13}} B^{13}\nonumber \\&\quad + \frac{\partial \tilde{f}}{\partial D^{22}} B^{22} + \frac{\partial \tilde{f}}{\partial D^{23}} B^{23} + \frac{\partial \tilde{f}}{\partial D^{33}} B^{33} = \frac{\partial \tilde{f}}{\partial \mathbf D}.\mathbf{B} \end{aligned}$$

    and therefore this expression may be also written as an inner product in the space of symmetric tensors, i.e., as a scalar product (denoted by dot) of 6-dimensional vectors. This way is also often used; then, of course, we understand (in (3.146) etc.) \(f\) as a function in the space of symmetric tensor \(\mathbf{D}\), i.e., as \(\tilde{f}\).

    Similarly it may be proved that the derivative of a scalar function with respect to a skew-symmetric tensor is again skew-symmetric.

  34. 34.

    Construction of \(\rho (\mathbf{y},\tau )\) in Euler description is more complicated: in principle we can use current deformation of the body in present time \(t\) (assumed to be known as well as density fields \(\rho (\mathbf{y},t)\) in it) as the reference, calculate relative deformation function \(\mathbf{y} = \underline{\chi }_t({\mathbf{x}},\tau )\) (cf. (3.1)) by integration of velocity field (3.152) and in turn the relative deformation gradient \(\mathbf{F}_t = \text {grad}\underline{\chi }_t\) (see [8] p. 9 for details or [7, 10]). Then \(\rho (\mathbf{y},\tau ) = \rho (\mathbf{y},t)/ \mid \text {det} \mathbf{F}_t(\mathbf{y},\tau ) \mid \) following analogy with (3.65).

  35. 35.

    Consider an example of the non-Newtonian liquid (e.g., solutions and melts of polymers, suspensions, etc.), isothermal and without heat conduction for simplicity. Isotropic nonequilibrium stress fulfils (cf. (3.177))

    $$\begin{aligned} \mathbf{Q} \hat{\mathbf{T}}_N(\rho ,\mathbf{D})\mathbf{Q}^T = \hat{\mathbf{T}}_N(\rho ,\mathbf{QDQ}^T) \end{aligned}$$

    for any \(\mathbf{Q} \in \mathcal{O}\). Representation theorem of this symmetric isotropic nonlinear function of symmetric tensor is (see [9, 12, 64])

    $$\begin{aligned} \mathbf{T}_N = \gamma _0 \mathbf{1} + \gamma _1 \mathbf{D} + \gamma _2 \mathbf{D}^2 \end{aligned}$$

    where coefficients \(\gamma _0,\gamma _1,\gamma _2\) are (nonlinear) functions of \(\rho \) (\(T\) is constant) and tr\(\mathbf{D}\), tr\(\mathbf{D}^2\), tr\(\mathbf{D}^3\). Such nonNewtonian liquid is practically incompressible (tr\(\mathbf{D}= 0\), see Rem. 26, (3.17), (3.16), below and end of Sect. 3.7), \(\gamma _0\) may be included in the undetermined pressure and for small velocity gradients the last member may be neglected. Constitutive equation for nonequilibrium stress is reduced to [81]

    $$\begin{aligned} \mathbf{T}_N = \gamma _1 \mathbf{D} \end{aligned}$$

    where \(\gamma _1\) depends nonlinearly on tr\(\mathbf{D}^2\) (and \(\rho , T\)). For more complicated models see [8, 10, 82].

  36. 36.

    Namely, neglecting the motion and external fields (\(\mathbf{v},\dot{\mathbf{v}},\mathbf{b},\mathbf{i}\) are practically zeros) the momentum balance (3.81) of the thin layer along the real boundary reduces to \(\int _{\mathcal{{V}}}\mathbf{Tn}\,\text {d}v=\mathbf{o}\) with (mostly) pressure \(P\), \(\mathbf{T}= - P\mathbf{1}\) (cf. [84], figure on p. 108). In the limit of this narrow sub body this balance expresses the action-reaction law; therefore the pressure from the outside is given by the constitutive equation of the fluid inside (under the boundary). Pressure \(P\) in the model B is given by (2.7)\(_3\) (the pressure may contain here a nonequilibrium part (2.34) given (in linear approximation) by the volume viscosity, cf. Rems. 9 in Chap. 1, 1 and 8 in Chap. 2, 37 in this chapter).

    The equilibrium pressure part is given by the state equation, see (2.33), (2.32). This, in fact “equilibrium” pressure in “reversible” processes, forms the whole pressure (2.6)\(_3\) of the “classical” thermodynamic model A (density of uniform body with constant mass is given by its volume \(V\)).

  37. 37.

    As we note at the end of Sect. 3.6 all this and the subsequent results follow if the assumption of linearity has been used in a constitutive relation of a nonsimple fluid with viscosity and heat conduction (3.127), (3.146) (i.e., before application of admissibility principle). These constitutive relations are scalar, vector and symmetric tensor isotropic functions (3.128) (including \(f\)) which are linear in vector \(\mathbf g\), \(\mathbf h\) and symmetrical tensor \(\mathbf D\).

    The representation theorems for such linear functions (A.67), (A.58), (A.68) from Appendix A.2 then gives for scalar functions

    $$\begin{aligned} s = s^{(0)} + s^{(1)} \text {tr} \mathbf{D} \;, \qquad u = u^{(0)} + u^{(1)} \text {tr} \mathbf{D} \end{aligned}$$
    $$\begin{aligned} f = f^{(0)} + f^{(1)} \text {tr} \mathbf{D} \end{aligned}$$

    and (3.183) and (3.184) for vector and tensor functions. Similarly, as scalar coefficients here, the scalars \(s^{(0)},u^{(0)},f^{(0)} \equiv u^{(0)} - Ts^{(0)}\), \(s^{(1)},u^{(1)},f^{(1)} \equiv u^{(1)} - Ts^{(1)}\) are (generally nonlinear) functions of density \(\rho \) and temperature \(T\). Using them in the reduced inequality (3.113) and by the admissibility principle, we obtain all the results (like (3.185), (3.186), etc.) of this section (namely \(s^{(1)},u^{(1)},f^{(1)}\) are zeros identically), see [14, 27, 84].

  38. 38.

    As the configurational or material forces [4, 87] (note that the density of chemical potential \(\rho g\) has a pressure dimension). An analog of chemical potential is the Eshelby tensor (of chemical potential) \(\underline{\Gamma }\) defined as (\(\mathbf{F}^{-T} \equiv (\mathbf{F}^T)^{-1}\))

    $$\begin{aligned} \underline{\Gamma }\equiv f\mathbf{1} - (1/\rho ) \mathbf{F}^T \mathbf{T} \mathbf{F}^{-T} \end{aligned}$$

    Note, that if stress is reduced to pressure \(P\), \(\mathbf{T} = - P\mathbf{1}\), (usual in fluids) this definition gives the classical result (3.203) \(\underline{\Gamma }= g\mathbf{1}\), see (3.199). The Eshelby tensor, e.g. gives the condition of phase equilibria (Maxwell relation—equality of chemical potentials (2.116) in fluid phases), namely equality of \(\underline{\Gamma }\mathbf{n}\) on both sides of equilibrated solid phases (n is the normal to phase boundary) and may be also used to describe surface phenomena, dislocations, etc. [1, 4, 87]. Eshelby tensors may also be defined in mixtures [2, 3].

  39. 39.

    If we assume that the ideal gas studied fulfils the local equilibrium (and this is the usual case: ideal gas may be from the linear fluid models discussed here, but it may be also from some nonlinear models fulfilling this principle, e.g. those in [78]), then property (3.213) follows from state equation (3.212). Indeed, the local equilibrium means the validity of Gibbs equations (3.200)\(_1\), (3.198)\(_1\), from which \(\partial \hat{u}/\partial \rho = T \partial \hat{s}/\partial \rho + P/\rho ^2\) and \(\partial \hat{s}/\partial \rho = -\partial \hat{\overline{(P/\rho ^2)}}/\partial T\). By their combination and using state equation (3.212) we obtain identically \(\partial \hat{u}/\partial \rho = 0\). Cf. also [27, Sect. 16], [90, 91].

  40. 40.

    Cf. (3.18), Rem. 5 and deduction of (3.25). Because the change of frame describes the change of frame in a rigid motion to another one the result (3.223) is intuitively clear. Formally, inserting rigid motion from Rem. 5 into (3.25) we seek the (starred) frame in which \({\mathbf{x}}^* = \mathbf{X}\) (and therefore \(\mathbf{v}^* = \mathbf{o}\), i.e., (3.223)) through the body. It may be seen that this need the change of frame by time functions \(\mathbf{Q}= \underline{\Theta }^T\) and \(\mathbf{c}=-\underline{\Theta }^T \underline{\gamma }\).

  41. 41.

    Note that an equilibrium process (as the time succession of states with (3.220)–(3.222)) with nonzero radiation \(Q\not = 0\), which is even reversible, is possible: in the “straight” part of the process the heating (defined by the first two members on the right-hand side of (3.97)) is given as \(\int _V Q\,\text {d}v\) by (3.226) (\(V\) is the volume of the body); see also Sect. 1.2, models A, B in Sect. 2.2 and Rems. 12 in Chap. 1, 48 in this chapter.

    Temperature may change in time but not in space (3.221) during such an equilibrium process in the rigid and not moving body, density does not change in time (3.227) (but may change in space); \(u,s\) change as the temperature changes, similarly \(P\) changes by a corresponding time change of \(\mathbf{b}+\mathbf{i}\) (say by (3.192), (3.191), (3.194) in the linear fluid model). The reverse process may be imagined to exist as going through the same states of the equilibrium process, power and entropy production are again zero, heating is of reverse sign \(-\int _V Q \,\text {d}v\) (in comparison with the appropriate instant of “straight” equilibrium process).

    Even this reversible process is rather a special one. We note it here to demonstrate that in the model of the linear fluid equality (in entropic inequality) is possible, see (1.35), and to show that entropy may be calculated with the precision of a constant, see (1.40), cf. application of reversible processes in Sect. 1.4. An equilibrium state is also an equilibrium process formed by a unique state with (3.231), cf. definition below (2.11).

  42. 42.

    Similarly as in Rem. 11 in Chap. 1 and in Sects. 2.1, 2.2, we try to avoid in this way the unusual, often “pathological” situations of real complex materials in our simple models (as, e.g. zero values of some transport coefficients (3.197) at certain \(\rho ,T\)); other motivation is the “practical realization of the persistence of the equilibrium state” which may be achieved through its stability (discussed below), e.g. regularity conditions (3.233), (3.234) are even intensified in such a stable equilibrium state (both derivatives are positive, see (3.256), (3.257) below).

    Again we assume that the constitutive model together with regularities introduced is valid in all situations, e.g. the model of fluid with linear transport properties with regular response is assumed to be valid for all values of \(\rho ,T\). Namely, we study the (properties of) model even though we know that there are values of \(\rho ,T\) for which a real fluid does not fulfil some regularities assumed (e.g. stability in the region of phase transformations); as usually, such difficulties are resolved by the appropriate limiting applications of the model studied.

  43. 43.

    Calculation of tr\((\mathop {\mathbf{D}}\limits ^{\circ })^2\) in (3.196) gives tr\((\mathop {\mathbf{D}}\limits ^{\circ })^2 = (\mathop {D^{11}}\limits ^{\circ })^2 + (\mathop {D^{22}}\limits ^{\circ })^2 + (\mathop {D^{33}}\limits ^{\circ })^2 + 2(\mathop {D^{12}}\limits ^{\circ })^2 + 2(\mathop {D^{13}}\limits ^{\circ })^2 + 2(\mathop {D^{23}}\limits ^{\circ })^2 \) and therefore zero entropy production (3.222) and positivity (3.232) give from (3.196) the result (3.221) as well as tr\(\mathbf{D}=0\) and \(\mathop {D^{11}}\limits ^{\circ } =\ \mathop {D^{22}}\limits ^{\circ } =\ \mathop {D^{33}}\limits ^{\circ } =\ \mathop {D^{12}}\limits ^{\circ } =\ \mathop {D^{13}}\limits ^{\circ } =\ \mathop {D^{23}}\limits ^{\circ } = 0\) which with definition (3.188) of \(\mathop {\mathbf{D}}\limits ^{\circ }\) gives (3.220).

  44. 44.

    This result (3.239) may be generalized for Eshelby tensor \(\underline{\Gamma }\) (generalization of chemical potential, e.g. for solids, see Rem. 38) as

    $$\begin{aligned} \text {Div}(\underline{\Gamma } + \Phi \mathbf{1}) = \mathbf{o} \end{aligned}$$

    cf. [1, 96] (Div is the divergence in referential description).

  45. 45.

    But we omit the generalizations of equilibrium stabilities for phase transitions [1, 103–106] (for them typically criteria stability like (3.256), (3.257) are not valid), for more general materials (say solids), and the more complicated problem of stability of nonequilibrium states (e.g. the vast field of dissipative structures [24, 37, 80, 107–109]) because most of these issues do not concern our (one-phase) model or are now in the stage of intensive and not completely resolved research; see also Rem. 31 in Chap. 4.

  46. 46.

    It follows from our intention to use the theorem of concave function from Appendix A.3 for the proof. This assumes the negative (or positive) definiteness of a matrix composed from second derivatives of such a function. This property has, besides (3.247), e.g. function \(\tilde{g}(T,P)\) (3.205) (used also below in this section) but unfortunately not the more natural \(\hat{f}(\rho ,T)\) (3.190) or even \(\check{f}(v,T)\) (see (3.199)); cf. [113, Sect. 39].

  47. 47.

    Although assumptions giving (3.267), (3.268) look natural, this is not so, e.g. such \(S(t)\) fulfilling (3.264), (3.265) may exist where \(\dot{S}(t)>0\) changes oscillatorily for any time and therefore a limit does not exist. Similarly the existence of limit (3.268) is not clear, e.g. \(\sigma ^o\) in (3.267) may be nonzero on surfaces or lines (sets of zero measure) and such a situation may be obtained even by limitation from smooth function \(\sigma \).

    However, these difficulties may be avoided by other means, e.g. it is possible to prove (often with special types of material or with other potentials instead of entropy) the dynamical stability (even asymptotical one) but mostly in integral form (deviations are expressed by integral through the body). For further discussions see [1, 18, 93–95, 97–103, 114, 115].

  48. 48.

    Namely, we discuss two examples of equilibrium reversible processes: the isothermal and then those which are adiabatic. Such processes with ideal gas (i.e., with real stable gas at sufficiently low pressures) are used in the Carnot cycle in Appendix A.1.

    The uniform process described here for linear fluid (see below (3.239) and (3.211)) which is isothermal (temperature \(T=T^0\) is permanently the same constant) may be considered as a special case of equilibrium reversible processes in the fluid model B of Sect. 2.2 if the entropy production (given by (2.36) or (3.196)) may be neglected. A stable equilibrium state in a given instant has (besides the constant temperature \(T^0\)) the volume V (with zero velocity everywhere (3.223)). The change of this state to another one with the volume \(V+\mathrm{d}V\) (and the same temperature \(T^0\) and zero velocity) by such a reversible process can be imagined as a sudden change of the volume by a small d\(V\) and as a development of this perturbed state isothermally to a new stable equilibrium state as described above (the second example without the body force: \(\mathbf{b} = \mathbf{o}\) in (3.270)). A new equilibrium state will be practically achieved after a time interval much greater then the typical time scale in model B. Therefore the reversible process composed from sequences of such \(V\) to \(V+\mathrm{d}V\) changes must be slow \(\dot{V}\) is zero as well as the entropy production (2.36) and all this happens in the time scale of the model B. Heat exchange is nonzero and gives the entropy change, i.e., both members on the left-hand side of (3.274) compensate (similarly as in (2.10)) because the entropy production is zero (in (3.196) the second order contributions of heat and viscosity are neglected in fact, while in (3.274) the not neglected first order heat contribution is compensated).

    Quite analogously we can discuss the adiabatic reversible equilibrium process using the perturbation of the isolated body described above (the first example with persistent \(\mathbf{i}= \mathbf{b}= \mathbf{o}\), \(Q=0\), \(\mathbf{q}= \mathbf{o}\)). Starting with a corresponding equilibrium state in a given instant with the volume \(V\) (with zero velocity everywhere (3.223)) we obtain the new perturbed state changing suddenly the volume to \(V+\mathrm{d}V\) which develops as an isolated body into an equilibrium state with the new volume \(V+\mathrm{d}V\), isolated and with zero velocity everywhere. An adiabatic reversible process is obtained, continuing in this way sequentially (analogously as in the previous isothermal example). Such \(V\) to \(V+\mathrm{d}V\) changes must be again slow, with \(\dot{V}\) nearly zero as well as the entropy production (all members in (3.196) are neglected, i.e., equality in (2.36) is valid in the time scale of model B). But then the left-hand side of (3.265) is nearly zero and entropy remains constant during such a reversible adiabatic process.

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  1. Faculty of Chemistry, Brno University of Technology, Brno, Czech Republic

    Miloslav Pekař

  2. Institute of Chemical Technology, Prague 6, Czech Republic

    Ivan Samohýl

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Pekař, M., Samohýl, I. (2014). Continuum Thermodynamics of Single Fluid. In: The Thermodynamics of Linear Fluids and Fluid Mixtures. Springer, Cham. https://doi.org/10.1007/978-3-319-02514-8_3

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