Thermodynamics of Class I and Class II Classical Mixtures

Abstract

In this chapter, two versions of mixtures of Boltzmann-type continua are subject to thermodynamic analyses for viscous fluids. Of the two forms of the Second Law that were introduced—the Clausius–Duhem inequality applied to open systems and the entropy principle of Müller—the latter principle is employed in the process of deduction of the implications revealed by the particular Second Law. The goal in the two parts of the chapter is to derive the ultimate forms of the governing equations, which describe the thermomechanical response of the postulated constitutive behavior without violation of the Second Law of thermodynamics. The versions of mixtures which are analyzed are

• Diffusion of tracers in a classical fluid: The conceptual prerequisites of this type of processes are mixtures of class I, in which the major component is the bearer fluid within which a finite number of constituents with minute concentration are suspended or solved in the bearer fluid. The motion of these tracers is described by the difference of the constituent velocities relative to the barycentric velocity of the mixture as a whole. For the dissipative constitutive class applied to the entropy principle, the existence of the Kelvin temperature is proved, the form of the Gibbs relation could be determined as could the conditions of thermodynamic equilibrium and the constitutive behavior in its vicinity.

• Thermodynamics of a saturated mixture of nonpolar solid–fluid constituents: Conceptually these systems are treated as classical mixtures of class II, in which the individual motions of the constituents are separately accounted for by their own balances of mass and momentum, but subject to a common temperature. The analysis of the dissipation inequality is performed subject to the assumption of constant true density of all constituents and the supposition of saturation of the mixture. The constitutive relations are postulated for a mixture of viscous heat conducting fluids. The explanation of the entropy principle is structurally analogous to that of the class I-diffusion theory, but is analytically much more complex. Unfortunately, intermediate ad hoc assumptions must be introduced to deduce concrete results that will lead to fully identifiable fluid dynamical equations, which are in conformity with the Second Law for the presented type of mixtures.

This chapter is a revised and somewhat extended version of Chap. 7 “Theory of Mixtures” of [20], which was fraught with a number of slips and computational errors. The text is also pruned in certain places and extended in others.

Appendices

Appendix 23.A   Some Details on the Poincaré Theorem and the Condition of Frobenius

In this appendix, we shall prove the correctness of the following statements:

Proposition 23.4

• (i) In the one-dimensional case ($n = 1$) every total differential is complete.

• (ii) In two dimensions ($n = 2$), every differential can be transformed into a total differential by multiplying it with an integrating factor.

• (iii) In three-dimensional space ($n = 3$), the condition that an arbitrary differential $\mathrm {d}F = X_{i}\mathrm {d}x_{i}$ can be made complete is the requirement that the vector field ${\varvec{X}}$ stays perpendicular to its vorticity, $\mathrm {curl}\,{\varvec{X}}$.    $\blacksquare$

Proof

• (i) In the one-dimensional case, there is no possibility to select between different paths of integration between two points to reach point b from point a. Every Riemann integral from a to a certainly vanishes.

• (ii) In two dimensions, one has $\mathrm {d}F = X_{1}\mathrm {d}x_{1} + X_{2}\mathrm {d}x_{2}$. If Poincaré’s theorem holds, then

  $$\begin{aligned} \frac{\partial X_{1}}{\partial x_{2}} = \frac{\partial X_{2}}{\partial x_{1}} \end{aligned}$$

  (23.174)

  and F is a potential such that

  $$\begin{aligned} X_{1} = \frac{\partial F}{\partial x_{1}}, \quad X_{2} = \frac{\partial F}{\partial x_{2}} \quad {\mathop {\Longrightarrow }\limits ^{(23.147)}} \quad \frac{\partial ^{2}F}{\partial x_{2}\partial x_{1}} = \frac{\partial ^{2}F}{\partial x_{1}\partial x_{2}}. \end{aligned}$$

  (23.175)

  The value $F({\varvec{x}})$ can be determined by integrating from ${\varvec{x}_{0}}$ to ${\varvec{x}}$ along any continuous curve,

  $$\begin{aligned} F({\varvec{x}}) = \int _{{\varvec{x}}_{0}}^{{\varvec{x}}}\left( X_{1}(x_{1}, x_{2})\mathrm {d}x_{1} + X_{2}(x_{1}, x_{2})\mathrm {d}x_{2}\right) . \end{aligned}$$

  (23.176)

  It follows that the integral along any curve from ${\varvec{a}}$ to ${\varvec{b}}$ yields

  $$\begin{aligned} \int _{{\varvec{a}}}^{{\varvec{b}}}\left( X_{1}(x_{1}, x_{2})\mathrm {d}x_{1} + X_{2}(x_{1}, x_{2})\mathrm {d}x_{2}\right) = \int _{{\varvec{a}}}^{{\varvec{b}}}\mathrm {d}F = F({\varvec{b}}) - F({\varvec{a}})\quad \end{aligned}$$

  (23.177)

  and is independent of the path chosen between ${\varvec{a}}$ and ${\varvec{b}}$, see Fig.  23.4 .

Fig. 23.4

Explaining the path independence of the integration of a differential in two dimensions

If the Poincaré theorem is not fulfilled, one may try its satisfaction by an integrating factor. In that case, one writes

$$\begin{aligned} \mathrm {d}F_{\lambda } = \lambda \,X_{1}\mathrm {d}x_{1} +\lambda \,X_{2}\mathrm {d}x_{2}. \end{aligned}$$

(23.178)

Satisfaction of the Poincaré theorem for this differential requires

$$\begin{aligned}&\frac{\partial (\lambda \,X_{1})}{\partial x_{2}} = \frac{\partial (\lambda \,X_{2})}{\partial x_{1}}, \qquad \mathrm {or} \end{aligned}$$

(23.179)

$$\begin{aligned}&\lambda \underbrace{\left( \frac{\partial X_{1}}{\partial x_{2}} - \frac{\partial X_{2}}{\partial x_{1}}\right) }_{\ne 0} = \frac{\partial \lambda }{\partial x_{1}}X_{2} - \frac{\partial \lambda }{\partial x_{2}}X_{1}, \end{aligned}$$

(23.180)

in which the underbraced term differs from zero since the starting equation does not satisfy the Poincaré condition. Evidently, $\lambda = \mathrm {const.}$ is not a possible integrating factor. However, the determination of a nonconstant integrating factor is not unique.

• If $X_{2} \ne 0$, one may, for instance, select $\lambda = \lambda _{1}(x_{1})$; then it follows from (23.180) that

  

  (23.181)

  This formula shows that $\lambda _{1}$ is a function of $x_{1}$ alone only, if the integrand function is independent of $x_{2}$

• If $X_{1} \ne 0$ and $\lambda = \lambda _{2}(x_{2})$ is assumed, the analogous procedure leads to

  

  (23.182)

  which shows that $\lambda = \lambda _{2}(x_{2})$ is only a function of $x_{2}$, if the integrand function does not depend on $x_{1}$.

• In all other cases, $\lambda$ must be determined by solving the partial differential equation (23.180).

• (iii) In the three-dimensional case, it is known that a vector field ${\varvec{X}}$ can be written as the gradient of a scalar potential field only, if $\mathrm {curl}\,{\varvec{X}} = {\varvec{0}}$. In Cartesian component form, this then yields the Poincaré condition

  $$\begin{aligned} \frac{\partial X_{i}}{\partial x_{j}} - \frac{\partial X_{j}}{\partial x_{i}} = 0, \quad (i, j = 1, 2, 3). \end{aligned}$$

  If $\mathrm {curl}\,{\varvec{X}} \ne {\varvec{0}}$, one may try to construct a total differential

  $$\begin{aligned} \mathrm {d}F_{\lambda } = \lambda X_{1}\mathrm {d}x_{1} + \lambda X_{2}\mathrm {d}x_{2} + \lambda X_{3}\mathrm {d}x_{3}, \end{aligned}$$

  (23.183)

  by multiplying $\mathrm {d}F$ with an integrating factor. The condition for this to be successful is

  $$\begin{aligned} {\varvec{0}} {\mathop {=}\limits ^{!}} \mathrm {curl}\,(\lambda {\varvec{X}}) = \lambda (\mathrm {curl}\,{\varvec{X}}) + (\mathrm {grad}\,\lambda )\times {\varvec{X}}. \end{aligned}$$

  (23.184)

  Multiplying this equation scalarly with ${\varvec{X}}$ yields

  $$\begin{aligned} \mathrm {curl}\,{\varvec{X}}\cdot {\varvec{X}} = 0, \quad \mathrm {if}\; \lambda \ne 0. \end{aligned}$$

  (23.185)

  Thus, ${\varvec{X}}$ and $\mathrm {curl}\,{\varvec{X}}$ are perpendicular to one another; in Cartesian components, this reads

  $$\begin{aligned} \varepsilon _{ijk}X_{k,j}X_{i} = 0, \end{aligned}$$

  (23.186)

  which is nothing else than the Frobenius condition.

For a plane vector field, condition (23.185) is always satisfied; a fact that corroborates the statement that in two dimensions an integrating factor always exists.

Appendix 23.B   Proof of a Statement About a Complete Differential

In this Appendix, the following statement is proved:

Let $\mathrm {d}f$ be an incomplete differential of the variables $x_{i}$ $(i=1,\ldots ,n)$. Let, moreover, $g \ne 0$ be an integrating denominator, which makes

$$\begin{aligned} \mathrm {d}F := \frac{\mathrm {d}f}{g} \end{aligned}$$

(23.187)

a total (complete) differential. Then,

$$\begin{aligned} \mathrm {d}H = G(F)\frac{\mathrm {d}f}{g} \end{aligned}$$

(23.188)

is equally complete.

Proof

In long-hand notation, $\mathrm {d}H$ is written as

$$\begin{aligned} \mathrm {d}H = \frac{G(F)}{g}\left\{ \cdots +\frac{\partial f}{\partial x_{i}}\mathrm {d}x_i+ \cdots +\frac{\partial f}{\partial x_{j}}\mathrm {d}x_j+ \cdots \right\} . \end{aligned}$$

(23.189)

Hence,

$$\begin{aligned} \frac{\partial H}{\partial x_{i}} = \frac{G(F)}{g}\frac{\partial f}{\partial x_{i}}, \quad \frac{\partial H}{\partial x_{j}} = \frac{G(F)}{g}\frac{\partial f}{\partial x_{j}}. \end{aligned}$$

(23.190)

Testing Poincaré’s theorem, one deduces,

$$\begin{aligned}&\frac{\partial ^{2}H}{\partial x_{j}\partial x_{i}} = \left\{ \frac{\partial G}{\partial F}\frac{\partial F}{\partial x_{j}} \frac{1}{g}\frac{\partial f}{\partial x_{i}} + G(F)\frac{\partial }{\partial x_{j}}\left( \frac{1}{g} \frac{\partial f}{\partial x_{i}}\right) \right\} , \nonumber \\[-2.5mm] \\[-2.5mm]&\frac{\partial ^{2}H}{\partial x_{i}\partial x_{j}} = \left\{ \frac{\partial G}{\partial F}\frac{\partial F}{\partial x_{i}}\frac{1}{g}\frac{\partial f}{\partial x_{j}} + G(F)\frac{\partial }{\partial x_{i}}\left( \frac{1}{g} \frac{\partial f}{\partial x_{j}}\right) \right\} .\nonumber \end{aligned}$$

(23.191)

In these expressions, the second terms are equal, because, by prerequisite, $\mathrm {d}f/g$ is a complete differential (Poincaré theorem). This is also true for the first term, since according to (23.187)

$$\begin{aligned} \frac{\partial G}{\partial F}\frac{\partial F}{\partial x_{j}}\frac{\partial F}{\partial x_{i}} = \frac{\partial G}{\partial F}\frac{\partial F}{\partial x_{i}}\frac{\partial F}{\partial x_{j}}. \end{aligned}$$

(23.192)

It has, therefore, been proved that the two mixed derivatives (23.191) have the same value. It follows that $\mathrm {d}H$ is necessarily a complete differential, if $\mathrm {d}F$ is one.

Appendix 23.C   Proof of Some Rules of Differentiation

• (i) The formulae (23.95)$_{1,2,3}$ follow from the definition of the barycentric velocity, ${\varvec{v}}=\sum _{\gamma =1}^{N}\xi ^{\gamma }{\varvec{v}}^{\gamma }$ by applying the product rule of differentiation. For instance,

  $$\begin{aligned} \mathrm {grad}\,{\varvec{v}} = \sum _{\gamma =1}^{N}\left\{ \xi ^{\gamma }\mathrm {grad}\,{\varvec{v}}^{\gamma } + {\varvec{v}}^{\gamma }\otimes \mathrm {grad}\,\xi ^{\gamma }\right\} , \end{aligned}$$

  (23.193)

  from which expressions for ${\varvec{D}}$ and ${\varvec{W}}$ may be derived by applying the operators “sym” and “skw”,

  $$\begin{aligned} {\varvec{D}}= & {} \sum _{\gamma =1}^{N}\left\{ \xi ^{\gamma }{\varvec{D}}^{\gamma } + \mathrm {sym} \left( {\varvec{v}}^{\gamma }\otimes \mathrm {grad}\,\xi ^{\gamma }\right) \right\} , \nonumber \\[-2.5mm] \\[-2.5mm] {\varvec{W}}= & {} \sum _{\gamma =1}^{N}\left\{ \xi ^{\gamma }{\varvec{W}}^{\gamma } + \mathrm {skw}\left( {\varvec{v}}^{\gamma }\otimes \mathrm {grad}\,\xi ^{\gamma }\right) \right\} . \nonumber \end{aligned}$$

  (23.194)

• (ii) From the definition of the diffusion velocity, there follows

  $$\begin{aligned} \frac{\partial {\varvec{u}}^{\beta }}{\partial {\varvec{v}}^\alpha } = \frac{\partial }{\partial {\varvec{v}}^{\alpha }} \left( {\varvec{v}}^{\beta } - \sum _{\gamma =1}^{N}\xi ^{\gamma }{\varvec{v}}^{\gamma }\right) = \delta _{\alpha \beta } {\varvec{I}} - \xi ^{\alpha }{\varvec{I}} = \left( \delta _{\alpha \beta } - \xi ^{\alpha }\right) {\varvec{I}},\qquad \end{aligned}$$

  (23.195)

  in which ${\varvec{I}}$ is the $3\times 3$ unit rank-2 tensor. Similarly,

  $$\begin{aligned} \frac{\partial {\varvec{U}}^{\beta }}{\partial {\varvec{W}}^{\alpha }}= & {} \frac{\partial }{\partial {\varvec{W}}^\alpha }\left\{ \mathrm {grad}\,{\varvec{v}}^{\beta } - \sum _{\gamma =1}^{N}\left( \xi ^{\gamma }{\varvec{W}}^{\gamma } + \mathrm {skw}\left( {\varvec{v}}^{\gamma }\otimes \mathrm {grad}\,\xi ^{\gamma }\right) \right) \right\} \nonumber \\= & {} \delta _{\alpha \beta }{\varvec{I}}_{4}-\xi ^{\alpha }{\varvec{I}}_{4} = \left( \delta _{\alpha \beta }- \xi ^{\alpha }\right) {\varvec{I}}_{4}, \end{aligned}$$

  (23.196)

  in which ${\varvec{I}}_{4}$ is the unit rank-4 tensor with $({\varvec{I}}_{4})_{ijkl} = \delta _{ij}\delta _{kl}$.

• (iii) With the results (23.195) and (23.196), we may now deduce

  $$\begin{aligned} \frac{\partial f}{\partial {\varvec{v}}^{\alpha }}= & {} \sum _{\beta }\frac{\partial f}{\partial {\varvec{u}}^{\beta }} \frac{\partial {\varvec{u}}^{\beta }}{\partial {\varvec{v}}^{\alpha }} =\sum _{\beta }\frac{\partial f}{\partial {\varvec{u}}^{\beta }}\left( \delta _{\alpha \beta } - \xi ^{\alpha }\right) {\varvec{I}} \nonumber \\= & {} \frac{\partial f}{\partial {\varvec{u}}^{\alpha }} - \xi ^{\alpha }\sum _{\beta } \frac{\partial f}{\partial {\varvec{u}}^{\beta }}, \end{aligned}$$

  (23.197)
  $$\begin{aligned} \frac{\partial f}{\partial {\varvec{W}}^{\alpha }}= & {} \sum _{\beta }\frac{\partial f}{\partial {\varvec{U}}^{\beta }}\frac{\partial {\varvec{U}}^{\beta }}{\partial {\varvec{W}}^{\alpha }} = \sum _{\beta } \frac{\partial f}{\partial {\varvec{U}}^{\beta }}\left( \delta _{\alpha \beta } - \xi ^{\alpha }\right) {\varvec{I}}_{4} \nonumber \\= & {} \frac{\partial f}{\partial {\varvec{U}}^{\alpha }} - \xi ^{\alpha }\sum _{\beta }\frac{\partial f}{\partial {\varvec{U}}^{\beta }}. \end{aligned}$$

  (23.198)

  This result can now be used in the evolution of $\partial f/\partial \mathrm {grad}\,{\varvec{v}}^{\alpha }$

  

  (23.199)

where $\mathrm {grad}\,{\varvec{v}}^{\alpha } = {\varvec{D}}^{\alpha } + {\varvec{W}}^{\alpha }$ has been used.