Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics

Abstract

The decisive word in this postulate is the quantifier all it makes the postulate a restrictive condition on the internal constitutive assumptions that can be imposed on systems of the type under consideration.

Appendix

The purpose of this appendix is to record the derivation of (30) and some related auxiliary results. Recall that ϖ_(a) denotes a generic component-specific property such as ${{v}_{{(a)}}}$ or ${{\varepsilon}_{{(a)}}}$and we seek a simple formula for $\sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{\rm{D}}^a}{{\varpi}_{{(a)}}}}}{{{\rm{D}}t}}$ to be used in determining the continuum level form of the conservation laws by summing over the single constituent continuum forms of the conservation laws. A formula relating the density-weighted sum of the time derivatives of the selected components to the sum of the density-weighted time derivatives is desired. Recall that the sum of generic constituent-specific quantity per unit mass ϖ_(a) is related to its density-weighted sum ϖ by (10.29). The time derivative of (10.29) with respect to the selected component is given by

$$ \rho \frac{{{{\rm{D}}^s}\varpi }}{{{\rm{D}}t}} + \varpi \frac{{{{\rm{D}}^s}\rho }}{{{\rm{D}}t}} = \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{\rm{D}}^s}{{\varpi}_{{(a)}}}}}{{{\rm{D}}t}} + \sum\limits_{{a = 1}}^N {{{\varpi}_{{(a)}}}\frac{{{{\rm{D}}^s}{{\rho}_{{(a)}}}}}{{{\rm{D}}t}}} $$

(10.95)

which may be solved for $\sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{\rm{D}}^s}{{\varpi}_{{(a)}}}}}{{{\rm{D}}t}}$, thus

$$ \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{\rm{D}}^s}{{\varpi}_{{(a)}}}}}{{{\rm{D}}t}} = \rho \frac{{{{\rm{D}}^s}\varpi }}{{{\rm{D}}t}} + \varpi \frac{{{{\rm{D}}^s}\rho }}{{{\rm{D}}t}} - \sum\limits_{{a = 1}}^N {{{\varpi}_{{(a)}}}\frac{{{{\rm{D}}^s}{{\rho}_{{(a)}}}}}{{{\rm{D}}t}}}. $$

(10.96)

The relationship between the time derivatives with respect to the selected component and with respect to the “a” component is obtained using (10.12)

$$\begin{array}{llllllllll} \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{\rm{D}}^s}{{\varpi}_{{(a)}}}}}{{{\rm{D}}t}} = \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{\rm{D}}^a}{{\varpi}_{{(a)}}}}}{{{\rm{D}}t}} + \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} {{v}_{{(s)}}} \cdot \nabla {{\varpi}_{{(a)}}} - \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} {{v}_{{(a)}}} \cdot \nabla {{\varpi}_{{(a)}}}; \end{array}$$

(10.97)

this is used to rewrite (10.96) as

$$\begin{array}{lllllllll}\sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{\rm{D}}^a}{{\varpi}_{{(a)}}}}}{{{\rm{D}}t}} = \rho \frac{{{{\rm{D}}^s}\varpi }}{{{\rm{D}}t}} + \varpi \frac{{{{\rm{D}}^s}\rho }}{{{\rm{D}}t}} - \sum\limits_{{a = 1}}^N {{{\varpi}_{{(a)}}}\frac{{{{\rm{D}}^s}{{\rho}_{{(a)}}}}}{{{\rm{D}}t}}} - \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} {{v}_{{(s)}}} \cdot \nabla {{\varpi}_{{(a)}}}\cr\quad + \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} {{v}_{{(a)}}} \cdot \nabla {{\varpi}_{{(a)}}}. \end{array}$$

(10.98)

The following relationships, the first obtained from the conservation of mass for the mixture (10.25),

$$ \frac{{{{\rm{D}}^s}\rho }}{{{\rm{D}}t}} = \frac{{\partial \rho }}{{\partial t}} + {{v}_{{(s)}}} \cdot \nabla \rho = {{\mathord{\buildrel{\text{$\scriptscriptstyle\smile$}}\over s} }}(t) - \nabla \cdot (\rho v) + {{v}_{{(s)}}} \cdot \nabla \rho $$

(10.99)

and the second obtained from the conservation of mass for the constituent (10.23)

$$ \frac{{{{\rm{D}}^s}{{\rho}_{{(a)}}}}}{{{\rm{D}}t}} = \frac{{\partial {{\rho}_{{(a)}}}}}{{\partial t}} + {{v}_{{(s)}}} \cdot \nabla {{\rho}_{{(a)}}} = {{{{\mathord{\buildrel{\text{$\scriptscriptstyle\smile$}}\over s} }}}_{{(a)}}}(t) - \nabla \cdot ({{\rho}_{{(a)}}}{{v}_{{(a)}}}) + {{v}_{{(s)}}} \cdot \nabla {{\rho}_{{(a)}}} $$

(10.100)

will now be used in (10.98). However, before using (10.100) it is multiplied by and summed over all values of “a,” thus

$$ \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{D}^a}{{\varpi}_{{(a)}}}}}{{Dt}} = \rho \frac{{{{D}^s}\varpi }}{{Dt}} + \varpi ({{\mathord{\buildrel{\text{$\scriptscriptstyle\smile$}}\over s} }}(t) - \nabla \cdot (\rho v) + {{v}_{{(s)}}} \cdot \nabla \rho ) - \sum\limits_{{a = 1}}^N {{{\varpi}_{{(a)}}}} {{{{\mathord{\buildrel{\text{$\scriptscriptstyle\smile$}}\over s} }}}_{{(a)}}}(t) + \sum\limits_{{a = 1}}^N {{{\varpi}_{{(a)}}}} \nabla \cdot ({{\rho}_{{(a)}}}{{v}_{{(a)}}}) - \sum\limits_{{a = 1}}^N {{{\varpi}_{{(a)}}}} {{v}_{{(s)}}} \cdot \nabla {{\rho}_{{(a)}}} - \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} {{v}_{{(s)}}} \cdot \nabla {{\varpi}_{{(a)}}} + \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} {{v}_{{(a)}}} \cdot \nabla {{\varpi}_{{(a)}}}. $$

(10.101)

The second line of the result above is condensed

$$\begin{array}{lllllll} \sum\limits_{{a = 1}}^N {{{\rho}_{{(a)}}}} \frac{{{{\rm{D}}^a}{{\varpi}_{{(a)}}}}}{{{\rm{D}}t}} =\ \rho \frac{{{{\rm{D}}^s}\varpi }}{{{\rm{D}}t}} + \varpi ({{\mathord{\buildrel{\text{$\scriptscriptstyle\smile$}}\over s} }}(t) - \nabla \cdot (\rho v) + {{v}_{{(s)}}} \cdot \nabla \rho ) - \sum\limits_{{a = 1}}^N {{{\varpi}_{{(a)}}}} {{{{\mathord{\buildrel{\text{$\scriptscriptstyle\smile$}}\over s} }}}_{{(a)}}}(t) \cr+ \sum\limits_{{a = 1}}^N {\nabla \cdot ({{\varpi}_{{(a)}}}{{\rho}_{{(a)}}}{{v}_{{(a)}}})} - \sum\limits_{{a = 1}}^N {{{v}_{{(s)}}} \cdot \nabla {{\varpi}_{{(a)}}}{{\rho}_{{(a)}}}}\end{array} $$

(10.102)

and then the entire equation is algebraically reduced to (10.30).