Second-order constitutive theory of fluids

Abstract

A fully second-order continuum theory of fluids is developed. The conventional balance equations of mass, linear momentum, energy and entropy are used. Constitutive equations are assumed to depend on density, temperature and velocity, and their derivatives up to second order. The principle of equipresence is used along with the Coleman–Noll procedure to derive restrictions on the constitutive equations by utilizing the second law. The entropy flux is not assumed to be equal to the heat flux over the temperature. We obtain explicit results for all constitutive quantities up to quadratic nonlinearity so as to satisfy the Clausius–Duhem inequality. Our results are shown to be consistent but more general than other published results.

Appendices

Appendix A

The following tables are limited to quadratic terms and are obtained from those given in Tables 5.1–5.3 of Paolucci [36] (see also [19, 30, 61, 62]), while those associated with the third rank tensor are given by Ahmad [2] (see also [12, 13]).

Table 6 Complete and irreducible function basis of isotropic scalar invariants of vectors $u_{i}$ and $v_{i}$, symmetric tensors $A_{ij}$ and $B_{ij}$, and third rank tensor $H_{ijk}$, up to quadratic order

Table 7 Generators for a vector-valued isotropic function of vector $u_{i}$, symmetric tensor $A_{ij}$, and third rank tensor $H_{ijk}$, up to quadratic order

Table 8 Generators for a symmetric tensor-valued isotropic function of vectors $u_{i}$ and $v_{i}$, symmetric tensors $A_{ij}$ and $B_{ij}$, and third rank tensor $H_{ijk}$, up to quadratic order

Appendix B

Below we provide the general quadratic nonlinear forms of the constitutive quantities $\eta$, $h_{i}$ and $\sigma _{ij}$ obtained using Tables 6, 7 and 8:

$$\begin{aligned} \eta&= \eta (\rho , \theta , r_{j}, R_{jk}, g_{j}, G_{jk}, A^{(1)}_{jk}, H_{jkl}, {\dot{\theta }}, \ddot{\theta }, {\dot{\theta }}_{,j}, A^{(2)}_{jk}), \end{aligned}$$

(B.1)

$$\begin{aligned}&=\nu _{0} + \nu _{1} {\dot{\theta }} + \frac{1}{2}\, \nu _{2} {\dot{\theta }}^{2} + \nu _{3} \ddot{\theta } + \frac{1}{2}\, \nu _{4} \ddot{\theta }^{2} + \nu _{5} {\dot{\theta }}\,\ddot{\theta } + \frac{1}{2}\, \nu _{6} r_{k} r_{k} + \frac{1}{2}\, \nu _{7} g_{k} g_{k} + \frac{1}{2}\, \nu _{8} {\dot{\theta }}_{,k} {\dot{\theta }}_{k} \nonumber \\&\quad +\nu _{9} r_{k} g_{k} + \nu _{10} r_{k} {\dot{\theta }}_{,k} + \nu _{11} g_{k} {\dot{\theta }}_{,k} + \nu _{12} R_{kk} + \nu _{13} G_{kk} + \nu _{14} A^{(1)}_{kk} + \nu _{15} A^{(2)}_{kk} + \nu ^{(16)} {\dot{\theta }}\, R_{kk} \nonumber \\&\quad +\nu _{17} {\dot{\theta }}\, G_{kk} + \nu _{18} {\dot{\theta }}\, A^{(1)}_{kk} + \nu _{19} {\dot{\theta }}\, A^{(2)}_{kk} + \nu _{20} \ddot{\theta }\, R_{kk} + \nu _{21} \ddot{\theta }\, G_{kk} + \nu _{22} \ddot{\theta }\, A^{(1)}_{kk} \nonumber \\&\quad +\nu _{23} \ddot{\theta }\, A^{(2)}_{kk} + \frac{1}{2}\, \nu _{24} R_{kk}^{2} + \frac{1}{2}\, \nu _{25} G_{kk}^{2} + \frac{1}{2}\, \nu _{26} {A^{(1)}_{kk}}^{2} + \frac{1}{2}\, \nu _{27} {A^{(2)}_{kk}}^{2} + \nu _{28} R_{kl} R_{kl} \nonumber \\&\quad +\nu _{29} G_{kl} G_{kl} + \nu _{30} A^{(1)}_{kl} A^{(1)}_{kl} + \nu _{31} A^{(2)}_{kl} A^{(2)}_{kl} + \nu _{32} H_{llk} H_{mmk} + \nu _{33} H_{kll} H_{kmm} \nonumber \\&\quad +\nu _{34} H_{klm} H_{klm} + \nu _{35} H_{klm} H_{lkm} + \nu _{36} H_{llk} H_{kmm} + \nu _{37} R_{kk} G_{ll} + \nu _{38} R_{kk} A^{(1)}_{ll} \nonumber \\&\quad +\nu _{39} R_{kk} A^{(2)}_{ll} + \nu _{40} G_{kk} A^{(1)}_{ll} + \nu _{41} G_{kk} A^{(2)}_{ll} + \nu _{42} A^{(1)}_{kk} A^{(2)}_{ll} + \nu _{43} R_{kl} G_{kl} \nonumber \\&\quad +\nu _{44} R_{kl} A^{(1)}_{kl} + \nu _{45} R_{kl} A^{(2)}_{kl} + \nu _{46} G_{kl} A^{(1)}_{kl} + \nu _{47} G_{kl} A^{(2)}_{kl} + \nu _{48} A^{(1)}_{kl} A^{(2)}_{kl} \nonumber \\&\quad +\nu _{49} r_{k} H_{kll} + \nu _{50} r_{k} H_{llk} + \nu _{51} g_{k} H_{kll} + \nu _{52} g_{k} H_{llk} + \nu _{53} {\dot{\theta }}_{,k} H_{kll} + \nu _{54} {\dot{\theta }}_{,k} H_{llk}, \end{aligned}$$

(B.2)

$$\begin{aligned} h_{i}&= h_{i}(\rho , \theta , r_{j}, R_{jk}, g_{j}, G_{jk}, A^{(1)}_{jk}, H_{jkl}, {\dot{\theta }}, \ddot{\theta }, {\dot{\theta }}_{,j}, A^{(2)}_{jk}), \end{aligned}$$

(B.3)

$$\begin{aligned}&=\left[ \left( \gamma _{0} + \gamma _{1} {\dot{\theta }} + \gamma _{2} \ddot{\theta } + \gamma _{3} R_{kk} + \gamma _{4} G_{kk} + \gamma _{5} A^{(1)}_{kk} + \gamma _{6} A^{(2)}_{kk} \right) \delta _{il}\, \right. \nonumber \\&\quad +\left. \gamma _{7} R_{il} + \gamma _{8} G_{il} + \gamma _{9} A^{(1)}_{il} + \gamma _{10} A^{(2)}_{il} \right] r_{l}\, \nonumber \\&\quad +\left[ \left( \gamma _{11} + \gamma _{12} {\dot{\theta }} + \gamma _{13} \ddot{\theta } + \gamma _{14} R_{kk} + \gamma _{15} G_{kk} + \gamma _{16} A^{(1)}_{kk} + \gamma _{17} A^{(2)}_{kk} \right) \delta _{il}\, \right. \nonumber \\&\quad + \left. \gamma _{18} R_{il} + \gamma _{19} G_{il} + \gamma _{20} A^{(1)}_{il} + \gamma _{21} A^{(2)}_{il} \right] g_{l}\, \nonumber \\&\quad +\left[ \left( \gamma _{22} + \gamma _{23} {\dot{\theta }} + \gamma _{24} \ddot{\theta } + \gamma _{25} R_{kk} + \gamma _{26} G_{kk} + \gamma _{27} A^{(1)}_{kk} + \gamma _{28} A^{(2)}_{kk} \right) \delta _{il}\, \right. \nonumber \\&\quad +\left. \gamma _{29} R_{il} + \gamma _{30} G_{il} + \gamma _{31} A^{(1)}_{il} + \gamma _{32} A^{(2)}_{il} \right] {\dot{\theta }}_{,l}\, \nonumber \\&\quad +\left( \gamma _{33} + \gamma _{34} {\dot{\theta }} + \gamma _{35} \ddot{\theta } + \gamma _{36} R_{kk} + \gamma _{37} G_{kk} + \gamma _{38} A^{(1)}_{kk} + \gamma _{39} A^{(2)}_{kk} \right) H_{ill}\, \nonumber \\&\quad +\left( \gamma _{40} + \gamma _{41} {\dot{\theta }} + \gamma _{42} \ddot{\theta } + \gamma _{43} R_{kk} + \gamma _{44} G_{kk} + \gamma _{45} A^{(1)}_{kk} + \gamma _{46} A^{(2)}_{kk} \right) H_{lli}\, \nonumber \\&\quad +\left( \gamma _{47} H_{ijk} + \gamma _{48} H_{jki} \right) R_{jk} + \left( \gamma _{49} H_{ijk} + \gamma _{50} H_{jki} \right) G_{jk} + \left( \gamma _{51} H_{ijk} + \gamma _{52} H_{jki} \right) A^{(1)}_{jk}\, \nonumber \\&\quad +\left( \gamma _{53} H_{ijk} + \gamma _{54} H_{jki} \right) A^{(2)}_{jk}, \end{aligned}$$

(B.4)

$$\begin{aligned} \sigma _{ij}&= \sigma _{i j}(\rho , \theta , r_{j}, R_{jk}, g_{j}, G_{jk}, A^{(1)}_{jk}, H_{jkl}, {\dot{\theta }}, \ddot{\theta }, {\dot{\theta }}_{,j}, A^{(2)}_{jk}), \end{aligned}$$

(B.5)

$$\begin{aligned}&= \left( \alpha _{0} + \alpha _{1} {\dot{\theta }} + \frac{1}{2}\, \alpha _{2} {\dot{\theta }}^{2} + \alpha _{3} \ddot{\theta } + \frac{1}{2}\, \alpha _{4} \ddot{\theta }^{2} + \alpha _{5} {\dot{\theta }}\,\ddot{\theta } + \frac{1}{2}\, \alpha _{6} r_{k} r_{k} + \frac{1}{2}\, \alpha _{7} g_{k} g_{k} \right. \nonumber \\&\quad +\frac{1}{2}\, \alpha _{8} {\dot{\theta }}_{,k} {\dot{\theta }}_{,k} + \alpha _{9} r_{k} g_{k} + \alpha _{10} r_{k} {\dot{\theta }}_{,k} + \alpha _{11} g_{k} {\dot{\theta }}_{,k} + \alpha _{12} R_{kk} + \alpha _{13} G_{kk} + \alpha _{14} A^{(1)}_{kk} \nonumber \\&\quad +\alpha _{15} A^{(2)}_{kk} + \alpha _{16} {\dot{\theta }}\, R_{kk} + \alpha _{17} {\dot{\theta }}\, G_{kk} + \alpha _{18} {\dot{\theta }}\, A^{(1)}_{kk} + \alpha _{19} {\dot{\theta }}\, A^{(2)}_{kk} \nonumber \\&\quad +\alpha _{20} \ddot{\theta }\, R_{kk} + \alpha _{21} \ddot{\theta }\, G_{kk} + \alpha _{22} \ddot{\theta }\, A^{(1)}_{kk} + \alpha _{23} \ddot{\theta }\, A^{(2)}_{kk} + \frac{1}{2}\, \alpha _{24} R_{kk}^{2} \nonumber \\&\quad +\frac{1}{2}\, \alpha _{25} G_{kk}^{2} + \frac{1}{2}\, \alpha _{26} {A^{(1)}_{kk}}^{2} + \frac{1}{2}\, \alpha _{27} {A^{(2)}_{kk}}^{2} + \alpha _{28} R_{kl} R_{kl} + \alpha _{29} G_{kl} G_{kl} \nonumber \\&\quad +\alpha _{30} A^{(1)}_{kl} A^{(1)}_{kl} + \alpha _{31} A^{(2)}_{kl} A^{(2)}_{kl} + \alpha _{32} H_{llk} H_{mmk} + \alpha _{33} H_{kll} H_{kmm} + \alpha _{34} H_{klm} H_{klm} \nonumber \\&\quad +\alpha _{35} H_{klm} H_{lkm} + \alpha _{36} H_{llk} H_{kmm} + \alpha _{37} R_{kk} G_{ll} + \alpha _{38} R_{kk} A^{(1)}_{ll} + \alpha _{39} R_{kk} A^{(2)}_{ll} \nonumber \\&\quad +\alpha _{40} G_{kk} A^{(1)}_{ll} + \alpha _{41} G_{kk} A^{(2)}_{ll} + \alpha _{42} A^{(1)}_{kk} A^{(2)}_{ll} + \alpha _{43} R_{kl} G_{kl} + \alpha _{44} R_{kl} A^{(1)}_{kl} \nonumber \\&\quad +\alpha _{45} R_{kl} A^{(2)}_{kl} + \alpha _{46} G_{kl} A^{(1)}_{kl} + \alpha _{47} G_{kl} A^{(2)}_{kl} + \alpha _{48} A^{(1)}_{kl} A^{(2)}_{kl} + \alpha _{49} r_{k} H_{kll} \nonumber \\&\quad +\alpha _{50} r_{k} H_{llk} + \alpha _{51} g_{k} H_{kll} + \alpha _{52} g_{k} H_{llk} + \alpha _{53} {\dot{\theta }}_{,k} H_{kll} + \alpha _{54} {\dot{\theta }}_{,k} H_{llk} \bigg ) \delta _{ij}\, \nonumber \\&\quad +\left( \alpha _{55} + \alpha _{56} {\dot{\theta }} + \alpha _{57} \ddot{\theta } + \alpha _{58} R_{kk} + \alpha _{59} G_{kk} + \alpha _{60} A^{(1)}_{kk} + \alpha _{61} A^{(2)}_{kk} \right) R_{ij} \nonumber \\&\quad +\left( \alpha _{62} + \alpha _{63} {\dot{\theta }} + \alpha _{64} \ddot{\theta } + \alpha _{65} R_{kk} + \alpha _{66} G_{kk} + \alpha _{67} A^{(1)}_{kk} + \alpha _{68} A^{(2)}_{kk} \right) G_{ij} \nonumber \\&\quad +\left( \alpha _{69} + \alpha _{70} {\dot{\theta }} + \alpha _{71} \ddot{\theta } + \alpha _{72} R_{kk} + \alpha _{73} G_{kk} + \alpha _{74} A^{(1)}_{kk} + \alpha _{75} A^{(2)}_{kk} \right) A^{(1)}_{ij} \nonumber \\&\quad +\left( \alpha _{76} + \alpha _{77} {\dot{\theta }} + \alpha _{78} \ddot{\theta } + \alpha _{79} R_{kk} + \alpha _{80} G_{kk} + \alpha _{81} A^{(1)}_{kk} + \alpha _{82} A^{(2)}_{kk} \right) A^{(1)}_{ij} \nonumber \\&\quad +\alpha _{83} r_{i} r_{j} + \alpha _{84} g_{i} g_{j} + \alpha _{85} {\dot{\theta }}_{,i} {\dot{\theta }}_{,j} + \alpha _{86} \overline{r_{i} g_{j}} + \alpha _{87} \overline{r_{i} {\dot{\theta }}_{,j}} + \alpha _{88} \overline{g_{i} {\dot{\theta }}_{,j}} + \alpha _{89} R_{ik} R_{kj} \nonumber \\&\quad +\alpha _{90} G_{ik} G_{kj} + \alpha _{91} A^{(1)}_{ik} A^{(1)}_{kj} + \alpha _{92} A^{(2)}_{ik} A^{(2)}_{kj} + \alpha _{93} H_{kki} H_{llj} + \alpha _{94} H_{ikk} H_{jll} \nonumber \\&\quad + \alpha _{95} H_{kil} H_{kjl} + \alpha _{96} H_{ikl} H_{jkl} + \alpha _{97} \overline{H_{kki} H_{jll}} + \alpha _{98} \overline{H_{ikl} H_{klj}} + \alpha _{99} \overline{R_{ik} G_{kj}} \nonumber \\&\quad +\alpha _{100} \overline{R_{ik} A^{(1)}_{kj}} + \alpha _{101} \overline{R_{ik} A^{(2)}_{kj}} + \alpha _{102} \overline{G_{ik} A^{(1)}_{kj}} + \alpha _{103} \overline{G_{ik} A^{(2)}_{kj}} + \alpha _{104} \overline{A^{(1)}_{ik} A^{(2)}_{kj}} \nonumber \\&\quad +\alpha _{105} \overline{r_{k} H_{ijk}} + \alpha _{106} r_{k} H_{kij} + \alpha _{107} \overline{g_{k} H_{ijk}} + \alpha _{108} g_{k} H_{kij} + \alpha _{109} \overline{{\dot{\theta }}_{,k} H_{ijk}} \nonumber \\&\quad +\alpha _{110} {\dot{\theta }}_{,k} H_{kij}. \end{aligned}$$

(B.6)

Appendix C

We may write the following symmetric matrices as sums of traceless and diagonal matrices:

$$\begin{aligned} A^{(1)}_{ij} = A^{(1)}_{\left\langle ij\right\rangle } + \frac{1}{3}\, A^{(1)}_{ll} \delta _{ij} \qquad \mathrm{and} \qquad A^{(1)}_{\left\langle ij\right\rangle } A^{(1)}_{\left\langle jk\right\rangle } = A^{(1)}_{\left\langle \left\langle ij\right\rangle \right. } A^{(1)}_{\left. \left\langle jk\right\rangle \right\rangle } + \frac{1}{3}\, A^{(1)}_{\left\langle lj\right\rangle } A^{(1)}_{\left\langle jl\right\rangle } \delta _{ik}. \end{aligned}$$

(C.1)

With (C.1), we have

$$\begin{aligned} A^{(1)}_{\left\langle ij\right. } A^{(1)}_{\left. jk\right\rangle } = A^{(1)}_{ij} A^{(1)}_{jk} - \frac{1}{3}\, A^{(1)}_{lj} A^{(1)}_{jl} \delta _{ik} = A^{(1)}_{\left\langle \left\langle ij\right\rangle \right. } A^{(1)}_{\left. \left\langle jk\right\rangle \right\rangle } + \frac{2}{3}\, A^{(1)}_{ll} A^{(1)}_{\left\langle ik\right\rangle }. \end{aligned}$$

(C.2)

Also, with (C.1), (C.2) and (12), and noting that $A^{(1)}_{lj} W_{jl} = 0$, we obtain

$$\begin{aligned} A^{(1)}_{\left\langle ij\right. } L_{\left. jk\right\rangle } = \frac{1}{2} A^{(1)}_{\left\langle \left\langle ij\right\rangle \right. } A^{(1)}_{\left. \left\langle jk\right\rangle \right\rangle } + \frac{1}{3}\, A^{(1)}_{ll} A^{(1)}_{\left\langle ik\right\rangle } + A^{(1)}_{\left\langle ij \right\rangle } W_{jk} + \frac{1}{3}\, A^{(1)}_{ll} W_{ik}. \end{aligned}$$

(C.3)

Subsequently, from (11) and (C.3), and noting that $W_{ji} = - W_{ij}$, it follows that

$$\begin{aligned} A^{(2)}_{\left\langle ij \right\rangle }&= {\dot{A}}^{(1)}_{\left\langle ij \right\rangle } + A^{(1)}_{\left\langle il\right. } L_{\left. lj\right\rangle } + A^{(1)}_{\left\langle jl\right. } L_{\left. li\right\rangle } \nonumber \\&= {\dot{A}}^{(1)}_{\left\langle ij \right\rangle } + A^{(1)}_{\left\langle \left\langle il\right\rangle \right. } A^{(1)}_{\left. \left\langle lj\right\rangle \right\rangle } + \frac{2}{3} A^{(1)}_{ll} A^{(1)}_{\left\langle ij \right\rangle } - W_{il} A^{(1)}_{\left\langle lj \right\rangle } + A^{(1)}_{\left\langle il \right\rangle } W_{lj}. \end{aligned}$$

(C.4)