Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature

Abstract

A fluid occupying a mechanically isolated vessel with walls kept at spatially non-uniform temperature is in the long run expected to reach the spatially inhomogeneous steady state. Irrespective of the initial conditions the velocity field is expected to vanish, and the temperature field is expected to be fully determined by the steady heat equation. This simple observation is however difficult to prove using the corresponding governing equations. The main difficulties are the presence of the dissipative heating term in the evolution equation for temperature and the lack of control on the heat fluxes through the boundary. Using thermodynamical-based arguments, it is shown that these difficulties in the proof can be overcome, and it is proved that the velocity and temperature perturbations to the steady state actually vanish as the time goes to infinity.

Appendices

Derivation of the formulae for the time derivatives of the functionals

The derivation presented in the main text can be greatly simplified if we restrict ourselves to formal manipulations. We recall that the main objective is to find the time derivative of the functional

$$\begin{aligned}&{\mathscr {V}}_{\mathrm {meq}} \left( \left. \widetilde{\varvec{W}} \right\| \widehat{\varvec{W}} \right) = - \int _{\varOmega } \rho \left[ {\widehat{\theta }} \eta ({\widehat{\theta }} + {\widetilde{\theta }}) - {\widehat{\theta }} \eta ({\widehat{\theta }}) - {\widehat{\theta }} \left. \frac{\partial {\eta }}{\partial {\theta }} \right| _{\theta = {\widehat{\theta }}} {\widetilde{\theta }} - e({\widehat{\theta }} + {\widetilde{\theta }}) + e({\widehat{\theta }}) + \left. \frac{\partial {e}}{\partial {\theta }} \right| _{\theta = {\widehat{\theta }}} {\widetilde{\theta }} \right] \, \mathrm {d}\mathrm {v}\nonumber \\&\quad + \int _{\varOmega } \rho \frac{1}{2} \left| {\widetilde{\varvec{v}}}\right| ^2 \, \mathrm {d}\mathrm {v}= - \int _{\varOmega } \rho \left[ {\widehat{\theta }} \left( \eta ({\widehat{\theta }} + {\widetilde{\theta }}) - \eta ({\widehat{\theta }}) \right) - \left( e({\widehat{\theta }} + {\widetilde{\theta }}) - e({\widehat{\theta }}) \right) \right] \, \mathrm {d}\mathrm {v}+ \int _{\varOmega } \rho \frac{1}{2} \left| {\widetilde{\varvec{v}}}\right| ^2 \, \mathrm {d}\mathrm {v}, \end{aligned}$$

(A.1)

where we have used the identity ${\widehat{\theta }} \left. \frac{\partial {\eta }}{\partial {\theta }} \right| _{\theta = {\widehat{\theta }}} = \left. \frac{\partial {e}}{\partial {\theta }} \right| _{\theta = {\widehat{\theta }}}$, which holds universally for any material. In the particular case of the fluid described by the Helmholtz free energy in the form (2.2), we get the explicit formula for the functional ${\mathscr {V}}_{\mathrm {meq}}$, see Lemma 3, Eq. (3.1). It is convenient to rewrite (3.1) in terms of the relative entropy, $\eta _{\mathrm {diff}}= c_{\mathrm {V}, \mathrm {ref}}\ln \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}}\right)$, see (3.10) and (3.11). If we do so, the counterpart of (3.1) reads

$$\begin{aligned} {\mathscr {V}}_{\mathrm {meq}} \left( \left. \widetilde{\varvec{W}} \right\| \widehat{\varvec{W}} \right) =_{\mathrm {def}} \int _{\varOmega } \left[ \rho c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} \left[ {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} - \frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}} - 1 \right] + \frac{1}{2} \rho \left| {\widetilde{\varvec{v}}}\right| ^2 \right] \, \mathrm {d}\mathrm {v}. \end{aligned}$$

(A.2)

We know that the evolution equation for the entropy of the perturbed state reads

$$\begin{aligned} \rho \frac{\mathrm {d}{\eta }}{\mathrm {d}{t}} = \frac{ {{\,\mathrm{div}\,}}\left( \kappa ({\widehat{\theta }} + {\widetilde{\theta }}) \nabla \left( {\widehat{\theta }} + {\widetilde{\theta }} \right) \right) }{ {\widehat{\theta }} + {\widetilde{\theta }} } + \frac{\zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) }{{\widehat{\theta }} + {\widetilde{\theta }} } , \end{aligned}$$

(A.3)

which in our case of constant heat conductivity $\kappa$ and vanishing steady-state velocity ${\widehat{\varvec{v}}} = \varvec{0}$ simplifies to

$$\begin{aligned} \rho \frac{\partial {\eta {\left( {\widehat{\theta }} + {\widetilde{\theta }} \right) }}}{\partial {t}} + \rho {\widetilde{\varvec{v}}} \bullet \nabla \eta \left( {\widehat{\theta }} + {\widetilde{\theta }} \right) = \frac{ {{\,\mathrm{div}\,}}\left( \kappa _{\mathrm {ref}}\nabla \left( {\widehat{\theta }} \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) \right) \right) }{ {\widehat{\theta }} \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) } + \frac{ \zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) }{ {\widehat{\theta }} \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) } . \end{aligned}$$

(A.4)

The last equation can be again rewritten in terms of the relative entropy,

$$\begin{aligned} \rho \frac{\partial {\eta _{\mathrm {diff}}}}{\partial {t}} = \frac{ {{\,\mathrm{div}\,}}\left( \kappa _{\mathrm {ref}}\nabla \left( {\widehat{\theta }} {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right) }{ {\widehat{\theta }} {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} } + \frac{ \zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) }{ {\widehat{\theta }} {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} } - \rho {\widetilde{\varvec{v}}} \bullet \nabla \eta _{\mathrm {diff}} - \rho {\widetilde{\varvec{v}}} \bullet \nabla {\widehat{\eta }} . \end{aligned}$$

(A.5)

Note that at this point, we heavily exploit the fact that in our case the relative entropy is given by a formula that nicely matches with the structure of the heat flux term. Using the evolution equation for the relative entropy, one can derive the evolution equation for any quantity of the form $c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} f ( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} )$ , where f is a given function, see Lemma 7, Eq. (A.6). Using the lemma for particular choices of f then allows us to seamlessly derive formulae for the time derivatives of the functionals introduced in Sects. 3 and 4.

Lemma 7

(Pointwise evolution equation for functions of the exponential of the relative entropy) Let $\frac{\mathrm {d}{_{{\widetilde{\varvec{v}}}}}}{\mathrm {d}{t}}= \frac{\partial {}}{\partial {t}} + {\widetilde{\varvec{v}}} \bullet \nabla$ denote the material time derivative with respect to the perturbed velocity field ${\widetilde{\varvec{v}}}$, and let f denote a given function. The evolution equation for the quantity $c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} f ( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} )$ reads

$$\begin{aligned}&\rho \frac{\mathrm {d}{_{{\widetilde{\varvec{v}}}}}}{\mathrm {d}{t}} \left[ c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right] = {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left( {\widehat{\theta }} f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right) \right] - \kappa _{\mathrm {ref}}{\widehat{\theta }} f^{\prime \prime } \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \bullet \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \nonumber \\&\quad + f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) + \rho c_{\mathrm {V}, \mathrm {ref}}\left[ f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) - f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right] {\widetilde{\varvec{v}}} \bullet \nabla {\widehat{\theta }} . \end{aligned}$$

(A.6)

Proof

The proof is based on the direct computation. Using the evolution equation for the relative entropy, see (A.5), we see that

$$\begin{aligned}&\rho \frac{\partial {}}{\partial {t}} \left( {\widehat{\theta }} c_{\mathrm {V}, \mathrm {ref}}f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right) = \rho {\widehat{\theta }} f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \frac{\partial { \eta _{\mathrm {diff}}}}{\partial { t }} = f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left( {\widehat{\theta }} {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right] \nonumber \\&\quad + f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) - \rho {\widehat{\theta }} f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} {\widetilde{\varvec{v}}} \bullet \nabla \eta _{\mathrm {diff}} - \rho {\widehat{\theta }} f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} {\widetilde{\varvec{v}}} \bullet \nabla {\widehat{\eta }} ,\nonumber \\ \end{aligned}$$

(A.7)

where $f^\prime$ denotes the derivative of function f with respect to its argument. The first term on the right-hand side can be rewritten as a divergence term plus a source term,

$$\begin{aligned}&f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left( {\widehat{\theta }} {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right] = {{\,\mathrm{div}\,}}\left[ f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \kappa _{\mathrm {ref}}\nabla {\widehat{\theta }} + f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \kappa _{\mathrm {ref}}{\widehat{\theta }} \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right] \nonumber \\&\quad - \kappa _{\mathrm {ref}} \nabla {\widehat{\theta }} \bullet {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \nabla f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) - \kappa _{\mathrm {ref}}{\widehat{\theta }} \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \bullet \nabla f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) . \end{aligned}$$

(A.8)

Now we use the identity ${\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \nabla f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) = \nabla \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) - f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right)$, which allows us to rewrite the corresponding term on right-hand side of (A.8) as

$$\begin{aligned} \kappa _{\mathrm {ref}} \nabla {\widehat{\theta }} \bullet {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \nabla f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) = {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\left( \nabla {\widehat{\theta }} \right) \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) - f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right) \right] , \end{aligned}$$

(A.9)

where we have used the fact that ${{\,\mathrm{div}\,}}\left( \kappa _{\mathrm {ref}}\nabla {\widehat{\theta }} \right) = 0$. Using (A.9) on the right-hand side of (A.8) reveals that

$$\begin{aligned}&f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left( {\widehat{\theta }} {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right] \nonumber \\&\quad = {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left( {\widehat{\theta }} f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right) \right] - \kappa _{\mathrm {ref}}{\widehat{\theta }} f^{\prime \prime } \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \bullet \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} . \end{aligned}$$

(A.10)

This finishes the manipulation with the first term on the right-hand side of (A.7). Now we focus on the prospective convective term on the right-hand side of (A.7) . We see that

$$\begin{aligned} \rho {\widehat{\theta }} f^\prime \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} {\widetilde{\varvec{v}}} \bullet \nabla \eta _{\mathrm {diff}} = \rho {\widetilde{\varvec{v}}} \bullet \nabla \left[ c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) \right] - \rho c_{\mathrm {V}, \mathrm {ref}}f \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) {\widetilde{\varvec{v}}} \bullet \nabla {\widehat{\theta }} . \end{aligned}$$

(A.11)

Next we observe that $\nabla {{\widehat{\eta }}} = c_{\mathrm {V}, \mathrm {ref}}\frac{\nabla {{\widehat{\theta }}}}{{\widehat{\theta }}}$, and using (A.10) and (A.11) in (A.7) we finally obtain the evolution Eq. (A.6). $\square$

Now we are in the position to exploit Lemma 7 in the derivation of formulae for the time derivative of the functionals introduced in Lemmas 3 and 5. Indeed, if we set

$$\begin{aligned} f(y) =_{\mathrm {def}} y - \ln y - 1, \end{aligned}$$

(A.12)

then $f^\prime (y) = 1 - \frac{1}{y}$, $f^{\prime \prime }(y) = \frac{1}{y^2}$, and using Lemma 7 we obtain the pointwise evolution equation

$$\begin{aligned}&\rho \frac{\mathrm {d}{_{{\widetilde{\varvec{v}}}}}}{\mathrm {d}{t}} \left[ c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} - \frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}} - 1 \right) \right] = {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left( {\widehat{\theta }} \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} - \frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}} - 1 \right) \right) \right] - \kappa _{\mathrm {ref}}{\widehat{\theta }} \frac{ \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \bullet \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} }{ \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) ^2 } \nonumber \\&\quad + \zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) - \frac{ \zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) }{ {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} } - \rho \eta _{\mathrm {diff}} {\widetilde{\varvec{v}}} \bullet \nabla {\widehat{\theta }} . \end{aligned}$$

(A.13)

The last equation can be upon straightforward manipulations and upon using the fact that $\zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) = 2 \mu {\widetilde{\mathbb {D}}} : {\widetilde{\mathbb {D}}}$ rewritten in terms of the temperature

$$\begin{aligned}&\rho \frac{\mathrm {d}{_{{\widetilde{\varvec{v}}}}}}{\mathrm {d}{t}} \left[ c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} \left( \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} - \ln \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) \right) \right] = {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left\{ {\widehat{\theta }} \left( \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} - \ln \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) \right) \right\} \right] \nonumber \\&\quad - \kappa _{\mathrm {ref}}{\widehat{\theta }} \nabla \ln \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) \bullet \nabla \ln \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) + 2 \mu {\widetilde{\mathbb {D}}} : {\widetilde{\mathbb {D}}} - \frac{ 2 \mu {\widetilde{\mathbb {D}}} : {\widetilde{\mathbb {D}}} }{ 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} } - \rho c_{\mathrm {V}, \mathrm {ref}}\ln \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) {\widetilde{\varvec{v}}} \bullet \nabla {\widehat{\theta }} . \end{aligned}$$

(A.14)

This is the sought pointwise evolution equation for the thermal part of the integrand in the functional ${\mathscr {V}}_{\mathrm {meq}}$. Integrating (A.14) over the (material) domain $\varOmega$ leads directly to the result reported in Lemma 3. (Recall that the boundary condition $\left. {\widetilde{\theta }} \right| _{\partial \varOmega } = 0$ guarantees that the boundary terms vanish).

On the other hand, if we choose $m \in {{\mathbb {R}}}$, and if we set

$$\begin{aligned} f(y) =_{\mathrm {def}} y - \frac{1}{m} \left( y^m -1\right) - 1, \end{aligned}$$

(A.15)

then $f^\prime (y) = 1 - y^{m-1}$, $f^{\prime \prime }(y) = -(m-1) y^{m-2}$, and using Lemma 7 we obtain the pointwise evolution equation

$$\begin{aligned}&\rho \frac{\mathrm {d}{_{{\widetilde{\varvec{v}}}}}}{\mathrm {d}{t}} \left[ c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} - \frac{1}{m} \left( \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}}\right) ^m -1 \right) - 1 \right) \right] \nonumber \\&\quad = {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left\{ {\widehat{\theta }} \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} - \frac{1}{m} \left( \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}}\right) ^m -1 \right) - 1 \right) \right\} \right] \nonumber \\&\qquad +\, (m-1) \kappa _{\mathrm {ref}}{\widehat{\theta }} \frac{ \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \bullet \nabla {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} }{ \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}} \right) ^{2-m} } + \zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) - \frac{ \zeta _{\mathrm {mech}}\left( \widehat{\varvec{W}} + \widetilde{\varvec{W}}\right) }{ \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}}\right) ^{1-m} }\nonumber \\&\qquad +\, \rho c_{\mathrm {V}, \mathrm {ref}}\frac{m-1}{m} \left( \left( {\mathrm e}^{\frac{\eta _{\mathrm {diff}}}{c_{\mathrm {V}, \mathrm {ref}}}}\right) ^m-1 \right) {\widetilde{\varvec{v}}} \bullet \nabla {\widehat{\theta }} . \end{aligned}$$

(A.16)

Using the specific formula for the mechanical dissipation, and rewriting (A.13) in terms of the temperature we get

$$\begin{aligned}&\rho \frac{\mathrm {d}{_{{\widetilde{\varvec{v}}}}}}{\mathrm {d}{t}} \left[ c_{\mathrm {V}, \mathrm {ref}}{\widehat{\theta }} \left( \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} - \frac{1}{m} \left( \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) ^m -1 \right) \right) \right] = {{\,\mathrm{div}\,}}\left[ \kappa _{\mathrm {ref}}\nabla \left\{ {\widehat{\theta }} \left( \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} - \frac{1}{m} \left( \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) ^m -1 \right) \right) \right\} \right] \nonumber \\&\quad - 4 \frac{ 1-m }{ m^2 } \kappa _{\mathrm {ref}}{\widehat{\theta }} \nabla \left[ \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) ^{\frac{m}{2}} -1 \right] \bullet \nabla \left[ \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) ^{\frac{m}{2}} -1 \right] \nonumber \\&\quad + 2 \mu {\widetilde{\mathbb {D}}} : {\widetilde{\mathbb {D}}} - \frac{ 2 \mu {\widetilde{\mathbb {D}}} : {\widetilde{\mathbb {D}}} }{ \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) ^{1-m} } + \rho c_{\mathrm {V}, \mathrm {ref}}\frac{m-1}{m} \left( \left( 1 + \frac{{\widetilde{\theta }}}{{\widehat{\theta }}} \right) ^m-1 \right) {\widetilde{\varvec{v}}} \bullet \nabla {\widehat{\theta }} . \end{aligned}$$

(A.17)

This is the sought pointwise evolution equation for the thermal part of the integrand in the functional ${\mathscr {V}}_{\mathrm {meq}}^{\vartheta ,\, m}$. Integrating (A.14) over the (material) domain $\varOmega$ leads directly to the result reported in Lemma 5.

Auxiliary tools

Lemma 8

Let $m,n \in (0,1)$ and let $n>m$. Let $x \in (-1, +\infty )$ and let us define the function

$$\begin{aligned} f(x,m,n) =_{\mathrm {def}} \frac{1}{n} \left( 1 + x \right) ^{n} - \frac{1}{m} \left( 1 + x \right) ^{m} + \frac{n-m}{mn} . \end{aligned}$$

(B.1)

The function f(x, m, n) is for $x \in (-1, +\infty )$ a non-negative function which vanishes if and only if $x=0$.

Proof

Taking the derivative of (B.1) with respect to x yields $\frac{\mathrm {d}{}}{\mathrm {d}{x}} f(x,m,n) = \left( 1 + x \right) ^{m-1} \left[ \left( 1 + x \right) ^{n-m} - 1 \right]$. If $n>m$, then the derivative of f(x, m, n) is positive for $x>0$ and negative for $x<0$. Further the value of f(x, m, n) at $x=0$ is $f(0,m,n)=0$. Consequently, function f(x, m, n) is for all $x \in (-1, +\infty )$ a non-negative function which vanishes if and only if $x=0$. $\square$

Lemma 9

Let $m,n \in (0,1)$ and let $n>m> \frac{n}{2}$. Let $x \in (-1, +\infty )$ and let us define the function

$$\begin{aligned} g(x,m,n)= & {} {}_{\mathrm {def}} - \left( \left[ \left( 1 + x \right) ^{\frac{m}{2}} - 1 \right] ^2 + \left[ \left( 1 + x \right) ^{\frac{n}{2}} - 1 \right] ^2 \right) \nonumber \\&+\, n \left[ \frac{1}{n} \left( 1 + x \right) ^{n} - \frac{1}{m} \left( 1 + x \right) ^{m} + \frac{n-m}{mn} \right] . \end{aligned}$$

(B.2)

The function g(x, m, n) has the limits $\lim _{x \rightarrow -1+} g(x, m ,n) = - 3 + \frac{n}{m} < 0,$ and $\lim _{x \rightarrow + \infty } g(x, m ,n) = - \infty ,$ and the function g(x, m, n) is non-positive for all $x \in (-1, +\infty )$, and it vanishes if and only if $x=0$.

Proof

Apparently, the function vanishes for $x=0$, and the computation of the limits is straightforward. If we take the derivative of this function with respect to x we get

$$\begin{aligned} \frac{\mathrm {d}{}}{\mathrm {d}{x}} g(x, m ,n) = \frac{m \left( 1+x\right) ^{\frac{m}{2}} \left( 1 - \left( 1+x\right) ^{\frac{m}{2}} \right) + n \left( 1+x\right) ^{\frac{n}{2}} \left( 1 - \left( 1+x\right) ^{m-\frac{n}{2}} \right) }{1+x}, \end{aligned}$$

(B.3)

and we see that the derivative is negative for $x > 0$, and it is positive for $x < 0$. Consequently, we see that $g(x, m ,n) \le 0$ for all $x \in (-1, +\infty )$, and that it vanishes if and only if $x=0$. $\square$

Lemma 10

Let $m, n \in (0,1)$ and $n> m > \frac{n}{2}$, and let $f(x) =_{\mathrm {def}} \frac{1}{n} {\mathrm e}^{n x} - \frac{1}{m} {\mathrm e}^{m x} + \frac{n-m}{mn}$. Let $x_{\mathrm {crit}}$ is an arbitrary negative number, and let l is a given natural number, $l \ge 3$, then there exists a constant L (possibly large) such that $\frac{\left| x\right| ^l}{L} \le f(x)$ holds for all $x \in [x_{\mathrm {crit}}, + \infty )$.

Proof

Using the series expansion for the exponential, we can observe that $\frac{1}{n} {\mathrm e}^{n x} - \frac{1}{m} {\mathrm e}^{m x} + \frac{n-m}{mn} = \sum _{k=2}^{+\infty } \frac{n^{k-1}-m^{k-1}}{k!} x^k$, and since $n>m$, we see that the inequality

$$\begin{aligned} \frac{n^{l-1}-m^{l-1}}{l!} x^l \le \frac{1}{n} {\mathrm e}^{n x} - \frac{1}{m} {\mathrm e}^{m x} + \frac{n-m}{mn} \end{aligned}$$

(B.4)

holds for any non-negative x and arbitrary $l\in {{\mathbb {N}}}$, $l \ge 2$, and the equality occurs if and only if $x=0$. Hence if we define function h as

$$\begin{aligned} h(x) =_{\mathrm {def}} \frac{n^{l-1}-m^{l-1}}{l!} \left| x\right| ^l, \end{aligned}$$

(B.5)

we see that $h(x) < f(x)$ for all positive x and $h(x) = f(x)$ for $x=0$.

Moreover, a straightforward calculation also shows that the function $f(x)-h(x)$ has for $l \ge 3$ strict local minimum at zero, hence $h(x)<f(x)$ even in some left neighbourhood of zero, that is for some $x \in (-\varepsilon , 0)$. On the other hand, function h(x) is for $x<0$ unbounded, while f(x) is bounded, hence they must intersect, $h(x_{\mathrm {int}}) = f (x_{\mathrm {int}})$, at some point $x_{\mathrm {int}}$, where $x_{\mathrm {int}} < 0$. (Among possible intersecting points we chose the one with the smallest magnitude). If $x_{\mathrm {int}} \le x_{\mathrm {crit}}$, then we are done, and we have found the desired function. In this case L is given by the formula $\frac{1}{L}= \frac{n^{l-1}-m^{l-1}}{l!}$.

If $x_{\mathrm {crit}} < x_{\mathrm {int}}$, it remains to flatten the graph of h(x) such that the intersection point is moved sufficiently far to the left. This can be done by the means of the transformation $g(x) =_{\mathrm {def}} \frac{1}{K} h \left( x \right) ,$ where K is a sufficiently large positive constant. If we choose $\frac{1}{K} = _{\mathrm {def}} \frac{h(x_{\mathrm {int}})}{h(x_{\mathrm {crit}})}$, then $\frac{1}{K} < 1$. (Function h is for $x<0$ a strictly decreasing function). We can observe that $g(x) < f(x)$ for $x \in [x_{\mathrm {int}}, + \infty )$. On the other hand, if $x \in [x_{\mathrm {crit}}, x_{\mathrm {int}})$ then

$$\begin{aligned} g(x) = \frac{h(x_{\mathrm {int}})}{h(x_{\mathrm {crit}})} h(x)< h(x_{\mathrm {int}}) \le f(x_{\mathrm {int}}) < f(x) , \end{aligned}$$

(B.6)

where we have exploited the fact that f is for $x<0$ a strictly decreasing function. Consequently $g(x) \le f(x)$ holds for all $x \in [x_{\mathrm {crit}}, + \infty )$ as desired. Moreover, the constant L is given by the formula $\frac{1}{L} = \frac{f(x_{\mathrm {int}})}{\left| x_{\mathrm {crit}}\right| ^l}$. $\square$

Auxiliary tools—inequalities in function spaces

For further reference we recall the standard embedding theorems for Lebesgue and Sobolev spaces, as well as other standard inequalities, for proofs see for example [8] or [1].

Lemma 11

(Trivial embedding of Lebesgue spaces) Let $\varOmega$ be a domain with a finite volume, and let $p_1, p_2 \in [1, + \infty ]$ such that $p_2 \ge p_1$, then $\left\| f\right\| _{L^{p_1} \left( \varOmega \right) } \le \left| \varOmega \right| ^{\frac{p_2 - p_1}{p_1 p_2}} \left\| f\right\| _{L^{p_2} \left( \varOmega \right) }$.

Lemma 12

(Continuous embedding of Sobolev spaces into Lebesgue spaces) Let $\varOmega \subset {{\mathbb {R}}}^n$ be a domain with ${\mathscr {C}}^{0,1}$ boundary and let $p \in [1, n)$. Then for all $q \in [1, p^\star ]$ where $p^{\star } = _{\mathrm {def}} \frac{np}{n-p}$ there exists a constant $C_{\mathrm {S}}$ that depends on $\varOmega$, p, n and q such that $\left\| f\right\| _{L^{q} \left( \varOmega \right) } \le C_{\mathrm {S}} \left\| f\right\| _{W^{1, p} \left( \varOmega \right) }$. The constant $C_{\mathrm {S}}$ is referred to as the Sobolev embedding constant.

Lemma 13

(Poincaré inequality) Let $f \in W_{0}^{1, p} \left( \varOmega \right)$, where $p \in [1, +\infty )$ and $\varOmega \in {{\mathscr {C}}}^{0,1}$. Then there exists a constant C that depends only on $\varOmega$ and p such that $\left\| f\right\| _{L^{p} \left( \varOmega \right) } \le C(\varOmega , p) \left\| \nabla f\right\| _{L^{p} \left( \varOmega \right) }$. In particular, if $p=2$ then $\left\| f\right\| _{L^{2} \left( \varOmega \right) }^2 \le C_{\mathrm {P}} \left\| \nabla f\right\| _{L^{2} \left( \varOmega \right) }^2$, where $C_{\mathrm {P}}$ is referred to as the Poincaré constant.

Lemma 14

(Hölder inequality) Let $\left\{ p_i\right\} _{i=1}^k$ be a sequence of exponents such that $p_i \in [1, +\infty ]$ and $\sum _{k=1}^{+\infty } \frac{1}{p_k} = 1$. Let us consider functions $\left\{ f_i\right\} _{i=1}^k$, such that $f_i \in L^{p_i} \left( \varOmega \right)$, then $\left\| \prod _{i=1}^k f_i\right\| _{L^{1} \left( \varOmega \right) } \le \prod _{i=1}^k \left\| f_i\right\| _{L^{p_i} \left( \varOmega \right) }$.

Lemma 15

($\varepsilon$-Young inequality) Let $a, b \in {{\mathbb {R}}}^+$, $\varepsilon \in {{\mathbb {R}}}^+$ and $p,q \in (1, + \infty )$ such that $\frac{1}{p} + \frac{1}{q} = 1$. Then $ab \le \varepsilon a^p + \frac{1}{\left( \varepsilon p \right) ^{\frac{q}{p}}q} b^q$.

Lemma 16

(Korn equality) Let $\varOmega$ be a bounded domain with smooth boundary. Let $\varvec{v}$ be a smooth vector field that vanishes on the boundary $\left. \varvec{v} \right| _{\partial \varOmega }=0$, then $2 \int _{\varOmega } { \mathbb {D} : \mathbb {D} } \, \mathrm {d}\mathrm {v}= \int _{\varOmega } { \nabla \varvec{v} : \nabla \varvec{v} } \, \mathrm {d}\mathrm {v}+ \int _{\varOmega } { \left( {{\,\mathrm{div}\,}}\varvec{v} \right) ^2 } \, \mathrm {d}\mathrm {v}$.