# Nonlinear finite elements/Entropy inequality

Proof:
Assume that $\Omega$ is an arbitrary fixed control volume. Then $u_{n}=0$ and the derivative can be taken inside the integral to give
    $\int _{\Omega }{\frac {\partial }{\partial t}}(\rho ~\eta )~{\text{dV}}\geq -\int _{\partial \Omega }\rho ~\eta ~(\mathbf {v} \cdot \mathbf {n} )~{\text{dA}}-\int _{\partial \Omega }{\cfrac {\mathbf {q} \cdot \mathbf {n} }{T}}~{\text{dA}}+\int _{\Omega }{\cfrac {\rho ~s}{T}}~{\text{dV}}~.$
Using the divergence theorem, we get
    $\int _{\Omega }{\frac {\partial }{\partial t}}(\rho ~\eta )~{\text{dV}}\geq -\int _{\Omega }{\boldsymbol {\nabla }}\cdot (\rho ~\eta ~\mathbf {v} )~{\text{dV}}-\int _{\Omega }{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)~{\text{dV}}+\int _{\Omega }{\cfrac {\rho ~s}{T}}~{\text{dV}}~.$
Since $\Omega$ is arbitrary, we must have
    ${\frac {\partial }{\partial t}}(\rho ~\eta )\geq -{\boldsymbol {\nabla }}\cdot (\rho ~\eta ~\mathbf {v} )-{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.$
Expanding out
    ${\frac {\partial \rho }{\partial t}}~\eta +\rho ~{\frac {\partial \eta }{\partial t}}\geq -{\boldsymbol {\nabla }}(\rho _{\eta })\cdot \mathbf {v} -\rho ~\eta ~({\boldsymbol {\nabla }}\cdot \mathbf {v} )-{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}$
or,
    ${\frac {\partial \rho }{\partial t}}~\eta +\rho ~{\frac {\partial \eta }{\partial t}}\geq -\eta ~{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} -\rho ~{\boldsymbol {\nabla }}\eta \cdot \mathbf {v} -\rho ~\eta ~({\boldsymbol {\nabla }}\cdot \mathbf {v} )-{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}$
or,
    $\left({\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} +\rho ~{\boldsymbol {\nabla }}\cdot \mathbf {v} \right)~\eta +\rho ~\left({\frac {\partial \eta }{\partial t}}+{\boldsymbol {\nabla }}\eta \cdot \mathbf {v} \right)\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.$
Now, the material time derivatives of $\rho$ and $\eta$ are given by
    ${\dot {\rho }}={\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} ~;~~{\dot {\eta }}={\frac {\partial \eta }{\partial t}}+{\boldsymbol {\nabla }}\eta \cdot \mathbf {v} ~.$
Therefore,
    $\left({\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\cdot \mathbf {v} \right)~\eta +\rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.$
From the conservation of mass ${\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\cdot \mathbf {v} =0$. Hence,
    ${\rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.}$
Proof:
Using the identity ${\boldsymbol {\nabla }}\cdot (\varphi ~\mathbf {v} )=\varphi ~{\boldsymbol {\nabla }}\cdot \mathbf {v} +\mathbf {v} \cdot {\boldsymbol {\nabla }}\varphi$ in the Clausius-Duhem inequality, we get
    $\rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}\qquad {\text{or}}\qquad \rho ~{\dot {\eta }}\geq -{\cfrac {1}{T}}~{\boldsymbol {\nabla }}\cdot \mathbf {q} -\mathbf {q} \cdot {\boldsymbol {\nabla }}\left({\cfrac {1}{T}}\right)+{\cfrac {\rho ~s}{T}}~.$
Now, using index notation with respect to a Cartesian basis $\mathbf {e} _{j}$,
    ${\boldsymbol {\nabla }}\left({\cfrac {1}{T}}\right)={\frac {\partial }{\partial x_{j}}}\left(T^{-1}\right)~\mathbf {e} _{j}=-\left(T^{-2}\right)~{\frac {\partial T}{\partial x_{j}}}~\mathbf {e} _{j}=-{\cfrac {1}{T^{2}}}~{\boldsymbol {\nabla }}T~.$
Hence,
    $\rho ~{\dot {\eta }}\geq -{\cfrac {1}{T}}~{\boldsymbol {\nabla }}\cdot \mathbf {q} +{\cfrac {1}{T^{2}}}~\mathbf {q} \cdot {\boldsymbol {\nabla }}T+{\cfrac {\rho ~s}{T}}\qquad {\text{or}}\qquad \rho ~{\dot {\eta }}\geq -{\cfrac {1}{T}}\left({\boldsymbol {\nabla }}\cdot \mathbf {q} -\rho ~s\right)+{\cfrac {1}{T^{2}}}~\mathbf {q} \cdot {\boldsymbol {\nabla }}T~.$
Recall the balance of energy
    $\rho ~{\dot {e}}-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} +{\boldsymbol {\nabla }}\cdot \mathbf {q} -\rho ~s=0\qquad \implies \qquad \rho ~{\dot {e}}-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} =-({\boldsymbol {\nabla }}\cdot \mathbf {q} -\rho ~s)~.$
Therefore,
    $\rho ~{\dot {\eta }}\geq {\cfrac {1}{T}}\left(\rho ~{\dot {e}}-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \right)+{\cfrac {1}{T^{2}}}~\mathbf {q} \cdot {\boldsymbol {\nabla }}T\qquad \implies \qquad \rho ~{\dot {\eta }}~T\geq \rho ~{\dot {e}}-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} +{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}~.$
Rearranging,
    ${\rho ~({\dot {e}}-T~{\dot {\eta }})-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \leq -{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}~.}$