The Clausius-Duhem inequality can be expressed in integral form as
![{\displaystyle {\cfrac {d}{dt}}\left(\int _{\Omega }\rho ~\eta ~{\text{dV}}\right)\geq \int _{\partial \Omega }\rho ~\eta ~(u_{n}-\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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c80a5ac5e06b678dbefe164ea8398a877a82874)
In differential form the Clusius-Duhem inequality can be written as
![{\displaystyle \rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/947a27ec0e448823f8486e9f0671ba037f15c614)
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Proof:
Assume that
is an arbitrary fixed control volume. Then
and the derivative can be taken inside the integral to give
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb99507cbccb44061a192e8fe58c936f28b97bd0)
Using the divergence theorem, we get
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a25f084c5d7b371cd81562ff51bbdc05d9285b4a)
Since
is arbitrary, we must have
![{\displaystyle {\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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4083afd80d8af890dcc90ec68a754d90409651e1)
Expanding out
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57da30b43ee690d7a1494745d965a0a07eabe7b3)
or,
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/143e62d6936eb5779ed96cecf5290c365416d229)
or,
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/351b62e17340b4b0cdf88290c1806cde2002cae0)
Now, the material time derivatives of
and
are given by
![{\displaystyle {\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} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1375cef7c6175dd2fe9498b580ad9c6837890c)
Therefore,
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94181d76cfbb786a6c31f80e96646b842bf4f63a)
From the conservation of mass
. Hence,
![{\displaystyle {\rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df04f935067f8131d62af8fb8aba312551250822)
In terms of the specific entropy, the Clausius-Duhem inequality is written as
![{\displaystyle \rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2bf51b5615c4b6ad2064551a1e59fc7a82eaf0)
Show that the inequality can be expressed in terms of the internal energy as
![{\displaystyle \rho ~({\dot {e}}-T~{\dot {\eta }})-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \leq -{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a95d449140d8f9deb411c75b63406a22e0437e5)
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Proof:
Using the identity
in the Clausius-Duhem inequality, we get
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5eda71df376dbd1fa5a2ea2d4187ce3c61b40e96)
Now, using index notation with respect to a Cartesian basis
,
![{\displaystyle {\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~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e5ed013d484bcffa44c8749fa23eb1417d82a8)
Hence,
![{\displaystyle \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~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26bc57405d86a88583ef744818b270803d3ac460)
Recall the balance of energy
![{\displaystyle \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)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f26af4e5a07fce1afd1ed197f2c702631509e69)
Therefore,
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55cd57ea423695028245760901a541263306b1)
Rearranging,
![{\displaystyle {\rho ~({\dot {e}}-T~{\dot {\eta }})-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \leq -{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a66ac7dbc04bc632c903ac8e4cd113e92c112ed4)