Clausius-Duhem inequality
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The Clausius-Duhem inequality can be expressed in integral form as
In differential form the Clusius-Duhem inequality can be written as
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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
Using the divergence theorem, we get
Since is arbitrary, we must have
Expanding out
or,
or,
Now, the material time derivatives of and are given by
Therefore,
From the conservation of mass . Hence,
Clausius-Duhem inequality in terms of internal energy
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In terms of the specific entropy, the Clausius-Duhem inequality is written as
Show that the inequality can be expressed in terms of the internal energy as
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Proof:
Using the identity
in the Clausius-Duhem inequality, we get
Now, using index notation with respect to a Cartesian basis ,
Hence,
Recall the balance of energy
Therefore,
Rearranging,