Continuum mechanics/Entropy inequality

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Clausius-Duhem inequality[edit]

The Clausius-Duhem inequality can be expressed in integral form as

In differential form the Clusius-Duhem inequality can be written as

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[edit]

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

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,