ClausiusDuhem inequalityThe ClausiusDuhem inequality can be expressed in integral form as In differential form the ClusiusDuhem inequality can be written as 
Proof:
Assume that Ω is an arbitrary fixed control volume. Then u_{n} = 0 and the derivative can be taken inside the integral to give
Using the divergence theorem, we get
Since Ω is arbitrary, we must have
Expanding out
or,
or,
Now, the material time derivatives of ρ and η are given by
Therefore,
From the conservation of mass . Hence,
ClausiusDuhem inequality in terms of internal energyIn terms of the specific entropy, the ClausiusDuhem inequality is written as Show that the inequality can be expressed in terms of the internal energy as 
Proof:
Using the identity in the ClausiusDuhem inequality, we get
Now, using index notation with respect to a Cartesian basis ,
Hence,
Recall the balance of energy
Therefore,
Rearranging,
