Two (equivalent?) versions of Clausius-Duhem inequality?

[honululu]

Homework Statement

Hello everyone,
I have a problem to solve, which I found hard and so looked for help on the web. Finally found this:
https://books.google.co.uk/books?id...E#v=onepage&q=clausius-duhem inviscid&f=false
which is great as on page 183 is the EXACT question I am supposed to solve. It's not fully shown on google but I can do the rest.

There a a few differences:
I have

T=T(F,Θ,gradΘ)
q=q(F,Θ,gradΘ)
η=η(F,Θ,gradΘ)

rather than
T=T(ρ,Θ,gradΘ) etc. as in the link above.

The Clausius-Duhem inequality shown throughout the book has terms a term

+T⋅D

in it. We learned this differently in class: Our inequality read the exact same as in the link above but had a term

+tr(TD)

instead.

I would really like to understand why these two forms are equivalent before just using the form I found on the web and assuming it's correct. Is this to do with T vs. T?

I feel like such a beginner (which I am!) :D

Thank you so much for any advice!

 

[Asim Riaz]

Homework EquationsClausius-Duhem Inequality: ΔS ≥ ∑ i (∂q i /∂T) ΔT + ∑ i (∂q i /∂x j ) Δx j where S is entropy, q i is heat flux in direction i, T is temperature and x j is the spatial coordinate in direction j.The Attempt at a SolutionThe two forms of the Clausius-Duhem inequality are equivalent because they both account for the same physical processes. The first form has a term +T⋅D which accounts for the change in entropy due to a change in temperature. The second form has a term +tr(TD) which accounts for the change in entropy due to a change in the heat flux, which is related to the temperature gradient. Thus, both forms are equivalent as they both account for the same physical processes.