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THETHERMODYNAMICSOFLINEARFLUIDSANDFLUIDMIXTURES
Provisionally, the discussion simply means sending e-mails to me and then posting here. In the future more sophisticated ways may appear.
Question 1. My question concerns the proof of equations 4.147 and 4.148. On page 179, this is explained as being a result of the fact that, for sufficiently large driving forces, the third-order terms in equation 4.139 will dominate and potentially give negative entropy generation unless these terms always have coefficient zero. The part of this that I don't understand is that I thought that the fluxes could only be assumed to be linear functions of the driving forces (equations 4.130-4.133 and 4.136-4.138) when the driving forces are small (1 >> x >> x2 >> x3 >> ...). It seems like a paradox to me that the driving forces can be small (to justify a linear law) and large (to justify the dominance of a third-order term) at the same time. Is there a subtlety that I'm missing, or another way of justifying equations 4.147 and 4.148? Response.
Question 2. The part that I'm still curious about is whether the linear dependence on tensor and vector arguments in 4.130-4.138 can be justified somehow rather than assumed. I suppose that this is really outside of the scope of your book ("Linear" is in the title, after all!), but I'm interested in understanding it better nonetheless. It seems to me that one could do the following:
- Define "physical equilibrium" as a state in which all of the vector and tensor arguments to F in equation 4.129 are zero.
- Expand function F as a Taylor series in the vector and tensor arguments (but not the scalar arguments) using physical equilibrium as the starting point.
- Consider the limit where the vector and tensor arguments are close to their equilibrium values [zero], and thus second-order and higher terms are negligible.
- Follow the arguments of Appendix A.2 to eliminate terms which would produce non-isotropic functions.
The other approach that I can see would be to simply argue that a linear relationship has been observed experimentally, even at "high" deviation from equilibrium. This would allow 4.130-4.133 and 4.136-4.138 to be justified at all values of the vector and tensor arguments, and would therefore work with the argument that third-degree terms in 4.139 can be neglected. I wonder whether there is sufficient experimental data to support this, however... I imagine that it would be very difficult to control all of the vector and tensor variables independently and simultaneously probe the system with sufficient accuracy to validate/invalidate the linearity assumptions. Response.