As a hobby, I am trying to better understand the theory around objective rates, frame invariance and objective constitutive equations in non-Newtonian fluid mechanics. Unfortunately, I have been unable to find a reference at the level I'm interested in a language I understand.
My background is in applied mathematics, specialising in fluid mechanics for my Master's, though I have taken a couple of courses in General Relativity. I feel like the machinery developed in that context — metric tensors, co-/contravariance and so on — is particularly relevant to understanding things like objective rates. I would like to understand this theory using the language used by people studying GR.
In particular:
I do not want something written for the differential geometer, i.e. coordinate-free.
When I say GR I do not mean to imply I want a GR-compatible theory, with spacetime and so on. In fact I expect indices to go over the three spatial dimensions only. I simply mean that I would like the approach to differential geometry to be at the same level and with the same notation as that used by undergrad/postgrad introductions to GR.
I would like to understand the upper-/lower-convected derivatives in terms of Christoffel symbols.
I'm not interested in treatments with matrices — I really want to think of the relevant objects as tensors.
My request is one that I struggle to articulate (if I could say it better, perhaps I would be able to find a suitable reference on my own), so I hope that it is coherent and makes sense. I also hope such a reference exists.
