Geodesic
On the Navier--Stokes equations with the energy-dependent nonlocal viscosities
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 71-91

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We discuss the mathematical derivation of incompressible viscous flows where the viscosity depends on the total dissipation energy. In the two-dimensional periodic case, we consider first the case of temperature dependent viscosities with very large thermal conductivity in the heat convective equation, in which we obtain as an asymptotic limit the Navier–Stokes system coupled with and ordinary differential equation involving the dissipation energy. Letting further the latent heat vanish, we derive the Navier–Stokes equations with a nonlocal viscosity depending on the total dissipation of energy. 
@article{ZNSL_2003_306_a3,
     author = {L. Consiglieri and J.-F. Rodrigues and T. N. Shilkin},
     title = {On the {Navier--Stokes} equations with the energy-dependent nonlocal viscosities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {71--91},
     publisher = {mathdoc},
     volume = {306},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a3/}
}
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L. Consiglieri; J.-F. Rodrigues; T. N. Shilkin. On the Navier--Stokes equations with the energy-dependent nonlocal viscosities. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 71-91. http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a3/