Geodesic
On stationary flows with energy dependent nonlocal viscosities
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 99-117

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A nonlocal constitutive law for an incompressible viscous flow in which the viscosity depends on the total dissipation energy of the fluid is obtained as a limit case of very large thermal conductivity when the viscosity varies with the temperature. A rigorous analysis is illustrated in an Hilbertian framework for unidirectional stationary flows of Newtonian and Bingham fluids with heating by viscous dissipation. The extension to quasi-Newtonian fluids of power law type and with temperature dependent viscosities is obtained in the framework of the heat equation with a $L^1$-term. The nonlocal model proposed by Ladyzenskaya in 1966 as a modification of Navier–Stokes equations, in particular, may be obtained with this procedure. 
@article{ZNSL_2003_295_a4,
     author = {L. Consiglieri and J.-F. Rodrigues},
     title = {On stationary flows with energy dependent nonlocal viscosities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {99--117},
     publisher = {mathdoc},
     volume = {295},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a4/}
}
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L. Consiglieri; J.-F. Rodrigues. On stationary flows with energy dependent nonlocal viscosities. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 99-117. http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a4/