On stationary flows with energy dependent nonlocal viscosities

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.