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 the limit case of very large thermal conductivity when the viscosity varies with the temperature. A rigorous analysis is illustrated within the Hilbertian framework for unidirectional stationary flows of Newtonian and Bingham fluids with heating by viscous dissipation. An extension to quasi-Newtonian fluids of power law type and with temperature dependent viscosities is obtained in the context of the heat equation with an L¹-term. The nonlocal model proposed by Ladyzhenskaya in 1966 as a modification of Navier-Stokes equations can be, in particular, obtained with this procedure. Bibliography: 14 titles.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 99–117.