On the Navier–Stokes equations with the energy-dependent nonlocal viscosities

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.