Planar flows of incompressible heat-conducting shear-thinning fluids—Existence analysis

We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called L∞-truncation method, used to obtain the strong convergence of the velocity gradient. The important point of the approach consists in the choice of an appropriate form of the balance of energy.