Some models and problems for laminar and turbulent flows of viscous and nonlinear viscous fluids

Theories of flow of viscous and nonlinear viscous fluids are based on the Navier-Stokes equations and their modifications for which the viscosity of the fluid depends on rates of strains (derivatives of the velocity function). These models describe satisfactorily the slow laminar flows of some fluids and the beginning of loss of their stability, but they are not fit to compute and explore the flows with large gradients and the turbulent flows. For the rapid and turbulent flows of viscous fluids, special models were developed, which are on the whole directed at the flow in circular pipes. We introduce and study a general nonlocal model describing the steady and nonsteady flows of the viscous and nonlinear viscous fluids for both laminar and turbulent flows. In case of a turbulent flow, the model describes the averaged velocities. For slow laminar flows, our model turns into the Navier-Stokes equations or into equations of nonlinear viscous fluids. The model describes the main observed phenomena and gives rise to correct mathematical problems in the sense of the existence of a solution for wide classes of data and for various formulations, including the mixed one, when the velocities and surface forces are prescribed on different parts of the boundary.