General nonlocal model describing the laminar and turbulent flows of viscous and nonlinear viscous fluids and its investigation

A general nonlocal model describing the flows of viscous and nonlinear Visc ous fluids for both laminar and turbulent flows is introduced and studied. For this model, the viscosity of the fluid depends on the second invariant of the rate of the strain tensor and on a nonlocal (integral) characteristi c of the flow. This characteristic is a vector that, in the simplest case, is an analog of the Reynolds number. For slow flows, the model turns into t he Navier-Stokes equations or into the equations of a nonlinear viscous flu id. Problems on steady and nonsteady flows with mixed boundary conditions w hen velocities and surface forces are prescribed on different parts of the boundary are studied. Existence results without restrictions on the smallne ss of data and on the length of the interval of time are proved.