Model for laminar and turbulent flows of viscous and nonlinear viscous non-Newtonian fluids

Abstract

We develop a model that describes laminar and turbulent flows of viscous and nonlinear viscous fluids. A basis for the model is a characteristic, which we call a local Reynolds number, and which is calculated at each point of the domain of flow. In the subdomain wherein the local Reynolds number does not exceed some value, the flow is laminar; otherwise it is turbulent, i.e., the model identifies the areas of laminar and turbulent flows. The model predicts the existence of a laminar boundary layer at turbulent flows. It enables us to describe special features of turbulent flows such as a drastic increase in the resistance to flow and the variation of the velocity profile with the increase of the Reynolds number, and so on. The model is mathematically grounded. We prove the existence of global solutions to stationary and nonstationary flow problems with the nonhomogeneous Dirichlet and mixed boundary conditions, where velocities and surface forces are given on different parts of the boundary. Numerical methods for solving stationary and nonstationary flow problems are investigated.