On modified Navier-Stokes equations for viscous non-Newtonian incompressible fluid motion and their numerical solution

The aim of this study is to investigate numerically some modified Navier-Stokes equations, namely O. A. Ladyzhenskaya's set of equations, which describe viscous non-Newtonian incompressible fluid motion. Employing pseudo-spectral methods (Fourier-Galerkin method) Navier-Stokes equations and Ladyzhenskaya equations are solved in a cubic domain assuming periodic boundary conditions and the Taylor-Green Vortex as initial condition, in order to compare solutions and explore alternative fluid dynamics models whose hypotheses are less restrictive. The study consists of two Parts named Theoretical Formulation and Numerical Investigation. In Part I governing equations of Fluid Dynamics are derived using the continuum hypothesis and the conservation laws; then equations are specialized for incompressible flows, defining also the conditions under which flows can be considered incompressible and the role of the pressure. Viscous behavior is analyzed in Chapter 2 where expressions for the viscous stress tensor are derived for the cases of Newtonian fluids and general Reiner-Rivlin fluids. These expressions are then used in Chapter 3 to obtain incompressible Navier-Stokes equations and Ladyzhenskaya's models which contain non-linear additional viscous terms. In the same Chapter the global regularity problem for Navier-Stokes equations and regularity results for Ladyzhenksaya equations are presented pointing out turbulence's possible role in the Millennium Problem. Chapter 4 then covers Turbulence, its characteristics, its statistical description and statistical symmetries. Part I ends with some considerations on Navier-Stokes equations and the reasons why one should investigate also alternative models. In Part II the mathematical problems to solve are presented defining equations, domain and initial conditions. In Chapter 6 spectral methods are then introduced and some applications of the Fourier-Galerkin method to linear and non-linear partial differential equations are shown. Assuming the arising turbulence to be homogeneous and isotropic, Navier-Stokes and Ladyzhenskaya equations are discretized in space through the Fourier-Galerkin method, whereas advancement in time is realized through the fourth-order Runge-Kutta scheme. The MPI program which implements the numerical scheme described is presented. Simulations' results are shown and analyzed in Chapter 7.