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.