Navier-Stokes Equations

The choice of independent variables called spatial/Eulerian and material/Lagrangean description is explained and the incompressible and compressible versions of the Navier-Stokes pdes are established. Symmetries of the Euler and Navier-Stokes pdes in the spatial description are introduced based on the work of Oberlack and his coworkers, reference in book. Fundamental properties of the IBVP for the Navier-Stokes pdes and the major theorems from functional anaysis are summarized and a fundamental assumption on the well-posedness of the IBVP problems for Navier-Stokes is introduced. Vector and tensor fields relevant to turbulence: vorticity, rate of strain, and Beltrami and Trkalian and the Lamb vector fields are defined in the spatial description. The material description is introduced and the Navier-Stokes pdes are established in this description. Rotation and vorticity are analyzed in the material description; the velocity gradient tensor is introduced in the spatial description.