2. Fluid Equations

The chapter begins with the derivation of the Navier-Stokes equations from the principles of mass, momentum, and energy conservation. It introduces both Lagrangian and Eulerian descriptions of fluid motion and discusses the continuity equation for both compressible and incompressible fluids. The Navier-Stokes equations are then formulated for Newtonian fluids, with a focus on the incompressible case. The chapter also explores different boundary conditions, including stick, slip, and threshold conditions, as well as those for symmetric planes, inlets, outlets, and free surfaces. It concludes with a discussion of variational formulations for the Navier-Stokes equations with mixed boundary conditions, highlighting the use of Dirichlet, strain, and vorticity bilinear forms. This comprehensive overview makes the chapter a valuable resource for understanding the mathematical foundations of fluid dynamics.