Euler-Lagrangian Approach to Fluid Dynamics and the Incompleteness of the Navier-Stokes Equations

Navier-Stokes equations are based on Newton’s second law and the Stokes hypothesis. In this paper, we have derived the fluid dynamic equations by applying the powerful tool of the Euler-Lagrangian approach, based on the principle of least action. The new equation highlights the incompleteness of the Navier-Stokes equations. The main reason is that the Stokes hypothesis uses engineering shear strain concept (through an average procedure for shear strain) to model the viscous stresses instead of using the tensorial shear strain. The general velocity gradient (tensorial shear strains) contains stretch, shear, and rotation deformations. The average procedure, based on the Stokes hypothesis, can only partially account for the shear strains. This deficiency should be remedied by adding an extra term – a pure spin tensor. Geometric interpretations and geometric algebra explanations are provided to show this deficiency and its counterbalance. One of the notable findings is that, both fluid flow and electromagnetic fields are, in essence, the same. All of them can be described by the same mathematical tools.