A revisit of Navier–Stokes equation

Abstract

Abstract An effort has been recently paid to derive and to better understand the Navier–Stokes (N–S) equation, and it is found that, although the N–S equation has been proven to be correct by numerous examples, some concepts and principles behind the equation may not be correct or consistent. For instance, from an analysis of the simple classic Couette flow, the requirement of the symmetric stress tensor is in fact conflicting with the solution of the Couette flow. To solve the inconsistencies identified in this research, a reformulation of the total tensor is suggested for accommodating the fluid friction which bears a solid physics, and the new total tensor could resolve all the inconsistencies and conflicts identified. The newly defined fluid friction tensor is then used to derive N–S equation, and as expected, the same N–S equation as the original form of N–S equation for incompressible flows is obtained. For compressible flows, to achieve the same N–S equation as the original N–S equation, a slightly different assumption but yet in a very similar manner as Stokes made in 1845 is needed. It is the author’s intention that the N–S equation under the new defined total tensor has different, but yet more physical background concepts and principles. It is hoped that the revisit of the N–S equation could shed some light to better understand the dynamic flows and lead to establish new and better approaches to solve the complicated flow problems in future.