More considerations about the symmetry of the stress tensor of fluids.

Abstract

Regarding a recent dispute about the symmetry of the stress tensor of fluids, more considerations are presented. The usual proofs of this symmetry are reviewed, and contradictions between this symmetry and the mechanism of gas viscosity are analyzed for simple gas flows. It is emphasized that these proofs depend on the theorem of angular momentum that assumes that internal forces between any two fluid particles are always along the line connecting them. Without this assumption, from Newton's laws of motion one can only obtain the theorem of angular momentum with an additional term. It is proved within classical continuum mechanics that this additional term represents the total moment of internal forces, and its volume density is similar to the body couple introduced in generalized continuum mechanics. When discrete structure of matter is considered, this term corresponds to the total moment of the forces exerted on the nuclei by the internal electrons. This moment of internal forces may lead to nonsymmetry of the stress tensor and is, in general, nonzero as long as shear stress exists. A nonsymmetrical stress tensor suggested in the literature is discussed in terms of its effects in eliminating the contradictions and simplifying the Navier-Stokes equation. The derivation of this stress tensor for ideal gases based on the kinetic theory of gas molecules is presented, and its form in a general orthogonal curvilinear coordinate system is given. Finally, a possible experimental verification of this nonsymmetrical stress tensor is discussed.