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 may not be valid because they depend on the theorem of angular momentum which preassumes that the internal forces between any two fluid particles are always along the line connecting them. From Newton's laws of motion, however, one can only obtain the theorem of angular momentum with an additional term. It is proved that this additional term represents the total moment of internal forces in a fluid, both by using continuum model and by considering the microscopic structure of the fluid. In the latter case, this term has the form of the total moment of the forces exerted on the nuclei by the 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 effect 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.