Non-Riemannian geometrical asymmetrical damping stresses on the Lagrange instability of shear flows

TL;DR

This paper explores how non-Riemannian geometrical asymmetrical damping stresses, modeled via Cartan torsion, can stabilize shear flows by damping instabilities, extending the geometric approach to fluid dynamics and magnetohydrodynamics.

Contribution

It introduces a non-Riemannian geometric framework to analyze shear flow stability, highlighting the role of Cartan torsion in damping flow instabilities.

Findings

Cartan torsion induces asymmetric stresses that damp flow instability.

Flow stability is linked to zero sectional non-Riemannian curvature.

Flow speed inversely proportional to torsion affects stability.

Abstract

It is shown that the physical interpretation of Elie Cartan three-dimensional space torsion as couple asymmetric stress, has the effect of damping, previously Riemannian unstable Couette planar shear flow, leading to stability of the flow in the Lagrangean sense. Actually, since the flow speed is inversely proportional to torsion, it has the effect of causing a damping in the planar flow atenuating the instability effect. In this sense we may say that Cartan torsion induces shear viscous asymmetric stresses in the fluid, which are able to damp the instability of the flow. The stability of the flow is computed from the sectional curvature in non-Riemannian three-dimensional manifold. Marginal stability is asssumed by making the sectional non-Riemannian curvature zero, which allows us to determine the speeds of flows able to induce this stability. The ideas discussed here show that torsion plays the geometrical role of magnetic field in hydromagnetic instability of Couette flows recently investigated by Bonnano and Urpin (PRE, (2007,in press) can be extended and applied to plastic flows with microstructure defects. Recently Riemannian asymmetric stresses in magnetohydrodynamics (MHD) have been considered by Billig (2004).