On the Stretching of Line Elements in Fluids: an Approach from Differential Geometry

Summary

The rate of stretching of line elements is studied for an incompressible ideal fluid, based on the frame of differential geometry of a group of diffeomorphisms. Riemannian curvature is closely connected with the time evolution of distance between two mappings of fluid particles. Exponential stretching of line elements in time is considered in the context of negative curvature in turbulent flows. The corresponding two-dimensional MHD problem of a perfectly conducting fluid with the current perpendicular to the plane of motion is also investigated. Simultaneous concentration of vortex and magnetic tubes is presented first.

INTRODUCTION

Stretching of line elements in fluids with or without conductivity is studied from various points of view. Firstly, simultaneous concentration of vortex and magnetic tubes is considered in section 2 by presenting an exact solution of the axisymmetric MHD equation for a viscous, incompressible, conducting fluid. This solution (Kambe 1985) tends to a stationary state that results from complete balance of convection, diffusion and stretching. Sections 3 and 4 are concerned with mathematical formulation based on Riemannian differential geometry, and global (mean) stretching of line elements is considered.

The general form of the Riemannian curvature tensor for any Lie group was derived by Arnold (1966), where explicit formulae for T² (two-torus) were given. Explicit expressions for diffeomorphism curvatures on Tⁿ (and even on any locally flat manifold) were described by Lukatskii (1981). Recently, Nakamura et al. (1992) considered the curvature form on T³ corresponding to three-dimensional motion of an ideal fluid with periodic boundary conditions in a cubic space. An immediate consequence is the property that the section curvature in the space of ABC diffeomorphisms is a negative constant for all the sections (Kambe et al. 1992).