Singularities in Fluid Mechanics

TL;DR

This paper discusses the nature of singularities in fluid mechanics, especially in the Navier-Stokes equations, highlighting the existence of finite-time physical singularities at high Reynolds numbers and their implications for turbulence.

Contribution

It introduces a new analytical approach demonstrating the existence of physical singularities in Navier-Stokes flows at high Reynolds numbers, even with viscous effects considered.

Findings

Finite-time singularities can occur in Navier-Stokes flows at high Reynolds numbers.

Viscous effects do not necessarily prevent vorticity amplification to infinite levels.

Implications for turbulence and fluid interface phenomena are discussed.

Abstract

Singularities of the Navier-Stokes equations occur when some derivative of the velocity field is infinite at any point of a field of flow (or, in an evolving flow, becomes infinite at any point within a finite time). Such singularities can be mathematical (as e.g. in two-dimensional flow near a sharp corner, or the collapse of a Mobius-strip soap film onto a wire boundary) in which case they can be resolved by refining the geometrical description; or they can be physical (as e.g. in the case of cusp singularities at a fluid/fluid interface) in which case resolution of the singularity involves incorporation of additional physical effects; these examples will be briefly reviewed. The finite-time singularity problem for the Navier-Stokes equations will then be discussed and a recently developed analytical approach will be presented; here it will be shown that, even when viscous vortex reconnection is taken into account, there is indeed a physical singularity, in that, at sufficiently high Reynolds number, vorticity can be amplified by an arbitrarily large factor in an extremely small point-neighbourhood within a finite time, and this behaviour is not resolved by viscosity. Similarities with the soap-film-collapse and free-surface--cusping problems are noted in the concluding section, and the implications for turbulence are considered.