Towards a finite-time singularity of the Navier–Stokes equations Part 1. Derivation and analysis of dynamical system

Abstract

The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity$\unicode[STIX]{x1D708}$is studied, starting from a finite-energy configuration of two vortex rings of circulation$\pm \unicode[STIX]{x1D6E4}$and radius$R$, symmetrically placed on two planes at angles$\pm \unicode[STIX]{x1D6FC}$to a plane of symmetry$x=0$. The minimum separation of the vortices,$2s$, and the scale of the core cross-section,$\unicode[STIX]{x1D6FF}$, are supposed to satisfy the initial inequalities$\unicode[STIX]{x1D6FF}\ll s\ll R$, and the vortex Reynolds number$R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the ‘tipping points’) is determined solely by the curvature$\unicode[STIX]{x1D705}(\unicode[STIX]{x1D70F})$at the tipping points and by$s(\unicode[STIX]{x1D70F})$and$\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$, where$\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D6E4}/R^{2})t$is a dimensionless time variable. The Biot–Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot–Savart law breaks down just before this singularity is realised, when$\unicode[STIX]{x1D705}s$and$\unicode[STIX]{x1D6FF}/\!s$become of order unity. The dynamical system admits ‘partial Leray scaling’ of just$s$and$\unicode[STIX]{x1D705}$, and ultimately full Leray scaling of$s,\unicode[STIX]{x1D705}$and$\unicode[STIX]{x1D6FF}$, conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter$\unicode[STIX]{x1D705}$; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot–Savart approach just before the incipient singularity is realised. The Euler flow situation ($\unicode[STIX]{x1D708}=0$) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.