Singularities in fluid mechanics

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 Möbius-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-neighborhood within a finite time, and this behavior 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.

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Figure 1

(a) Corner eddies driven by antisymmetric excitation far from the corner. (b) This flow is driven by a rotating cylinder; a primary eddy and a secondary eddy are clearly visible. This is a Stokes flow, in which flow separation and reattachment are evident. (c) Similar Stokes separation in shear flow over a cylinder near a plane boundary; (d) the same with the cylinder very near the plane. (Photographs from Ref. [6], © 1979 the Physical Society of Japan.)

Figure 2

(a) A one-sided soap-film in the form of a Möbius strip of minimal area is supported by a wire boundary, which may be slowly untwisted until the instability causes the hole to collapse suddenly to the boundary; (b) the twist singularity, observed by reflection from the twisted surface immediately after the collapse; at this stage the surface is two-sided. (Photographs are from Ref. [9].)

Figure 3

(a) A cusp at the free surface of a viscous liquid induced by subsurface counter-rotating cylinders; (b) flow modeled by a vortex dipole at depth $d$ below the undisturbed free surface; the cusp appears at depth ${\displaystyle \frac{2}{3}d}$. (Similar to figure in Ref. [13].)

Figure 4

(a) Two circular vortices of radius $R={\kappa }_0^{-1}$ propagate toward each other at angle $2\alpha$, the arrows indicate the direction of vorticity $\mathbf{\omega }$, the planes $x=0$ and $y=0$ are planes of symmetry, the vortices are assumed to have Gaussian core cross sections of scale ${\delta }_0$, the minimum separation is $2s_0$, and it is assumed that ${\delta }_0\ll s_0\ll R$. (b) Early deformation of the vortices, the dashed curves indicating the positions at $t=0$; although the vortices on the whole propagate downward, the tipping points (i.e., the points of nearest approach) move upward and toward the plane $x=0$.

Figure 5

Sketch indicating that, no matter how small ${\delta }^2/s^2$ may be initially, there is always a neighborhood of the singularity time ${\tau }_c$ where ${\delta }^2/s^2=O\left(1\right)$; at this stage (indicated by the shaded circle), the vortices begin to overlap and viscous reconnection must be taken into account.

Figure 6

Sketch indicating the pyramid reconnection process very near to the apex of the pyramid; the initial circulations $\pm \mathrm{\Gamma }$ split into reconnected circulations $\pm {\mathrm{\Gamma }}_r$ (red) that propagate away from the apex and surviving circulations $\pm {\mathrm{\Gamma }}_s\equiv \pm \gamma \left(\tau \right)\mathrm{\Gamma }$ (blue) that continue to propagate toward the apex; black arrows indicate the direction of vorticity and colored arrows indicate the direction of propagation.

Figure 7

Evolution of ${\delta }^2\left(\tau \right)$ and $\gamma \left(\tau \right)$, for initial conditions $s\left(0\right)=0.1,\delta \left(0\right)=0.01,\kappa \left(0\right)=1,\gamma \left(0\right)=1$ and for ${\epsilon }^{-1}=3000$; the initial rise of ${\delta }^2\left(\tau \right)$ is due to viscous diffusion; the subsequent fall occurs when vortex stretching becomes dominant; ${\delta }^2\left(\tau \right)$ appears to fall to zero, but in fact attains a positive minimum ${\delta }_{min}^2=2.727803\times {10}^{-24}$ as described in the text; the subsequent rise is again due to viscous diffusion acting on residual un-reconnected vorticity. (Figure from Ref. [2].)

Figure 8

(a) The function (12) showing the extremely rapid decrease of ${\delta }_{min}^2$ with increasing $R_{\mathrm{\Gamma }}$; (b) the function (13) showing the corresponding sharp increase of ${\omega }_{max}$. (Figures modified from Ref. [2].)