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

2018 ◽  
Vol 861 ◽  
pp. 930-967 ◽  
Author(s):  
H. K. Moffatt ◽  
Yoshifumi Kimura
Keyword(s):  
Dynamical System ◽  
Finite Time ◽  
Stokes Equations ◽  
Rate Of Change ◽  
Navier Stokes ◽  
Tipping Points ◽  
Dimensionless Time ◽  
Time Singularity ◽  
Navier Stokes Equations ◽  
Finite Time Singularity

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.

scholarly journals Towards a finite-time singularity of the Navier–Stokes equations. Part 2. Vortex reconnection and singularity evasion

2019 ◽  
Vol 870 ◽  
Author(s):  
H. K. Moffatt ◽  
Yoshifumi Kimura
Keyword(s):  
Dynamical System ◽  
Reynolds Number ◽  
Finite Time ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Time Singularity ◽  
Navier Stokes Equations ◽  
Reconnection Process ◽  
Vortex Reconnection ◽  
Finite Time Singularity

In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier–Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute the maximum vorticity $\unicode[STIX]{x1D714}_{max}$ as a function of vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}$ in the range $2000\leqslant R_{\unicode[STIX]{x1D6E4}}\leqslant 3400$, and deduce a compatible behaviour $\unicode[STIX]{x1D714}_{max}\sim \unicode[STIX]{x1D714}_{0}\exp [1+220(\log [R_{\unicode[STIX]{x1D6E4}}/2000])^{2}]$ as $R_{\unicode[STIX]{x1D6E4}}\rightarrow \infty$. This may be described as a physical (although not strictly mathematical) singularity, for all $R_{\unicode[STIX]{x1D6E4}}\gtrsim 4000$.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

The interaction of skewed vortex pairs: a model for blow-up of the Navier–Stokes equations

2000 ◽  
Vol 409 ◽  
pp. 51-68 ◽  
Author(s):  
H. K. MOFFATT
Keyword(s):  
Blow Up ◽  
Stokes Equations ◽  
Vortex Pair ◽  
Navier Stokes ◽  
Interaction Zone ◽  
Navier Stokes Equations ◽  
Vortex Pairs ◽  
Burgers Vortex ◽  
High Reynolds ◽  
Finite Time Singularity

The interaction of two propagating vortex pairs is considered, each pair being initially aligned along the positive principal axis of strain associated with the other. As a preliminary, the action of accelerating strain on a Burgers vortex is considered and the conditions for a finite-time singularity (or ‘blow-up’) are determined. The asymptotic high Reynolds number behaviour of such a vortex under non-axisymmetric strain, and the corresponding behaviour of a vortex pair, are described. This leads naturally to consideration of the interaction of the two vortex pairs, and identifies a mechanism by which blow-up may occur through self-similar evolution in an interaction zone where scale decreases in proportion to (t* − t)1/2, where t* is the singularity time. The relevance of Leray scaling in this interaction zone is discussed.

Finite Time Lyapunov Exponent for Micro Chaotic Mixer Design

2003 ◽  
Author(s):  
Xi-Ze Niu ◽  
Patrick Tabeling ◽  
Yi-Kuen Lee
Keyword(s):  
Lyapunov Exponent ◽  
Finite Time ◽  
Stokes Equations ◽  
Channel Length ◽  
Navier Stokes ◽  
Chaotic Mixing ◽  
Mixing Efficiency ◽  
Navier Stokes Equations ◽  
Finite Time Lyapunov Exponent ◽  
Mixer Design

In this paper, the Finite Time Lyapunov Exponent (FTLE) approach is used to analyze and optimize chaotic mixing in an active microchannel and a static mixer. The characteristics of FTLE related to chaotic mixing are discussed. By comparing the similarity of Poincare´ mapping and FTLE contour, it is shown that FTLE can be used to evaluate the chaotic mixing of liquid in the micromixer qualitatively and quantitatively. The minimum channel length needed for full mixing in the mixers can be estimated by the mean FTLE. The results are consistent with CFD simulations directly solving the Navier-Stokes equations coupled with the diffusion equation. More than 3 orders of CPU time can be saved by using FTLE compared with the classical infinite time Lyapunov exponent approach. Moreover, the FTLE is used to optimize the design and operation of the chaotic micromixers to improve the mixing efficiency for the first time.

Hydrodynamic Induced Vibrations in Accelerated Thrust Bearings

1974 ◽  
Vol 16 (6) ◽  
pp. 357-366 ◽  
Author(s):  
C. F. Kettleborough
Keyword(s):  
Stokes Equations ◽  
Axial Direction ◽  
Rate Of Change ◽  
Navier Stokes ◽  
Slider Bearing ◽  
Navier Stokes Equations ◽  
Thrust Bearings ◽  
State Position ◽  
Tangential Acceleration ◽  
Time Dependent Solution

The transient response of an infinitely-wide slider bearing subject to tangential acceleration of the thrust ring has been investigated numerically. The reduced Navier-Stokes equations and the continuity equation are solved for a fixed-shoe bearing to produce a time-dependent solution for the pressure distribution, film geometry and velocity distributions for different types of runner acceleration. For a gradual increase or decrease in acceleration the new supporting film is found to be fully developed in the course of a few seconds. When the acceleration or deceleration is very rapid, an oscillation of the runner in the axial direction takes place. In the cases reported, this oscillation is always damped. As the ‘jerk’ (rate of change of acceleration) increases, the vibration increases in frequency, it has a greater amplitude and takes longer to die out. In all cases the resulting runner oscillation takes place about the steady-state position compatible with the new runner velocity.

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