scholarly journals No Finite Time Blowup for 3D Incompressible Navier Stokes Equations via Scaling Invariance

2021 ◽  
Vol 9 (3) ◽  
pp. 386-393
Author(s):  
Terry E. Moschandreou
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Scaling Invariance ◽  
Finite Time Blowup ◽  
Time Blowup

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.

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.

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 Maximum palinstrophy amplification in the two-dimensional Navier–Stokes equations

2018 ◽  
Vol 837 ◽  
pp. 839-857 ◽  
Author(s):  
Diego Ayala ◽  
Charles R. Doering ◽  
Thilo M. Simon
Keyword(s):  
Initial Data ◽  
Finite Time ◽  
Stokes Equations ◽  
Optimization Problems ◽  
Navier Stokes ◽  
Instantaneous Growth Rate ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Wide Range

We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the two-dimensional incompressible Navier–Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number $Re$ and palinstrophy ${\mathcal{P}}$ which maximize $\text{d}{\mathcal{P}}/\text{d}t$ is constructed by numerically solving suitable optimization problems for a wide range of $Re$ and ${\mathcal{P}}$, providing numerical evidence for the sharpness of the analytic estimate $\text{d}{\mathcal{P}}/\text{d}t\leqslant (a+b\sqrt{\ln Re+c}){\mathcal{P}}^{3/2}$ with respect to both $Re$ and ${\mathcal{P}}$. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations, and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e. $\sup _{t>0}{\mathcal{P}}(t)$, are observed to scale with the magnitude of the initial palinstrophy ${\mathcal{P}}(0)$ in accord with the corresponding a priori estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier–Stokes equation are discussed.

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 $R_{\unicode[STIX]{x1D6E4}}\gtrsim 4000$.

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