scholarly journals A Hopf Bifurcation in the Planar Navier–Stokes Equations

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Gianni Arioli ◽  
Hans Koch
Keyword(s):  
Hopf Bifurcation ◽  
Viscous Fluid ◽  
Kinematic Viscosity ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Computer Assisted ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Navier Boundary Conditions

AbstractWe consider the Navier–Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

The exponential behavior and stabilizability of the stochastic 3D Navier–Stokes equations with damping

2019 ◽  
Vol 31 (07) ◽  
pp. 1950023 ◽  
Author(s):  
Hui Liu ◽  
Lin Lin ◽  
Chengfeng Sun ◽  
Qingkun Xiao
Keyword(s):  
Weak Solutions ◽  
Multiplicative Noise ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Exponential Behavior ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
The Mean ◽  
The Stability

The stochastic 3D Navier–Stokes equation with damping driven by a multiplicative noise is considered in this paper. The stability of weak solutions to the stochastic 3D Navier–Stokes equations with damping is proved for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text]. The weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions are proved for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text]. The stabilization of these equations is obtained for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text].

scholarly journals Generalized Navier–Stokes Equations and Dynamics of Plane Molecular Media

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 288
Author(s):  
Alexei Kushner ◽  
Valentin Lychagin
Keyword(s):  
Carbon Dioxide ◽  
Internal Structure ◽  
Stokes Equation ◽  
Hydrogen Cyanide ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by chiral, planar and rigid molecules and propose some kind of Navier–Stokes equations for their description. Examples of such media are water, ozone, carbon dioxide and hydrogen cyanide.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

scholarly journals Disconnected stationary solutions for 3D Kolmogorov flow problem: preliminary results

2021 ◽  
Vol 2090 (1) ◽  
pp. 012046
Author(s):  
Nikolay M. Evstigneev
Keyword(s):  
Flow Problem ◽  
Stokes Equations ◽  
Continuation Method ◽  
Fourier Method ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
System Of Nonlinear Equations ◽  
Arc Length ◽  
Navier Stokes Equations ◽  
Numerical Bifurcation

Abstract The extension of the classical A.N. Kolmogorov’s flow problem for the stationary 3D Navier-Stokes equations on a stretched torus for velocity vector function is considered. A spectral Fourier method with the Leray projection is used to solve the problem numerically. The resulting system of nonlinear equations is used to perform numerical bifurcation analysis. The problem is analyzed by constructing solution curves in the parameter-phase space using previously developed deflated pseudo arc-length continuation method. Disconnected solutions from the main solution branch are found. These results are preliminary and shall be generalized elsewhere.

The Onset of Turbulence: Insights Into the Navier-Stokes Equations

2017 ◽  
Author(s):  
Carl E. Rathmann
Keyword(s):  
Stokes Equations ◽  
Direct Consequence ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Fluid Behavior ◽  
Onset Of Turbulence ◽  
And Mathematics ◽  
High Reynolds

For well over 150 years now, theoreticians and practitioners have been developing and teaching students easily visualized models of fluid behavior that distinguish between the laminar and turbulent fluid regimes. Because of an emphasis on applications, perhaps insufficient attention has been paid to actually understanding the mechanisms by which fluids transition between these regimes. Summarized in this paper is the product of four decades of research into the sources of these mechanisms, at least one of which is a direct consequence of the non-linear terms of the Navier-Stokes equation. A scheme utilizing chaotic dynamic effects that become dominant only for sufficiently high Reynolds numbers is explored. This paper is designed to be of interest to faculty in the engineering, chemistry, physics, biology and mathematics disciplines as well as to practitioners in these and related applications.

scholarly journals Dynamic analysis of submerged microscale plates: the effects of acoustic radiation and viscous dissipation

2016 ◽  
Vol 472 (2187) ◽  
pp. 20150728 ◽  
Author(s):  
Zhangming Wu ◽  
Xianghong Ma
Keyword(s):  
Viscous Fluid ◽  
Boundary Conditions ◽  
Viscous Dissipation ◽  
Stokes Equations ◽  
Acoustic Radiation ◽  
Navier Stokes ◽  
Rectangular Plates ◽  
Fluid Loading ◽  
Navier Stokes Equations ◽  
Loading Impedance

The aim of this paper is to study the dynamic characteristics of micromechanical rectangular plates used as sensing elements in a viscous compressible fluid. A novel modelling procedure for the plate–fluid interaction problem is developed on the basis of linearized Navier–Stokes equations and no-slip conditions. Analytical expression for the fluid-loading impedance is obtained using a double Fourier transform approach. This modelling work provides us an analytical means to study the effects of inertial loading, acoustic radiation and viscous dissipation of the fluid acting on the vibration of microplates. The numerical simulation is conducted on microplates with different boundary conditions and fluids with different viscosities. The simulation results reveal that the acoustic radiation dominates the damping mechanism of the submerged microplates. It is also proved that microplates offer better sensitivities (Q-factors) than the conventional beam type microcantilevers being mass sensing platforms in a viscous fluid environment. The frequency response features of microplates under highly viscous fluid loading are studied using the present model. The dynamics of the microplates with all edges clamped are less influenced by the highly viscous dissipation of the fluid than the microplates with other types of boundary conditions.

ON THE ASYMPTOTIC ANALYSIS OF THE BGK MODEL TOWARD THE INCOMPRESSIBLE LINEAR NAVIER–STOKES EQUATION

2010 ◽  
Vol 20 (08) ◽  
pp. 1299-1318 ◽  
Author(s):  
A. BELLOUQUID
Keyword(s):  
Asymptotic Analysis ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Uniqueness Theorems ◽  
Bgk Model ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stokes Systems ◽  
Global Strong Solutions

This paper deals with the analysis of the asymptotic limit for BGK model to the linearized Navier–Stokes equations when the Knudsen number ε tends to zero. The uniform (in ε) existence of global strong solutions and uniqueness theorems are proved for regular initial fluctuations. As ε tends to zero, the solution of BGK model converges strongly to the solution of the linearized Navier–Stokes systems. The validity of the BGK model is critically analyzed.

Large-frequency global regularity for the incompressible Navier–Stokes equation

1999 ◽  
Vol 129 (5) ◽  
pp. 903-912
Author(s):  
Joel D. Avrin
Keyword(s):  
Boundary Conditions ◽  
Global Regularity ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Thin Domains ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Incompressible Navier Stokes Equation

We obtain global existence and regularity of strong solutions to the incompressible Navier–Stokes equations for a variety of boundary conditions in such a way that the initial and forcing data can be large in the high-frequency eigenspaces of the Stokes operator. We do not require that the domain be thin as in previous analyses. But in the case of thin domains (and zero Dirichlet boundary conditions) our results represent a further improvement and refinement of previous results obtained.

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