scholarly journals Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2037
Author(s):  
Lin F. Liu ◽  
Juan J. Nieto
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Gronwall Lemma ◽  
Absorbing Set ◽  
Two Dimensional ◽  
Variable Delay ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Galerkin Approximations ◽  
Fractional Halanay Inequality

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem.

scholarly journals Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations

2013 ◽  
Vol 143 (5) ◽  
pp. 905-927 ◽  
Author(s):  
Margaret Beck ◽  
C. Eugene Wayne
Keyword(s):  
Decay Rate ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Long Time ◽  
High Reynolds

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows, where they often emerge on timescales much shorter than the viscous timescale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier–Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

scholarly journals Stochastic Navier-Stokes Equations with Artificial Compressibility in Random Durations

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
Hong Yin
Keyword(s):  
Incompressible Flow ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Bounded Domains ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Finite Dimensional

The existence and uniqueness of adapted solutions to the backward stochastic Navier-Stokes equation with artificial compressibility in two-dimensional bounded domains are shown by Minty-Browder monotonicity argument, finite-dimensional projections, and truncations. Continuity of the solutions with respect to terminal conditions is given, and the convergence of the system to an incompressible flow is also established.

scholarly journals Computer Simulation of Water Flow Animation Based on Two-Dimensional Navier-Stokes Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Rushi Cao ◽  
Ruyun Cao
Keyword(s):  
Computer Graphics ◽  
Boundary Conditions ◽  
Water Flow ◽  
Stokes Equations ◽  
Simulation Method ◽  
Fluid Simulation ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

Simulation of water flow animation is a significant and challenging subject in computer graphics. With the continuous development of computational fluid dynamics and computer graphics, many more effective simulation methods have been developed, and fluid animation simulation has developed rapidly. In order to obtain realistic flow animation, one of the key aspects is to simulate flow motion. Based on the two-dimensional Navier-Stokes equations, a mathematical model is established to solve the boundary conditions required by the physical flow field of water. The coordinate transformation formula is introduced to transform the irregular physical area into a regular square calculation area, and then, the specific expressions of the liberalized Navier-Stokes equation, continuity equation, pressure Poisson equation, and dimensionless boundary conditions are given. Using animation software to sequence graphics and images of all kinds of control and direct operation of the relevant instructions, through the computer technology to simulate the flow of animation, based on the stability of fluid simulation method and simulation efficiency, so as to make realistic flow animation. The results show that FluidsNet has considerable performance in accelerating large scene animation simulation on the basis of ensuring the rationality of prediction, and the motion of water wave is realistic. The application of computer successfully simulates water flow.

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 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.

Direct simulation of transition in an oscillatory boundary layer

1998 ◽  
Vol 371 ◽  
pp. 207-232 ◽  
Author(s):  
G. VITTORI ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Two Dimensional ◽  
Temporal Development ◽  
Direct Simulation ◽  
Navier Stokes Equations ◽  
Oscillatory Boundary Layer

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

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