scholarly journals Existence and stability of steady-state solutions with finite energy for the Navier–Stokes equation in the whole space

Nonlinearity ◽  
2009 ◽  
Vol 22 (7) ◽  
pp. 1615-1637 ◽  
Author(s):  
Clayton Bjorland ◽  
Maria E Schonbek
Keyword(s):  
Steady State ◽  
Stokes Equation ◽  
Finite Energy ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Existence And Stability ◽  
Whole Space ◽  
Steady State Solutions

Coriolis force influence on the AKA effect

2021 ◽  
Author(s):  
Peter Rutkevich ◽  
Georgy Golitsyn ◽  
Anatoly Tur
Keyword(s):  
Steady State ◽  
Stokes Equation ◽  
Nonlinear Equations ◽  
Coriolis Force ◽  
Large Scale ◽  
Fluid Velocity ◽  
Navier Stokes ◽  
Basic Solution ◽  
Navier Stokes Equation ◽  
Alpha Effect

<p>Large-scale instability in incompressible fluid driven by the so called Anisotropic Kinetic Alpha (AKA) effect satisfying the incompressible Navier-Stokes equation with Coriolis force is considered. The external force is periodic; this allows applying an unusual for turbulence calculations mathematical method developed by Frisch et al [1]. The method provides the orders for nonlinear equations and obtaining large scale equations from the corresponding secular relations that appear at different orders of expansions. This method allows obtaining not only corrections to the basic solutions of the linear problem but also provides the large-scale solution of the nonlinear equations with the amplitude exceeding that of the basic solution. The fluid velocity is obtained by numerical integration of the large-scale equations. The solution without the Coriolis force leads to constant velocities at the steady-state, which agrees with the full solution of the Navier-Stokes equation reported previously. The time-invariant solution contains three families of solutions, however, only one of these families contains stable solutions. The final values of the steady-state fluid velocity are determined by the initial conditions. After account of the Coriolis force the solutions become periodic in time and the family of solutions collapses to a unique solution. On the other hand, even with the Coriolis force the fluid motion remains two-dimensional in space and depends on a single spatial variable. The latter fact limits the scope of the AKA method to applications with pronounced 2D nature. In application to 3D models the method must be used with caution.</p><p>[1] U. Frisch, Z.S. She and P. L. Sulem, “Large-Scale Flow Driven by the Anisotropic Kinetic Alpha Effect,” Physica D, Vol. 28, No. 3, 1987, pp. 382-392.</p>

scholarly journals The Rise and Fall of α-Model Viscosity

1996 ◽  
Vol 158 ◽  
pp. 111-114
Author(s):  
F. V. Hessman ◽  
C. Obach
Keyword(s):  
Steady State ◽  
Stokes Equation ◽  
Accretion Disks ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Averaged Models

AbstractFor nearly two decades, our only useful model for the viscosity in accretion disks has been the so-called ‘α-model’. However, it has become clear that the simplest models –in which α is constant – are inadequate to explain the range of behaviours seen in real disks. We show that the properties of steady-state, vertically-averaged models can be determined without any assumptions other than that the disks obey the classical Navier-Stokes equation. These solutions have derived values of α which vary with radius by many orders of magnitude even in small CV disks.

scholarly journals Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

2020 ◽  
Vol 22 (2) ◽  
Author(s):  
Zdzisław Brzeźniak ◽  
Gaurav Dhariwal
Keyword(s):  
Invariant Measure ◽  
Stokes Equation ◽  
Diffusion Processes ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Uniqueness Of Solutions ◽  
Unique Strong Solution ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Whole Space

Abstract Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) proved the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space and for the periodic boundary case using a result from Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the $$L^4$$ L 4 -norm of the solution from the torus to $${\mathbb {R}}^3$$ R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on $${\mathbb {R}}^3$$ R 3 for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).

RELAXATION METHOD FOR NAVIER–STOKES EQUATION

2012 ◽  
Vol 23 (04) ◽  
pp. 1250021 ◽  
Author(s):  
P. M. C. DE OLIVEIRA
Keyword(s):  
Steady State ◽  
Stokes Equation ◽  
Wind Velocity ◽  
Linear Time ◽  
Navier Stokes ◽  
Cylinder Surface ◽  
Theoretical Problem ◽  
Navier Stokes Equation ◽  
Transient Time ◽  
Ball Speed

The motivation for this work was a simple experiment [P. M. C. de Oliveira, S. Moss de Oliveira, F. A. Pereira and J. C. Sartorelli, preprint (2010), arXiv:1005.4086], where a little polystyrene ball is released falling in air. The interesting observation is a speed breaking. After an initial nearly linear time-dependence, the ball speed reaches a maximum value. After this, the speed finally decreases until its final, limit value. The provided explanation is related to the so-called von Kármán street of vortices successively formed behind the falling ball. After completely formed, the whole street extends for some hundred diameters. However, before a certain transient time needed to reach this steady-state, the street is shorter and the drag force is relatively reduced. Thus, at the beginning of the fall, a small and light ball may reach a speed superior to the sustainable steady-state value. Besides the real experiment, the numerical simulation of a related theoretical problem is also performed. A cylinder (instead of a 3D ball, thus reducing the effective dimension to 2) is positioned at rest inside a wind tunnel initially switched off. Suddenly, at t = 0 it is switched on with a constant and uniform wind velocity [Formula: see text] far from the cylinder and perpendicular to it. This is the first boundary condition. The second is the cylinder surface, where the wind velocity is null. In between these two boundaries, the velocity field is determined by solving the Navier–Stokes equation, as a function of time. For that, the initial condition is taken as the known Stokes laminar limit V → 0, since initially the tunnel is switched off. The numerical method adopted in this task is the object of the current text.

History forces and the unsteady wake of a cylinder

1999 ◽  
Vol 393 ◽  
pp. 99-121 ◽  
Author(s):  
J. R. CHAPLIN
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Stokes Equation ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Asymptotic Properties ◽  
Quantitative Agreement ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Unsteady Wake

History forces on a stationary cylinder in arbitrary unsteady rectilinear flow are calculated by means of a model based on the asymptotic properties of the steady-state wake. The results capture many features found in numerical solutions of the Navier–Stokes equation for the same flows, though quantitative agreement deteriorates as the Reynolds number increases over the range 2 to 40. The cases studied are the impulsive start, stop, and reverse, and oscillatory flow.

Note on the fast decay property of steady Navier–Stokes flows in the whole space

Analysis ◽  
2018 ◽  
Vol 38 (2) ◽  
pp. 81-89
Author(s):  
Tomoyuki Nakatsuka
Keyword(s):  
Asymptotic Behavior ◽  
Fundamental Solution ◽  
Unique Solution ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Asymptotic Behavior Of Solutions ◽  
Fast Decay ◽  
Rapid Decay ◽  
Navier Stokes Equation ◽  
Whole Space

Abstract We investigate the pointwise asymptotic behavior of solutions to the stationary Navier–Stokes equation in {\mathbb{R}^{n}} ( {n\geq 3} ). We show the existence of a unique solution {\{u,p\}} such that {|\nabla^{j}u(x)|=O(|x|^{1-n-j})} and {|\nabla^{k}p(x)|=O(|x|^{-n-k})} ( {j,k=0,1,\ldots} ) as {|x|\rightarrow\infty} , assuming the smallness of the external force and the rapid decay of its derivatives. The solution {\{u,p\}} decays more rapidly than the Stokes fundamental solution.

Sign in / Sign up

Export Citation Format

Share Document