scholarly journals Asymptotic Behavior of Solutions of the Compressible Navier–Stokes Equation Around the Plane Couette Flow

2010 ◽  
Vol 13 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Yoshiyuki Kagei
Keyword(s):  
Asymptotic Behavior ◽  
Couette Flow ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Asymptotic Behavior Of Solutions ◽  
Plane Couette Flow ◽  
Navier Stokes Equation

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.

scholarly journals Doubly localized states in plane Couette flow

2014 ◽  
Vol 758 ◽  
pp. 1-4 ◽  
Author(s):  
Bruno Eckhardt
Keyword(s):  
Couette Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Linear Instability ◽  
Localized States ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Plane Couette Flow ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

AbstractMuch of our understanding of the transition to turbulence in flows without a linear instability came with the discovery and characterization of fully three-dimensional solutions to the Navier–Stokes equation. The first examples in plane Couette flow were periodic in both spanwise and streamwise directions, and could explain the transitions in small domains only. The presence of localized turbulent spots in larger domains, the spatiotemporal decoherence on larger scales and the ability to trigger turbulence with pointwise perturbations require solutions that are localized in both directions, like the one presented by Brand & Gibson (J. Fluid Mech., vol. 750, 2014, R3). They describe a steady solution of the Navier–Stokes equations and characterize in unprecedented detail, including an analytic computation of its localization properties. The study opens up new ways to describe localized turbulent patches.

scholarly journals Heteroclinic connections in plane Couette flow

2009 ◽  
Vol 621 ◽  
pp. 365-376 ◽  
Author(s):  
J. HALCROW ◽  
J. F. GIBSON ◽  
P. CVITANOVIĆ ◽  
D. VISWANATH
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Invariant Sets ◽  
Navier Stokes ◽  
Lower Branch ◽  
Plane Couette Flow ◽  
Navier Stokes Equation ◽  
Starting Point ◽  
Heteroclinic Connection ◽  
Flow Transitions

Plane Couette flow transitions to turbulence at Re ≈ 325 even though the laminar solution with a linear profile is linearly stable for all Re (Reynolds number). One starting point for understanding this subcritical transition is the existence of invariant sets in the state space of the Navier–Stokes equation, such as upper and lower branch equilibria and periodic and relative periodic solutions, that are distinct from the laminar solution. This article reports several heteroclinic connections between such objects and briefly describes a numerical method for locating heteroclinic connections. We show that the nature of streaks and streamwise rolls can change significantly along a heteroclinic connection.

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