Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

2020 ◽  
pp. 51-75
Author(s):  
Grigorij Kulinich ◽  
Svitlana Kushnirenko ◽  
Yuliya Mishura
Keyword(s):  
Asymptotic Analysis ◽  
Unstable Solutions

Evolution of unstable system.

2015 ◽  
Vol 9 (3) ◽  
pp. 2487-2502 ◽  
Author(s):  
Igor V. Lebed
Keyword(s):  
Reynolds Number ◽  
Vortex Shedding ◽  
Stokes Flow ◽  
Stable Solution ◽  
The State ◽  
Unstable System ◽  
Unstable Flow ◽  
Unstable Solutions ◽  
Vortex Shedding Modes ◽  
Hydrodynamics Equations

Scenario of appearance and development of instability in problem of a flow around a solid sphere at rest is discussed. The scenario was created by solutions to the multimoment hydrodynamics equations, which were applied to investigate the unstable phenomena. These solutions allow interpreting Stokes flow, periodic pulsations of the recirculating zone in the wake behind the sphere, the phenomenon of vortex shedding observed experimentally. In accordance with the scenario, system loses its stability when entropy outflow through surface confining the system cannot be compensated by entropy produced within the system. The system does not find a new stable position after losing its stability, that is, the system remains further unstable. As Reynolds number grows, one unstable flow regime is replaced by another. The replacement is governed tendency of the system to discover fastest path to depart from the state of statistical equilibrium. This striving, however, does not lead the system to disintegration. Periodically, reverse solutions to the multimoment hydrodynamics equations change the nature of evolution and guide the unstable system in a highly unlikely direction. In case of unstable system, unlikely path meets the direction of approaching the state of statistical equilibrium. Such behavior of the system contradicts the scenario created by solutions to the classic hydrodynamics equations. Unstable solutions to the classic hydrodynamics equations are not fairly prolonged along time to interpret experiment. Stable solutions satisfactorily reproduce all observed stable medium states. As Reynolds number grows one stable solution is replaced by another. They are, however, incapable of reproducing any of unstable regimes recorded experimentally. In particular, stable solutions to the classic hydrodynamics equations cannot put anything in correspondence to any of observed vortex shedding modes. In accordance with our interpretation, the reason for this isthe classic hydrodynamics equations themselves.

Three-dimensional asymptotic analysis of multiple-electroded piezoelectric laminates

AIAA Journal ◽  
2000 ◽  
Vol 38 ◽  
pp. 317-324 ◽  
Author(s):  
Zhen-Qiang Cheng ◽  
R. C. Batra
Keyword(s):  
Asymptotic Analysis ◽  
Three Dimensional
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