scholarly journals Long time estimate of solutions to 3d Navier–Stokes equations coupled with heat convection

2012 ◽  
Vol 39 (1) ◽  
pp. 23-41 ◽  
Author(s):  
Jolanta Socała ◽  
Wojciech M. Zajączkowski
Keyword(s):  
Stokes Equations ◽  
Heat Convection ◽  
Navier Stokes ◽  
Time Estimate ◽  
Navier Stokes Equations ◽  
Long Time

Three-dimensional impulsive vortex formation from slender orifices

2011 ◽  
Vol 666 ◽  
pp. 506-520 ◽  
Author(s):  
F. DOMENICHINI
Keyword(s):  
Stokes Equations ◽  
Intermediate Phase ◽  
Vortex Formation ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Vortex Ring Formation ◽  
Long Time ◽  
Definition Of

The vortex formation behind an orifice is a widely investigated phenomenon, which has been recently studied in several problems of biological relevance. In the case of a circular opening, several works in the literature have shown the existence of a limiting process for vortex ring formation that leads to the concept of critical formation time. In the different geometric arrangement of a planar flow, which corresponds to an opening with straight edges, it has been recently outlined that such a concept does not apply. This discrepancy opens the question about the presence of limiting conditions when apertures with irregular shape are considered. In this paper, the three-dimensional vortex formation due to the impulsively started flow through slender openings is studied with the numerical solution of the Navier–Stokes equations, at values of the Reynolds number that allow the comparison with previous two-dimensional findings. The analysis of the three-dimensional results reveals the two-dimensional nature of the early vortex formation phase. During an intermediate phase, the flow evolution appears to be driven by the local curvature of the orifice edge, and the time scale of the phenomena exhibits a surprisingly good agreement with those found in axisymmetric problems with the same curvature. The long-time evolution shows the complete development of the three-dimensional vorticity dynamics, which does not allow the definition of further unifying concepts.

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 On the Navier-Stokes equations with temperature-dependent transport coefficients

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
Eduard Feireisl ◽  
Josef Málek
Keyword(s):  
Stokes Equations ◽  
Transport Coefficients ◽  
Three Dimensional ◽  
Periodic Problem ◽  
Large Data ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Long Time ◽  
Spatially Periodic ◽  
Dependent Transport

We establish long-time and large-data existence of a weak solution to the problem describing three-dimensional unsteady flows of an incompressible fluid, where the viscosity and heat-conductivity coefficients vary with the temperature. The approach reposes on considering the equation for the total energy rather than the equation for the temperature. We consider the spatially periodic problem.

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