A connection between symmetry breaking for Sobolev minimizers and stationary Navier-Stokes flows past a circular obstacle

Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containi...

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Bibliographic Details
Published in:arXiv.org 2021-12
Main Authors: Gazzola, Filippo, Sperone, Gianmarco, Weth, Tobias
Format: Article
Language:English
Subjects:
Barriers
Broken symmetry
Dirichlet problem
Fluid dynamics
Fluid flow
Navier-Stokes equations
Stokes flow
Symmetry
Viscous flow
Online Access:Get full text
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Summary:Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity.
ISSN:2331-8422