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Fluid flows around a symmetric obstacle generate vortices which may lead tosymmetry breaking of the streamlines. We study this phenomenon for planarviscous flows governed by the stationary Navier-Stokes equations with constantinhomogeneous Dirichlet boundary data in a rectangular channel containing acircular obstacle. In such (symmetric) framework, symmetry breaking is strictlyrelated to the appearance of multiple solutions. Symmetry breaking propertiesof some Sobolev minimizers are studied and explicit bounds on the boundaryvelocity (in terms of the length and height of the channel) ensuring uniquenessare obtained after estimating some Sobolev embedding constants and constructinga suitable solenoidal extension of the boundary data. We show that, regardlessof the solenoidal extension employed, such bounds converge to zero at anoptimal rate as the length of the channel tends to infinity.

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