Fluid flows around an obstacle generate vortices which, in turn, generate forces on the obstacle. This phenomenon is studied for planar viscous flows governed by the stationary Navier–Stokes equations with inhomogeneous Dirichlet boundary data in a (virtual) square containing an obstacle. In a symmetric framework the appearance of forces is strictly related to the multiplicity of solutions. Precise bounds on the data ensuring uniqueness are then sought and several functional inequalities (concerning relative capacity, Sobolev embedding, solenoidal extensions) are analyzed in detail: explicit bounds are obtained for constant boundary data. The case of “almost symmetric” frameworks is also considered. A universal threshold on the Reynolds number ensuring that the flow generates no lift is obtained regardless of the shape and the nature of the obstacle. Based on the asymmetry/multiplicity principle, the performance of different obstacle shapes is then compared numerically. Finally, connections of the results with elasticity and mechanics are emphasized.

中文翻译：

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平面域中的稳态 Navier-Stokes 方程具有唯一可解性的障碍和显式边界

障碍物周围的流体流动会产生涡流，涡流又会在障碍物上产生力。这种现象是针对由静止 Navier-Stokes 方程控制的平面粘性流进行研究的，该方程在包含障碍物的（虚拟）正方形中具有非齐次 Dirichlet 边界数据。在对称框架中，力的出现与解的多样性密切相关。然后寻找确保唯一性的数据的精确边界，并详细分析几个功能不等式（关于相对容量、Sobolev 嵌入、螺线管扩展）：为恒定边界数据获得显式边界。还考虑了“几乎对称”框架的情况。无论障碍物的形状和性质如何，都可以获得雷诺数的通用阈值，确保流动不会产生升力。然后基于不对称/多重性原理，对不同障碍物形状的性能进行数值比较。最后，强调了结果与弹性和力学的联系。