The shape derivability analysis of the flow of a viscous and incompressible fluid surrounding a rigid body B is considered. The novelty being in the choice of the boundary condition on the body B where we impose the so-called Navier boundary condition. Well-posedness of the time-dependent Navier–Stokes equations with mixed boundary conditions, of Navier and Dirichlet type, is established under regularity and smallness assumptions. After proving the shape differentiability of the state system, we compute the first order necessary optimality condition associated to drag shape minimization problem.

中文翻译：

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Navier边界条件下粘性流的形状敏感性分析

考虑了围绕刚性体B的粘性和不可压缩流体的流动的形状导数分析。新颖之处在于选择了人体B的边界条件，我们在其中施加了所谓的Navier边界条件。在规则性和小假设下，建立了带有时域的，具有混合边界条件的Navier和Dirichlet类型的Navier-Stokes方程的适定性。在证明了状态系统的形状可微性之后，我们计算了与拖动形状最小化问题相关的一阶必要最优性条件。