Boundary variations in the Navier-Stokes equations and Lagrangian functionals. (English) Zbl 0988.76077

Boisgérault, Sébastien; Zolésio, Jean-Paul

Cagnol, John et al., Shape optimization and optimal design. Proceedings of the IFIP conference. Selected papers from the sessions ”Distributed parameter systems” and ”Optimization methods and engineering design” held within the 19th conference system modeling and optimization, Cambridge, GB, July 12-16, 1999. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 216, 7-26 (2001).

Summary: We study the shape sensitivity of stationary Navier-Stokes equations in the general case of non-homogeneous and shape-dependent forces and boundary conditions. Under the assumption of non-singularity of the equations, the shape differentiability of velocity and pressure are obtained in some Sobolev spaces. We study the influence of the regularity of geometrical and functional data on the best space for which the result holds. We apply these results to a class of shape functionals where a high regularity is required: Lagrangian functionals. Their main characteristic is to take into account the paths of fluid particles. The usual shape calculus is extended to take into account such features. We determine the shape derivative of a shape-dependent flow, and develop the methods to achieve an explicit calculation of shape gradient.
For the entire collection see [Zbl 0959.00036].

MSC:

76M30	Variational methods applied to problems in fluid mechanics
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

shape sensitivity; stationary Navier-Stokes equations; non-singularity; shape differentiability; Sobolev spaces; regularity; Lagrangian functionals; shape gradient