chapter  6
15 Pages

## Shape Derivative of Sharp Functionals Governed by Navier-Stokes Flow

ByS. Boisgérault, J.-P. Zolésio

The shape analysis of the Navier-Stokes equation has already been considered in the literature. Classical techniques, such as the Implicit Function Theorem, may be used to show that some functionals, the drag for example, are shape differentiable.

However, this property relies on results established for the basic regularity of the pressure and the velocity fields. Many other criteria of physical interest are out of this scope; we consider here the shape analysis of such functionals, for example, the (total) force exerted by the fluid on a body or the moment of these forces. The velocity and pressure fields u and p are assumed to be solutions of the stationary incompressible Navier-Stokes equation − v Δ u    +   [D u ] u   +   ∇ p   =   f in Ω with the boundary condition u | Γ = 0 , Γ=∂Ω

These new results are based on the so-called speed method which allows us to “bring back” vector fields from a perturbed domain to the initial one while preserving the divergence-free property. Regularity results are established for that correspondence and used to define and show some properties of the shape derivative u and of the boundary shape derivative u ′ Γ