A Formula for the Derivative with Respect to Domain Variations in Navier--Stokes Flow Based on an Embedding Domain Method

Fréchet differentiability and a formula for the derivative with respect to domain variation of a general class of cost functionals under the constraint of the two-dimensional stationary incompressible Navier-Stokes equations are shown. An embedding domain technique provides an equivalent formulation of the problem on a fixed domain and leads to a simple and computationally cheap line integral formula for the derivative of the cost functional with respect to domain variation. Existence of a solution to the corresponding domain optimization problems is proved. A numerical example shows the effectivity of the derivative formula. A formula for the derivative with respect to domain variations in Navier-Stokes flow based on an embedding domain method