Drag reduction for non-cylindrical Navier-Stokes flows

We address the minimal drag problem for a viscous and incompressible fluid contained in a moving domain Omega(t) whose boundary has a speed V(t) which turns to be the control parameter. The flow velocity u(t) verifies u(t) = V(t) on the boundary as the fluid sticks to the boundary. Then, the control acts on the non-cylindrical Navier-Stokes solution through two aspects. Directly on the Dirichlet non-homogeneous boundary condition and also through its flow mapping T-t(V) which builds the moving domain starting from the initial one. The drag functional J is given in terms of the dissipated viscous energy and depends on the control V. We give a sense to the gradient with respect to V by making use of the transverse field technique which enables us to obtain an explicit expression for the gradient. We give a feedback loop and an associated existence result in order to generate the best admissible boundary displacement. In this problem, we have no fluid-structure coupling nor free boundary. The moving boundary is an active (strongly) nonlinear control.