Microshape Control, Riblets, and Drag Minimization

Relying on the rugosity effect, we analyse the drag minimization problem in relation to the microstructure of the surface of a given obstacle. We construct a mathematical framework for the optimization problem, prove the existence of an optimal solution by $\Gamma$-convergence arguments, and analyze the stability of the drag with respect to the microstructure. For Stokes flows we justify why rugosity increases the drag, while for Navier--Stokes flows we give some numerical evidence supporting the thesis that adding rugosity on specific regions of the obstacle may contribute to decreasing the drag.