Journal of the
Korean Mathematical Society JKMS

This paper is concerned with an optimal shape control problem for the stationary Navier--Stokes system. A two--dimensional channel flow of an incompressible, viscous fluid is examined to determine the shape of a bump on a part of the boundary that minimizes the viscous drag. By introducing an artificial compressibility term to relax the incompressibility constraints, we take the penalty method. The existence of optimal solutions for the penalized problem will be shown. Next, by employing Lagrange multipliers method and the material derivatives, we derive the shape gradient for the minimization problem of the shape functional which represents the viscous drag.

Keywords: penalized stationary Navier--Stokes system, stress vector, Lagrange multipliers, sensitivity analysis, drag minimization, shape gradient

MSC numbers: 49J20, 76D05, 76M30