2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119853This paper is concerned with the problem of shape optimization of two-dimensional flows governed by the time-dependent Navier-Stokes equations. We derive the structures of shape gradients with respect to the shape of the variable domain for time-dependent cost functionals by using the state derivative with respect to the shape of the fluid domain and its associated adjoint state. Finally we apply a gradient type algorithm to our problem and numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible in low Reynolds number flow.22 pages, 5 figuresOptimization and ControlAnalysis of PDEs35B37; 35Q30; 49K40text