Shape optimization for non-Newtonian fluids in time-dependent domains

Jan Sokolowski (1) , Jan Stebel (2)

1 IECL - Institut Élie Cartan de Lorraine
2 Mathematical Institute [Praha] - Academy of Sciences

We study the model of an incompressible non-Newtonian fluid in a moving domain. The domain is defined as a tube built by the velocity field $\mathbf{V}$ and described by the family of domains $\Omega_t$ parametrized by $t\in[0,T]$. A new shape optimization problem associated with the model is defined for a family of initial domains $\Omega_0$ and admissible velocity vector fields. It is shown that such shape optimization problems are well posed under the classical conditions on compactness of the admissible shapes (Plotnikov, Sokolowski, Compressible Navier-Stokes Equations, Springer-Birkhauser, 2012}. For the state problem, we prove the existence of weak solutions and their continuity with respect to perturbations of the time-dependent boundary, provided that the power-law index $r\ge11/5$.

Mots clés: shape optimization; time-dependent domain; incompressible viscous fluid; Navier-Stokes equations

Domaines: Equations aux dérivées partielles [math.AP]

Jan Sokolowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations and Control Theory, 2014, 3 (2), pp.331-348. ⟨hal-00870119⟩