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

Jan Sokołowski; Jan Stebel

doi:10.3934/eect.2014.3.331

Article Contents

2014, Volume 3, Issue 2: 331-348. Doi: 10.3934/eect.2014.3.331

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

Jan Sokołowski^1, and
Jan Stebel^2,

1.
Institut Élie Cartan Nancy, UMR 7502((Université Lorraine, CNRS, INRIA), Laboratoire de Mathématiques, Université de Lorraine, B.P.239, 54506 Vandoeuvre-lès-Nancy Cedex
2.
Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 110 00 Praha 1

Received: August 31, 2013

Revised: January 31, 2014

Published: May 2014

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Abstract

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 [18]. 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 $r\ge11/5$.

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Mathematics Subject Classification: Primary: 35Q30, 76D55; Secondary: 35R37.

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