PARABOLIC AND NAVIER–STOKES EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 81 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2008 ON A CONSTRAINED MINIMIZATION PROBLEM ARISING IN HEMODYNAMICS JO ˜ AO JANELA Instituto Superior de Economia e Gest˜ao and CEMAT/IST, Universidade T´ ecnica de Lisboa Rua do Quelhas 6, 1200 Lisboa, Portugal E-mail: jjanela@iseg.utl.pt AD ´ ELIA SEQUEIRA Instituto Superior T´ ecnico and CEMAT/IST, Universidade T´ ecnica de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: adelia.sequeira@math.ist.utl.pt Abstract. Experimental evidence collected over the years shows that blood exhibits non- Newtonian characteristics such as shear-thinning, viscoelasticity, yield stress and thixotropic behaviour. Under certain conditions these characteristics become relevant and must be taken into consideration when modelling blood flow. In this work we deal with incompressible general- ized Newtonian fluids, that account for the non-constant viscosity of blood, and present a new numerical method to handle fluid-rigid body interaction problems. The work is motivated by the investigation of interaction problems occurring in the human cardiovascular system, where the rigid bodies may be blood particles, clots, valves or any structure that we may assume to move rigidly. This method is based on a variational formulation of the fully coupled problem in the whole fluid/solid domain, in which constraints are introduced to enforce the rigid motion of the body and the equilibria of forces and stresses at the interface. The main feature of the method consists in introducing a penalty parameter that relaxes the constraints and allows for the solution of an associated unconstrained problem. The convergence of the solution of the relaxed problem is established and some numerical simulations are performed using common benchmarks for this type of problems. 2000 Mathematics Subject Classification : Primary 65M60; Secondary 74F10, 76Z05. Key words and phrases : constrained minimization, hyper-viscosity method, fluid-rigid body interaction, shear-thinning fluid. Research funded by the Center for Mathematics and its Applications - CEMAT through FCT funding program. The project HPRN-CT-2002-00270 (Research Training Network ’Haemodel’ of the European Union) is also acknowledged. The paper is in final form and no version of it will be published elsewhere. [227] c  Instytut Matematyczny PAN, 2008