Abstract. Experimental evidence collected over the years shows that blood exhibits nonNewtonian 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 generalized 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. The paper is in final form and no version of it will be published elsewhere.[227] c Instytut Matematyczny PAN, 2008 228 J. JANELA AND A. SEQUEIRA 1. Introduction. The theoretical and numerical study of fluid-structure or fluid-particle interaction problems is of major importance when modelling several phenomena occuring in the human cardiovascular system, like the rolling of white blood cells (see [1]), or the motion and design of prostetic heart valves (see [9] ). There are also important industrial applications involving the settling and lift-off of particles in channel flows in the petroleum and coal industries (see [2,7] and references therein). The rheological characteristics of blood are determined by its complex morphology that we will briefly describe (for details see Sequeira and Janela [16], Robertson, Sequeira and Kameneva [15] and Robertson Sequeira and Owens [14]). Blood is a complex mixture consisting of different particles (erythrocytes, leukocytes, platelets and other matter) suspended in an aqueous polymer solution, the plasma (Newtonian fluid). These suspended particles, consisting mostly of erythrocytes (red blood cells) form about 45% of the volume of normal human blood and their effect should not be ignored. It is not feasible to track the individual behaviour of each RBC but it is possible to build homegenized models that reproduce the main non-Newtonian effects caused by their presence.