  Micro-organisms usually can swim in their liquid environment by flagellar or
ciliary beating. In this numerical work, we analyze the influence of flagellar
beating on the orbits of a swimming cell in a shear flow. We also calculate the
effect of the flagellar beating on the rheology of a dilute suspension of
micro-swimmers. A three-dimensional model is proposed for Chlamydomonas
Reinhardtii swimming with a breaststroke-like beating of two anterior flagella
modeled by two counter-rotating fore beads. The active swimmer model reveals
unusual angular orbits in a linear shear flow. Namely, the swimmer sustains
orientations transiently across the flow. Such behavior is a result of the
interplay between shear flow and swimmer's periodic beating motion of flagella
which exert internal torques on the cell body. This peculiar behavior has some
significant consequences on the rheological properties of the suspension. We
calculate the Einstein's viscosity of the suspension composed of such isolated
modeled microswimmers (dilute case) in a shear flow. We use numerical
simulations based on a Rotne-Prager like approximation for hydrodynamic
interaction between simplified flagella and the cell body. The results show an
increased intrinsic viscosity for active swimmer suspensions in comparison to
non-active ones as well as a shear thinning behavior in accordance with
previous experimental measurements [Phys. Rev. Lett. 104 , 098102 (2010)].
