Particle Orientation in a shear flow for power law and viscoelastic Fluids
Résumé
The first works devoted to the evolution of elongated particles are mainly due to Jeffery, which exhibits the evolution equation for orientation. In a simple shear flow and Newtonian fluid, the particle was found to undergo a periodic revolution, called the Jeffery orbit centered around the vorticity axis. This orbit is controlled by an aspect ratio β and the period of rotation is given by a simple formula T = 2 π (β^2 +1)/β (γ ) ̇ where γ ̇ is the applied macroscopic shear rate. Usually the studies are limited to rod-like particle (labelled as (c) in the Figure), but we extend our study to prolate ellipsoid (a), rod with rounded ends (d) and oblate ellipsoid (b). First, computations are made for Newtonian fluid to get the aspect ratio associated to particle shape. For ellipsoids, the numerical aspect ratio is very close to the ratio L/D, L being the length and D the diameter whereas for the two rod-like particle, there are slightly smaller. Secondly, the computations made for power law fluids show that Jeffery’s orbits are always observed. The main differences came from the period of rotation and the extreme values of the azimutal angle. The period of rotation increases for prolate spheroids whereas it decreases for oblate ones. It is possible to define an equivalent shear rate γ ̇_pow from the computed period T_Pow if the aspect ratios computed with a Newtonians fluid are always valid. For prolate spheroids, there are deviations with respect to the Jeffery's orbit as the minimal azimutal angle is bigger. For oblate spheroids, it is the maximal azimutal which is lower. Finally, computations are made with an Oldroyd-B fluid. For prolate spheroid, the fiber tends to align along the axis of vorticity. For oblate spheroid, the extremity of orientation vector evolves periodically towards the shear plane and finally the larger dimension of the particle is aligned along vorticity axis
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