A.Bernou, M. Duerinckx & A. Gloria: Homogenization of Active Suspensions and Reduction of Effective Viscosity

Abstract

We consider a suspension of active rigid particles (swimmers) in a steady Stokes flow, using for simplicity a steady-state model where particles are distributed according to a stationary ergodic random process, and we study its homogenization in the macroscopic limit. A key point in the model is that swimmers are allowed to adapt their propulsion to the surrounding fluid deformation: swimming forces are not prescribed a priori, but are rather obtained through the retroaction of the fluid. Qualitative homogenization of this nonlinear model requires an unusual proof that crucially relies on a semi-quantitative two-scale analysis. Thanks to the introduction of new correctors that accurately capture spatial oscillations created by swimming forces, we identify the contribution of the activity to the effective viscosity. In agreement with the physics literature, an analysis in the dilute regime shows that the activity of the particles can either increase or decrease the effective viscosity (depending on the swimming mechanism), in a way that strongly differs from the well-known effect of passive suspensions.

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