Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate concentrations

Recently, there has been a number of experimental studies convincingly demonstrating that a suspension of self-propelled bacteria (microswimmers in general) may have an effective viscosity significantly smaller than the viscosity of the ambient fluid. This is in sharp contrast with suspensions of hard passive inclusions, whose presence always increases the viscosity. Here we present a 2D model for a suspension of microswimmers in a fluid and analyze it analytically in the dilute regime (no swimmer-swimmer interactions) and numerically using a Mimetic Finite Difference discretization. Our analysis shows that in the dilute regime (in the absence of rotational diffusion) the effective shear viscosity is not affected by self-propulsion. But at the moderate concentrations (due to swimmer-swimmer interactions) the effective viscosity decreases linearly as a function of the propulsion strength of the swimmers. These findings prove that (i) a physically observable decrease of viscosity for a suspension of self-propelled microswimmers can be explained purely by hydrodynamic interactions and (ii) self-propulsion and interaction of swimmers are both essential to the reduction of the effective shear viscosity. We also performed a number of numerical experiments analyzing the dynamics of swimmers resulting from pairwise interactions. The numerical results agree with the physically observed phenomena (e.g., attraction of swimmer to swimmer and swimmer to the wall). This is viewed as an additional validation of the model and the numerical scheme.