The rheological response under simple shear of an active suspension of Escherichia coli is determined in a large range of shear rates and concentrations. The effective viscosity and the time scales characterizing the bacterial organization under shear are obtained. In the dilute regime, we bring evidences for a low shear Newtonian plateau characterized by a shear viscosity decreasing with concentration. In the semi-dilute regime, for particularly active bacteria, the suspension display a "super-fluid" like transition where the viscous resistance to shear vanishes, thus showing that macroscopically, the activity of pusher swimmers organized by shear, is able to fully overcome the dissipative effects due to viscous loss.Owing to its relevance in medicine, ecology and its importance for technological applications, the hydrodynamics of active suspensions is at the centre of many recent fundamental studies [1, 2]. In nature, wide classes of living micro-organisms move autonomously in fluids at very low Reynold numbers [3]. Their motility stems from a variety of propulsive flagellar systems powered by nanomotors. For bacteria such a bacillus subtilis or E.coli, the propulsion comes from the rotation of helix shaped flagella creating a propulsive force at the rear of the cell body [4]. Consequently, many original fluid properties stem from the swimming activity [5][6][7][8][9][10][11]. Due to hydrodynamic interactions bacteria may produce mesoscopic patterns of collective motion sometimes called "bio-turbulence" [13][14][15][16][17][18]. In a flow, these bacteria may organize spatially [12] and under shear, for pusher-swimmers, the swimming activity yields the possibility to decrease the macroscopic viscosity to values below the suspending fluid viscosity [5]. In the dilute regime, kinetic theories via a simple account of the dominant long range hydrodynamic field [19][20][21] provide closed-forms for shear viscosity as a function of shear rate. Remarkably, at low shear rate, these theories predict a Newtonian plateau with a viscosity decreasing linearly with concentration [19][20][21]. On the other hand, phenomenological theories were also proposed to describe macroscopically active suspensions via a coupling of hydrodynamic equations with polar and/or nematic order parameters [2, 5, 6,[22][23][24]. A striking outcome of these theories is that for a set of coupling parameters rendering essentially a high swimming activity, a self-organized motive macroscopic flow may show up in response to shear [22][23][24]. This onset of a dissipationless current is described in analogy with the super-fluidity transition [22,23] of liquids. Experimental evidences for viscosity reduction to values below the suspending fluid viscosity were brought for Bacillus subtilis [8] and E. coli [25] suspensions. However, no full rheological characterization (i.e. viscosity versus shear rate) under steady and uniform shear exists. Moreover, these pioneering experiments did not provide evidence for the low shear viscous plateau which is at the c...
