Figure 1

(Color online) Schematic example of the flow field surrounding tensile (left) and contractile (right) swimming microorganisms.Reuse & Permissions

Figure 2

(Color online) Regions of parameters where spontaneous flow occurs in an unsheared active film on a substrate. The regions of spontaneous flow are bounded by the critical activity ${\alpha }_{c1}\left(\beta \right)$ given in Eq. (8) (solid and dashed lines) and are shaded orange, with lighter shades corresponding to increasing values of $\beta$. The same critical activity also separates the regions $\left|\alpha \right|<{\alpha }_{c1}$ where the theoretical stress-strain curves are monotonic and the active suspension is either thinned or thickened by activity at small shear rates, as indicated, from the regions $\left|\alpha \right|>{\alpha }_{c1}$ where the theoretical stress-strain curves are nonmonotonic, with possible “superfluid” or hysteretic behavior.Reuse & Permissions

Figure 3

(Color online) Schematic representation of a quasi-one-dimensional film of thickness $L$. In our model, the film is sitting on a nonslipping surface and is sheared from the top at constant velocity $v_0$. The polar rods form an angle $\theta$ with respect to the infinite direction $x$ of the film. Because of the quasi-one-dimensional geometry, the system is invariant for translations along the $x$ axis.Reuse & Permissions

Figure 4

(Color online) Stress $\left(\sigma \right)$ vs strain $\left(\dot{\gamma }\right)$ for an active nematic $\left(\beta =w=0\right)$ suspension for various $\alpha$. Flow-tumbling systems with $\lambda =0.1$ are marked by circles and flow-aligning systems with $\lambda =1.9$ by triangles. Other parameters are set $L/\ell =5$, $\eta =1$, ${\varphi }_0=1$, $D=1$, and $\xi =0.3$. Inset shows the comparison to the analytical result given in Eq. (9).Reuse & Permissions

Figure 5

(Color online) Schematic example of the flow field surrounding tensile–flow-aligning (right) and contractile–flow-tumbling active particles. For the choice of the parameters $\alpha$ and $\lambda$ given in Eq. (12), the two flows are identical, leading to an equal apparent viscosity.Reuse & Permissions

Figure 6

(Color online) Apparent viscosity ${\eta }_{\text{app}}$ for active nematic (top) and polar (bottom) suspensions from Eq. (9). Solid (red) lines represent flow-tumbling systems $\left(\lambda <1\right)$ while dashed (blue) lines represent flow-aligning systems $\left(\lambda >1\right)$. The corresponding values of $\lambda$ are indicated next to the lines. In the bottom plot, $\alpha$ was set to zero. Top frame emphasizes the duality discussed in the text.Reuse & Permissions

Figure 7

(Color online) Top left frame displays a typical theoretical stress-strain curve of a nematic active suspension in the region ${\alpha }_{c1}<\left|\alpha \right|<{\alpha }_{c2}$. The theoretical curve is obtained by tuning $\dot{\gamma }$ and calculating the resulting $\sigma$ and exhibits a region of $d\sigma /d\dot{\gamma }<0$. The other three frames show three possible experimental stress-strain curves obtained by tuning $\sigma$ and measuring $\dot{\gamma }$ that could be consistent with the theoretical curve. Top right frame displays the superfluid scenario suggested in [17], with bulk shear bands accommodating different macroscopic shear rates and zero net stress, so that the apparent viscosity of the system is simply zero. Bottom left frame shows a yield-stress-like behavior with a yield stress ${\sigma }_y={\sigma }_m$. The last scenario is described in the bottom right frame and corresponds to a hysteretic stress-strain curve where the suspension can accommodate a range of macroscopic strain rates maintaining a constant total stress $\pm {\sigma }_0$.Reuse & Permissions

Figure 8

(Color online) Stress-strain curves of a nematic suspension $\left(\beta =w=0\right)$ obtained by numerical solution of the active hydrodynamic equations for several values of $\alpha$. ${\alpha }_{c1}=0.219$ and ${\alpha }_{c2}=0.877$ for the parameters chosen in the numerical solution.Reuse & Permissions

Figure 9

(Color online) Yield stress ${\sigma }_c$ as a function of $\alpha$ for a nematic suspension $\left(\beta =w=0\right)$ obtained by numerical solution of the active hydrodynamic equations.Reuse & Permissions

Figure 10

(Color online) Possible scenarios for the transition to the yield-stress regime at $\alpha >{\alpha }_{c2}$. The nonmonotonic curve obtained numerically is shown in the top left frame. In the superfluid scenario (top-right), the plateau at $\sigma =0$ divides into two disconnected branches terminating at $\sigma =\pm {\sigma }_c$. In this case, the yield stress is expected to grow monotonically from zero. In the yield-stress scenario (bottom-left), there is already a nonzero stress at $\dot{\gamma }=0$ and thus the yield stress simply continues increasing with no qualitative change in the behavior at ${\alpha }_{c2}$. In the hysteretic scenario (bottom-right), the loop intersects the positive $\sigma$ axis at $\pm {\sigma }_0$, with ${\sigma }_c\le {\sigma }_0\le {\sigma }_m$.Reuse & Permissions