Figure 1

Homogeneous constitutive curves.Reuse & Permissions

Figure 2

Top: Shear banded (or quiescent) profiles for $\zeta =0.01$. Values of $l=0.0005$, 0.00071, 0.001, 0.00113, 0.0014, 0.002, 0.0028, 0.004, 0.00476, 0.0057, 0.00673, 0.0073, 0.008, 0.00872, 0.00951, 0.016 increasing for decreasing $\mathrm{\gamma ̇}(y=0)$. Inset: Shear rate $\mathrm{\gamma ̇}(y=0)$ at the left edge of the cell extracted from such profiles, for various values of $\zeta ,l$, shown in the master scaling representation discussed in the main text. Bottom: Phase diagram for 1D runs showing the region of spontaneous shear banded flow (closed circles) and the region in which an initially heterogeneous state decays back to a quiescent state of zero flow (open circles). The dashed line is the power law $l^{*}=a{\zeta }^{1/2}$ using the value of the intercept $a$ extracted from the master scaling plot in the inset to the top figure.Reuse & Permissions

Figure 3

Phase diagram for 2D runs with free boundary conditions, each denoted by a square. Empty squares: quiescence. Elsewhere initial shear banded profile gives way to a state of 2D domains that is steady (crosses); oscillatory (shaded square); aperiodic (filled squares).Reuse & Permissions

Figure 4

Throughput versus time for $l=0.002$ and $\zeta =0.004$ (left), $\zeta =0.04$ (right).Reuse & Permissions

Figure 5

Snapshot at long time of 2D runs for $\zeta =0.002,0.004,0.01,0.02,0.04,0.1$ and $l=0.002$ (free boundary conditions, no external shear throughout). Grayscale on left shows $Q_{\mathit{xx}}$; while arrows on right show the fluid velocity. The $x$ direction is horizontal, $y$ vertical.Reuse & Permissions

Figure 6

Snapshot at long time of 2D runs for $\zeta =0.1$ and, from top to bottom, $l=0.016,0.008,0.004,0.002$ (free boundary conditions, no external shear throughout). Grayscale on left shows $Q_{\mathit{xx}}$; while arrows on right show the fluid velocity. The $x$ direction is horizontal, $y$ vertical.Reuse & Permissions

Figure 7

Top: Phase diagram for 1D runs with fixed boundary conditions, each denoted by a circle. Empty circles denote quiescence, filled ones spontaneously flowing states. Bottom: Phase diagram for 2D runs with fixed boundary conditions, each denoted by a square. Empty squares: quiescence. Elsewhere the stable 1D profile with a small inhomogeneity gives way to a state of 2D domains that is steady (crosses); oscillatory (shaded square); chaotic (filled squares).Reuse & Permissions

Figure 8

Snapshots corresponding to simulations with extensile fluids and fixed boundary conditions, for a system with $L_x=L_y=1$ and periodic boundary conditions along the flow direction. The snapshots show grayscale plots of $Q_{\mathit{xx}}$ (left column), and the velocity profile (right column). Maximum velocities are (in simulation units) (a) 0.00317, (b) 0.00346, (c) 0.00852, (d) 0.00060, and (e) 0.1008. The value of the activity is (from top to bottom): (a) 0.002, (b) 0.004, (c) 0.008, (d) 0.01, and (e) 0.08; throughout $l=0.002$.Reuse & Permissions

Figure 9

Top: Flow curve for $\zeta =0.001,l=0.002$. Lower: representative state snapshot at a late time (on the final attractor) for each of the lowest seven shear rates. (Shear rate increasing in snapshots downward. The $x$ direction is horizontal, $y$ vertical.) Grayscale shows the local shear stress. The solid line shows the 0D homogeneous constitutive curve.Reuse & Permissions

Figure 10

Top: Flow curve for $\zeta =0.005,l=0.002$. Lower: representative state snapshot at a late time (on the final attractor) for each of the lowest seven shear rates. (Shear rate increasing in snapshots downward. The $x$ direction is horizontal, $y$ vertical.) Grayscale shows the local shear stress. The solid line shows the 0D homogeneous constitutive curve.Reuse & Permissions

Figure 11

Top: Flow curve for $\zeta =0.04,l=0.002$. Lower: representative state snapshot at a late time (on the final attractor) for each of the lowest seven shear rates. (Shear rate increasing snapshots downward. The $x$ direction is horizontal, $y$ vertical.) Grayscale shows the local shear stress. The solid line shows the 0D homogeneous constitutive curve.Reuse & Permissions

Figure 12

Top: Flow curve for $\zeta =0.001,l=0.002$. The dashed curve is part of the homogeneous constitutive curve. Lower two panels: representative state snapshot at a late time for $\mathrm{\gamma ̇}={10}^{-6}$ (second row) and $\mathrm{\gamma ̇}={10}^{-4}$ (third row). Grayscale shows the local shear stress.Reuse & Permissions

Figure 13

Effective viscosity (normalized by $\eta$) versus time for an active gel with $l=0.002$, $\zeta =0.001$ with fixed boundary conditions. The solid line refers to $\mathrm{\gamma ̇}=0.0002$, the dashed one to $\mathrm{\gamma ̇}=0.001$. The steady-state effective viscosities are very similar in both cases, proving that rolls have a linear rheology regime.Reuse & Permissions

Figure 14

Top: Flow curve for $\zeta =0.04,l=0.002$. Lower three panels: representative snapshots at a late time for $\mathrm{\gamma ̇}=0.00005$ (second row), $\mathrm{\gamma ̇}=0.001$ (third row), and $\mathrm{\gamma ̇}=0.002$ (bottom row). Grayscale shows the local shear stress.Reuse & Permissions

Figure 15

Effective viscosity (normalized by $\eta$) versus time for an active gel with $l=0.002$, $\zeta =0.0005$, and fixed boundary conditions. The solid line refers to $\mathrm{\gamma ̇}=0.0002$, the dashed one to $\mathrm{\gamma ̇}=0.001$. The effective viscosities are smaller than $\eta$ in both cases, and nearly equal (demonstrating the existence of a linear regime). Note that to get an out-of-plane velocity field in the steady state one needs to start with an out-of-plane director profile.Reuse & Permissions